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21/11/2015

Liste des 20 000 premiers couples de nombres premiers jumeaux (p, p+2)

Source : http://arnflo.se/~site_files/Other/twinprimes

 

# 20000 first twin primes
# Calculated: 12/09-10 By Oscar Arnflo
# Processing time: 626.478574038 seconds

3,5 #1
5,7 #2
11,13 #3
17,19 #4
29,31 #5
41,43 #6
59,61 #7
71,73 #8
101,103 #9
107,109 #10
137,139 #11
149,151 #12
179,181 #13
191,193 #14
197,199 #15
227,229 #16
239,241 #17
269,271 #18
281,283 #19
311,313 #20
347,349 #21
419,421 #22
431,433 #23
461,463 #24
521,523 #25
569,571 #26
599,601 #27
617,619 #28
641,643 #29
659,661 #30
809,811 #31
821,823 #32
827,829 #33
857,859 #34
881,883 #35
1019,1021 #36
1031,1033 #37
1049,1051 #38
1061,1063 #39
1091,1093 #40
1151,1153 #41
1229,1231 #42
1277,1279 #43
1289,1291 #44
1301,1303 #45
1319,1321 #46
1427,1429 #47
1451,1453 #48
1481,1483 #49
1487,1489 #50
1607,1609 #51
1619,1621 #52
1667,1669 #53
1697,1699 #54
1721,1723 #55
1787,1789 #56
1871,1873 #57
1877,1879 #58
1931,1933 #59
1949,1951 #60
1997,1999 #61
2027,2029 #62
2081,2083 #63
2087,2089 #64
2111,2113 #65
2129,2131 #66
2141,2143 #67
2237,2239 #68
2267,2269 #69
2309,2311 #70
2339,2341 #71
2381,2383 #72
2549,2551 #73
2591,2593 #74
2657,2659 #75
2687,2689 #76
2711,2713 #77
2729,2731 #78
2789,2791 #79
2801,2803 #80
2969,2971 #81
2999,3001 #82
3119,3121 #83
3167,3169 #84
3251,3253 #85
3257,3259 #86
3299,3301 #87
3329,3331 #88
3359,3361 #89
3371,3373 #90
3389,3391 #91
3461,3463 #92
3467,3469 #93
3527,3529 #94
3539,3541 #95
3557,3559 #96
3581,3583 #97
3671,3673 #98
3767,3769 #99
3821,3823 #100
3851,3853 #101
3917,3919 #102
3929,3931 #103
4001,4003 #104
4019,4021 #105
4049,4051 #106
4091,4093 #107
4127,4129 #108
4157,4159 #109
4217,4219 #110
4229,4231 #111
4241,4243 #112
4259,4261 #113
4271,4273 #114
4337,4339 #115
4421,4423 #116
4481,4483 #117
4517,4519 #118
4547,4549 #119
4637,4639 #120
4649,4651 #121
4721,4723 #122
4787,4789 #123
4799,4801 #124
4931,4933 #125
4967,4969 #126
5009,5011 #127
5021,5023 #128
5099,5101 #129
5231,5233 #130
5279,5281 #131
5417,5419 #132
5441,5443 #133
5477,5479 #134
5501,5503 #135
5519,5521 #136
5639,5641 #137
5651,5653 #138
5657,5659 #139
5741,5743 #140
5849,5851 #141
5867,5869 #142
5879,5881 #143
6089,6091 #144
6131,6133 #145
6197,6199 #146
6269,6271 #147
6299,6301 #148
6359,6361 #149
6449,6451 #150
6551,6553 #151
6569,6571 #152
6659,6661 #153
6689,6691 #154
6701,6703 #155
6761,6763 #156
6779,6781 #157
6791,6793 #158
6827,6829 #159
6869,6871 #160
6947,6949 #161
6959,6961 #162
7127,7129 #163
7211,7213 #164
7307,7309 #165
7331,7333 #166
7349,7351 #167
7457,7459 #168
7487,7489 #169
7547,7549 #170
7559,7561 #171
7589,7591 #172
7757,7759 #173
7877,7879 #174
7949,7951 #175
8009,8011 #176
8087,8089 #177
8219,8221 #178
8231,8233 #179
8291,8293 #180
8387,8389 #181
8429,8431 #182
8537,8539 #183
8597,8599 #184
8627,8629 #185
8819,8821 #186
8837,8839 #187
8861,8863 #188
8969,8971 #189
8999,9001 #190
9011,9013 #191
9041,9043 #192
9239,9241 #193
9281,9283 #194
9341,9343 #195
9419,9421 #196
9431,9433 #197
9437,9439 #198
9461,9463 #199
9629,9631 #200
9677,9679 #201
9719,9721 #202
9767,9769 #203
9857,9859 #204
9929,9931 #205
10007,10009 #206
10037,10039 #207
10067,10069 #208
10091,10093 #209
10139,10141 #210
10271,10273 #211
10301,10303 #212
10331,10333 #213
10427,10429 #214
10457,10459 #215
10499,10501 #216
10529,10531 #217
10709,10711 #218
10859,10861 #219
10889,10891 #220
10937,10939 #221
11057,11059 #222
11069,11071 #223
11117,11119 #224
11159,11161 #225
11171,11173 #226
11351,11353 #227
11489,11491 #228
11549,11551 #229
11699,11701 #230
11717,11719 #231
11777,11779 #232
11831,11833 #233
11939,11941 #234
11969,11971 #235
12041,12043 #236
12071,12073 #237
12107,12109 #238
12161,12163 #239
12239,12241 #240
12251,12253 #241
12377,12379 #242
12539,12541 #243
12611,12613 #244
12821,12823 #245
12917,12919 #246
13001,13003 #247
13007,13009 #248
13217,13219 #249
13337,13339 #250
13397,13399 #251
13679,13681 #252
13691,13693 #253
13709,13711 #254
13721,13723 #255
13757,13759 #256
13829,13831 #257
13877,13879 #258
13901,13903 #259
13931,13933 #260
13997,13999 #261
14009,14011 #262
14081,14083 #263
14249,14251 #264
14321,14323 #265
14387,14389 #266
14447,14449 #267
14549,14551 #268
14561,14563 #269
14591,14593 #270
14627,14629 #271
14867,14869 #272
15137,15139 #273
15269,15271 #274
15287,15289 #275
15329,15331 #276
15359,15361 #277
15581,15583 #278
15641,15643 #279
15647,15649 #280
15731,15733 #281
15737,15739 #282
15887,15889 #283
15971,15973 #284
16061,16063 #285
16067,16069 #286
16139,16141 #287
16187,16189 #288
16229,16231 #289
16361,16363 #290
16451,16453 #291
16631,16633 #292
16649,16651 #293
16691,16693 #294
16829,16831 #295
16901,16903 #296
16979,16981 #297
17027,17029 #298
17189,17191 #299
17207,17209 #300
17291,17293 #301
17387,17389 #302
17417,17419 #303
17489,17491 #304
17579,17581 #305
17597,17599 #306
17657,17659 #307
17681,17683 #308
17747,17749 #309
17789,17791 #310
17837,17839 #311
17909,17911 #312
17921,17923 #313
17957,17959 #314
17987,17989 #315
18041,18043 #316
18047,18049 #317
18059,18061 #318
18119,18121 #319
18131,18133 #320
18251,18253 #321
18287,18289 #322
18311,18313 #323
18521,18523 #324
18539,18541 #325
18911,18913 #326
18917,18919 #327
19079,19081 #328
19139,19141 #329
19181,19183 #330
19211,19213 #331
19379,19381 #332
19421,19423 #333
19427,19429 #334
19469,19471 #335
19541,19543 #336
19697,19699 #337
19751,19753 #338
19841,19843 #339
19889,19891 #340
19961,19963 #341
19991,19993 #342
20021,20023 #343
20147,20149 #344
20231,20233 #345
20357,20359 #346
20441,20443 #347
20477,20479 #348
20507,20509 #349
20549,20551 #350
20639,20641 #351
20717,20719 #352
20747,20749 #353
20771,20773 #354
20807,20809 #355
20897,20899 #356
20981,20983 #357
21011,21013 #358
21017,21019 #359
21059,21061 #360
21191,21193 #361
21317,21319 #362
21377,21379 #363
21491,21493 #364
21521,21523 #365
21557,21559 #366
21587,21589 #367
21599,21601 #368
21611,21613 #369
21647,21649 #370
21737,21739 #371
21839,21841 #372
22037,22039 #373
22091,22093 #374
22109,22111 #375
22157,22159 #376
22271,22273 #377
22277,22279 #378
22367,22369 #379
22481,22483 #380
22541,22543 #381
22571,22573 #382
22619,22621 #383
22637,22639 #384
22697,22699 #385
22739,22741 #386
22859,22861 #387
22961,22963 #388
23027,23029 #389
23039,23041 #390
23057,23059 #391
23201,23203 #392
23291,23293 #393
23369,23371 #394
23537,23539 #395
23561,23563 #396
23627,23629 #397
23669,23671 #398
23687,23689 #399
23741,23743 #400
23831,23833 #401
23909,23911 #402
24107,24109 #403
24179,24181 #404
24371,24373 #405
24419,24421 #406
24917,24919 #407
24977,24979 #408
25031,25033 #409
25169,25171 #410
25301,25303 #411
25307,25309 #412
25409,25411 #413
25469,25471 #414
25577,25579 #415
25601,25603 #416
25799,25801 #417
25847,25849 #418
25931,25933 #419
25997,25999 #420
26111,26113 #421
26249,26251 #422
26261,26263 #423
26681,26683 #424
26699,26701 #425
26711,26713 #426
26729,26731 #427
26861,26863 #428
26879,26881 #429
26891,26893 #430
26951,26953 #431
27059,27061 #432
27107,27109 #433
27239,27241 #434
27281,27283 #435
27407,27409 #436
27479,27481 #437
27527,27529 #438
27539,27541 #439
27581,27583 #440
27689,27691 #441
27737,27739 #442
27749,27751 #443
27791,27793 #444
27917,27919 #445
27941,27943 #446
28097,28099 #447
28109,28111 #448
28181,28183 #449
28277,28279 #450
28307,28309 #451
28349,28351 #452
28409,28411 #453
28547,28549 #454
28571,28573 #455
28619,28621 #456
28661,28663 #457
28751,28753 #458
29021,29023 #459
29129,29131 #460
29207,29209 #461
29387,29389 #462
29399,29401 #463
29567,29569 #464
29669,29671 #465
29759,29761 #466
29879,29881 #467
30011,30013 #468
30089,30091 #469
30137,30139 #470
30269,30271 #471
30389,30391 #472
30467,30469 #473
30491,30493 #474
30557,30559 #475
30839,30841 #476
30851,30853 #477
30869,30871 #478
31079,31081 #479
31121,31123 #480
31151,31153 #481
31181,31183 #482
31247,31249 #483
31319,31321 #484
31391,31393 #485
31511,31513 #486
31541,31543 #487
31721,31723 #488
31727,31729 #489
31769,31771 #490
31847,31849 #491
32027,32029 #492
32057,32059 #493
32117,32119 #494
32141,32143 #495
32189,32191 #496
32297,32299 #497
32321,32323 #498
32369,32371 #499
32411,32413 #500
32441,32443 #501
32531,32533 #502
32561,32563 #503
32609,32611 #504
32717,32719 #505
32801,32803 #506
32831,32833 #507
32909,32911 #508
32939,32941 #509
32969,32971 #510
33071,33073 #511
33149,33151 #512
33179,33181 #513
33287,33289 #514
33329,33331 #515
33347,33349 #516
33587,33589 #517
33599,33601 #518
33617,33619 #519
33749,33751 #520
33767,33769 #521
33809,33811 #522
33827,33829 #523
34031,34033 #524
34127,34129 #525
34157,34159 #526
34211,34213 #527
34259,34261 #528
34301,34303 #529
34367,34369 #530
34469,34471 #531
34499,34501 #532
34511,34513 #533
34589,34591 #534
34649,34651 #535
34757,34759 #536
34841,34843 #537
34847,34849 #538
34961,34963 #539
35051,35053 #540
35081,35083 #541
35279,35281 #542
35447,35449 #543
35507,35509 #544
35531,35533 #545
35591,35593 #546
35729,35731 #547
35801,35803 #548
35837,35839 #549
35897,35899 #550
36011,36013 #551
36107,36109 #552
36341,36343 #553
36467,36469 #554
36527,36529 #555
36779,36781 #556
36791,36793 #557
36899,36901 #558
36929,36931 #559
37019,37021 #560
37199,37201 #561
37307,37309 #562
37337,37339 #563
37361,37363 #564
37547,37549 #565
37571,37573 #566
37589,37591 #567
37691,37693 #568
37781,37783 #569
37811,37813 #570
37991,37993 #571
38237,38239 #572
38327,38329 #573
38447,38449 #574
38459,38461 #575
38567,38569 #576
38609,38611 #577
38651,38653 #578
38669,38671 #579
38711,38713 #580
38747,38749 #581
38921,38923 #582
39041,39043 #583
39161,39163 #584
39227,39229 #585
39239,39241 #586
39341,39343 #587
39371,39373 #588
39509,39511 #589
39827,39829 #590
39839,39841 #591
40037,40039 #592
40127,40129 #593
40151,40153 #594
40427,40429 #595
40529,40531 #596
40637,40639 #597
40697,40699 #598
40847,40849 #599
41141,41143 #600
41177,41179 #601
41201,41203 #602
41231,41233 #603
41387,41389 #604
41411,41413 #605
41519,41521 #606
41609,41611 #607
41759,41761 #608
41849,41851 #609
41957,41959 #610
41981,41983 #611
42017,42019 #612
42071,42073 #613
42179,42181 #614
42221,42223 #615
42281,42283 #616
42407,42409 #617
42461,42463 #618
42569,42571 #619
42641,42643 #620
42701,42703 #621
42839,42841 #622
42899,42901 #623
43049,43051 #624
43319,43321 #625
43397,43399 #626
43541,43543 #627
43577,43579 #628
43607,43609 #629
43649,43651 #630
43781,43783 #631
43787,43789 #632
43889,43891 #633
43961,43963 #634
44027,44029 #635
44087,44089 #636
44129,44131 #637
44201,44203 #638
44267,44269 #639
44279,44281 #640
44381,44383 #641
44531,44533 #642
44621,44623 #643
44699,44701 #644
44771,44773 #645
45119,45121 #646
45137,45139 #647
45179,45181 #648
45317,45319 #649
45341,45343 #650
45587,45589 #651
45821,45823 #652
46049,46051 #653
46091,46093 #654
46181,46183 #655
46271,46273 #656
46307,46309 #657
46349,46351 #658
46439,46441 #659
46589,46591 #660
46679,46681 #661
46769,46771 #662
46817,46819 #663
46829,46831 #664
47057,47059 #665
47147,47149 #666
47351,47353 #667
47387,47389 #668
47417,47419 #669
47657,47659 #670
47699,47701 #671
47711,47713 #672
47741,47743 #673
47777,47779 #674
47807,47809 #675
48119,48121 #676
48311,48313 #677
48407,48409 #678
48479,48481 #679
48539,48541 #680
48647,48649 #681
48677,48679 #682
48731,48733 #683
48779,48781 #684
48821,48823 #685
48857,48859 #686
48869,48871 #687
48989,48991 #688
49031,49033 #689
49121,49123 #690
49169,49171 #691
49199,49201 #692
49277,49279 #693
49331,49333 #694
49367,49369 #695
49391,49393 #696
49409,49411 #697
49529,49531 #698
49547,49549 #699
49667,49669 #700
49739,49741 #701
49787,49789 #702
49919,49921 #703
49937,49939 #704
49991,49993 #705
50021,50023 #706
50051,50053 #707
50129,50131 #708
50261,50263 #709
50459,50461 #710
50549,50551 #711
50591,50593 #712
50891,50893 #713
50969,50971 #714
51059,51061 #715
51131,51133 #716
51197,51199 #717
51239,51241 #718
51341,51343 #719
51347,51349 #720
51419,51421 #721
51437,51439 #722
51479,51481 #723
51719,51721 #724
51767,51769 #725
51827,51829 #726
51869,51871 #727
51971,51973 #728
52067,52069 #729
52181,52183 #730
52289,52291 #731
52361,52363 #732
52541,52543 #733
52709,52711 #734
52859,52861 #735
52901,52903 #736
53087,53089 #737
53147,53149 #738
53171,53173 #739
53231,53233 #740
53267,53269 #741
53279,53281 #742
53549,53551 #743
53591,53593 #744
53609,53611 #745
53717,53719 #746
53897,53899 #747
54011,54013 #748
54401,54403 #749
54419,54421 #750
54497,54499 #751
54539,54541 #752
54581,54583 #753
54629,54631 #754
54917,54919 #755
55049,55051 #756
55217,55219 #757
55331,55333 #758
55337,55339 #759
55439,55441 #760
55619,55621 #761
55631,55633 #762
55661,55663 #763
55817,55819 #764
55901,55903 #765
55931,55933 #766
56039,56041 #767
56099,56101 #768
56207,56209 #769
56237,56239 #770
56267,56269 #771
56477,56479 #772
56501,56503 #773
56531,56533 #774
56597,56599 #775
56711,56713 #776
56807,56809 #777
56891,56893 #778
56909,56911 #779
56921,56923 #780
57191,57193 #781
57221,57223 #782
57269,57271 #783
57329,57331 #784
57347,57349 #785
57527,57529 #786
57557,57559 #787
57791,57793 #788
57899,57901 #789
58109,58111 #790
58151,58153 #791
58169,58171 #792
58229,58231 #793
58367,58369 #794
58391,58393 #795
58439,58441 #796
58451,58453 #797
58601,58603 #798
58787,58789 #799
58907,58909 #800
59009,59011 #801
59021,59023 #802
59051,59053 #803
59207,59209 #804
59219,59221 #805
59357,59359 #806
59417,59419 #807
59441,59443 #808
59471,59473 #809
59627,59629 #810
59669,59671 #811
60089,60091 #812
60101,60103 #813
60167,60169 #814
60257,60259 #815
60647,60649 #816
60659,60661 #817
60761,60763 #818
60887,60889 #819
60899,60901 #820
60917,60919 #821
61151,61153 #822
61331,61333 #823
61379,61381 #824
61469,61471 #825
61559,61561 #826
61979,61981 #827
62129,62131 #828
62141,62143 #829
62189,62191 #830
62297,62299 #831
62927,62929 #832
62969,62971 #833
62981,62983 #834
62987,62989 #835
63029,63031 #836
63197,63199 #837
63311,63313 #838
63389,63391 #839
63419,63421 #840
63587,63589 #841
63599,63601 #842
63647,63649 #843
63689,63691 #844
63839,63841 #845
64151,64153 #846
64187,64189 #847
64301,64303 #848
64451,64453 #849
64577,64579 #850
64661,64663 #851
64781,64783 #852
64877,64879 #853
64919,64921 #854
65027,65029 #855
65099,65101 #856
65171,65173 #857
65267,65269 #858
65447,65449 #859
65519,65521 #860
65537,65539 #861
65579,65581 #862
65699,65701 #863
65717,65719 #864
65729,65731 #865
65837,65839 #866
65927,65929 #867
65981,65983 #868
66107,66109 #869
66359,66361 #870
66569,66571 #871
66749,66751 #872
66851,66853 #873
66947,66949 #874
67139,67141 #875
67187,67189 #876
67211,67213 #877
67217,67219 #878
67271,67273 #879
67409,67411 #880
67427,67429 #881
67577,67579 #882
67757,67759 #883
67931,67933 #884
68111,68113 #885
68207,68209 #886
68279,68281 #887
68447,68449 #888
68489,68491 #889
68711,68713 #890
68819,68821 #891
68879,68881 #892
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145511,145513 #1652
145547,145549 #1653
145601,145603 #1654
145679,145681 #1655
145721,145723 #1656
145757,145759 #1657
145931,145933 #1658
145967,145969 #1659
146009,146011 #1660
146021,146023 #1661
146057,146059 #1662
146297,146299 #1663
146381,146383 #1664
146519,146521 #1665
146681,146683 #1666
146891,146893 #1667
146987,146989 #1668
147029,147031 #1669
147137,147139 #1670
147209,147211 #1671
147227,147229 #1672
147449,147451 #1673
147671,147673 #1674
148061,148063 #1675
148151,148153 #1676
148199,148201 #1677
148301,148303 #1678
148469,148471 #1679
148667,148669 #1680
148691,148693 #1681
148721,148723 #1682
148781,148783 #1683
148859,148861 #1684
148931,148933 #1685
149057,149059 #1686
149099,149101 #1687
149111,149113 #1688
149159,149161 #1689
149249,149251 #1690
149417,149419 #1691
149489,149491 #1692
149519,149521 #1693
149531,149533 #1694
149561,149563 #1695
149627,149629 #1696
149711,149713 #1697
149729,149731 #1698
149837,149839 #1699
149909,149911 #1700
149969,149971 #1701
150089,150091 #1702
150209,150211 #1703
150221,150223 #1704
150299,150301 #1705
150377,150379 #1706
150587,150589 #1707
150767,150769 #1708
150881,150883 #1709
150959,150961 #1710
150989,150991 #1711
151007,151009 #1712
151049,151051 #1713
151169,151171 #1714
151241,151243 #1715
151337,151339 #1716
151379,151381 #1717
151607,151609 #1718
151769,151771 #1719
151847,151849 #1720
151901,151903 #1721
151937,151939 #1722
151967,151969 #1723
152027,152029 #1724
152039,152041 #1725
152081,152083 #1726
152417,152419 #1727
152441,152443 #1728
152459,152461 #1729
152531,152533 #1730
152597,152599 #1731
152639,152641 #1732
152819,152821 #1733
152837,152839 #1734
152897,152899 #1735
152939,152941 #1736
153071,153073 #1737
153269,153271 #1738
153407,153409 #1739
153509,153511 #1740
153521,153523 #1741
153887,153889 #1742
153911,153913 #1743
153947,153949 #1744
154079,154081 #1745
154157,154159 #1746
154181,154183 #1747
154211,154213 #1748
154277,154279 #1749
154571,154573 #1750
154589,154591 #1751
154619,154621 #1752
154667,154669 #1753
154787,154789 #1754
154871,154873 #1755
155081,155083 #1756
155201,155203 #1757
155381,155383 #1758
155537,155539 #1759
155579,155581 #1760
155717,155719 #1761
155849,155851 #1762
155861,155863 #1763
155891,155893 #1764
156059,156061 #1765
156227,156229 #1766
156257,156259 #1767
156419,156421 #1768
156491,156493 #1769
156677,156679 #1770
156797,156799 #1771
156899,156901 #1772
156941,156943 #1773
157049,157051 #1774
157217,157219 #1775
157229,157231 #1776
157271,157273 #1777
157277,157279 #1778
157349,157351 #1779
157427,157429 #1780
157559,157561 #1781
157637,157639 #1782
157667,157669 #1783
157769,157771 #1784
157931,157933 #1785
158141,158143 #1786
158231,158233 #1787
158357,158359 #1788
158747,158749 #1789
158759,158761 #1790
159167,159169 #1791
159191,159193 #1792
159347,159349 #1793
159539,159541 #1794
159569,159571 #1795
159629,159631 #1796
159671,159673 #1797
159737,159739 #1798
159791,159793 #1799
159869,159871 #1800
159977,159979 #1801
160031,160033 #1802
160079,160081 #1803
160091,160093 #1804
160481,160483 #1805
160619,160621 #1806
160637,160639 #1807
160649,160651 #1808
160709,160711 #1809
160751,160753 #1810
160877,160879 #1811
160967,160969 #1812
161339,161341 #1813
161459,161461 #1814
161561,161563 #1815
161639,161641 #1816
161729,161731 #1817
161741,161743 #1818
161771,161773 #1819
161879,161881 #1820
161921,161923 #1821
161969,161971 #1822
162287,162289 #1823
162389,162391 #1824
162527,162529 #1825
162749,162751 #1826
162821,162823 #1827
162971,162973 #1828
163019,163021 #1829
163061,163063 #1830
163127,163129 #1831
163169,163171 #1832
163307,163309 #1833
163409,163411 #1834
163481,163483 #1835
163859,163861 #1836
163979,163981 #1837
163991,163993 #1838
164147,164149 #1839
164231,164233 #1840
164249,164251 #1841
164429,164431 #1842
164447,164449 #1843
164621,164623 #1844
164837,164839 #1845
164999,165001 #1846
165047,165049 #1847
165311,165313 #1848
165551,165553 #1849
165587,165589 #1850
165701,165703 #1851
165707,165709 #1852
165719,165721 #1853
166301,166303 #1854
166349,166351 #1855
166601,166603 #1856
166667,166669 #1857
166739,166741 #1858
166781,166783 #1859
166841,166843 #1860
166847,166849 #1861
167021,167023 #1862
167117,167119 #1863
167267,167269 #1864
167309,167311 #1865
167339,167341 #1866
167441,167443 #1867
167621,167623 #1868
167777,167779 #1869
167861,167863 #1870
168449,168451 #1871
168599,168601 #1872
168629,168631 #1873
168899,168901 #1874
169007,169009 #1875
169067,169069 #1876
169217,169219 #1877
169241,169243 #1878
169319,169321 #1879
169691,169693 #1880
169751,169753 #1881
169889,169891 #1882
170099,170101 #1883
170351,170353 #1884
170369,170371 #1885
170537,170539 #1886
170759,170761 #1887
171047,171049 #1888
171077,171079 #1889
171161,171163 #1890
171167,171169 #1891
171251,171253 #1892
171401,171403 #1893
171467,171469 #1894
171539,171541 #1895
171671,171673 #1896
171761,171763 #1897
172169,172171 #1898
172217,172219 #1899
172421,172423 #1900
172439,172441 #1901
172517,172519 #1902
173021,173023 #1903
173189,173191 #1904
173207,173209 #1905
173291,173293 #1906
173357,173359 #1907
173429,173431 #1908
173669,173671 #1909
173741,173743 #1910
173777,173779 #1911
174017,174019 #1912
174047,174049 #1913
174077,174079 #1914
174257,174259 #1915
174329,174331 #1916
174467,174469 #1917
174569,174571 #1918
174761,174763 #1919
174929,174931 #1920
174989,174991 #1921
175067,175069 #1922
175079,175081 #1923
175391,175393 #1924
175631,175633 #1925
175757,175759 #1926
175781,175783 #1927
175937,175939 #1928
175961,175963 #1929
175991,175993 #1930
176021,176023 #1931
176051,176053 #1932
176087,176089 #1933
176159,176161 #1934
176327,176329 #1935
176417,176419 #1936
176459,176461 #1937
176507,176509 #1938
176549,176551 #1939
176597,176599 #1940
176609,176611 #1941
176711,176713 #1942
176777,176779 #1943
176789,176791 #1944
176807,176809 #1945
176921,176923 #1946
177011,177013 #1947
177209,177211 #1948
177431,177433 #1949
177677,177679 #1950
177761,177763 #1951
177839,177841 #1952
177887,177889 #1953
178037,178039 #1954
178067,178069 #1955
178091,178093 #1956
178247,178249 #1957
178259,178261 #1958
178349,178351 #1959
178439,178441 #1960
178487,178489 #1961
178559,178561 #1962
178601,178603 #1963
178691,178693 #1964
178817,178819 #1965
178907,178909 #1966
178931,178933 #1967
179381,179383 #1968
179579,179581 #1969
179591,179593 #1970
179657,179659 #1971
179687,179689 #1972
179717,179719 #1973
179819,179821 #1974
179897,179899 #1975
179951,179953 #1976
179999,180001 #1977
180071,180073 #1978
180179,180181 #1979
180239,180241 #1980
180287,180289 #1981
180539,180541 #1982
180749,180751 #1983
180797,180799 #1984
181001,181003 #1985
181061,181063 #1986
181199,181201 #1987
181211,181213 #1988
181301,181303 #1989
181397,181399 #1990
181457,181459 #1991
181499,181501 #1992
181607,181609 #1993
181667,181669 #1994
181757,181759 #1995
181787,181789 #1996
181871,181873 #1997
181889,181891 #1998
182009,182011 #1999
182027,182029 #2000
182057,182059 #2001
182099,182101 #2002
182129,182131 #2003
182177,182179 #2004
182339,182341 #2005
182387,182389 #2006
182471,182473 #2007
182639,182641 #2008
182657,182659 #2009
182711,182713 #2010
182927,182929 #2011
183089,183091 #2012
183299,183301 #2013
183317,183319 #2014
183437,183439 #2015
183497,183499 #2016
183509,183511 #2017
183569,183571 #2018
183707,183709 #2019
183761,183763 #2020
183917,183919 #2021
183971,183973 #2022
184187,184189 #2023
184271,184273 #2024
184487,184489 #2025
184607,184609 #2026
184631,184633 #2027
184649,184651 #2028
184829,184831 #2029
184901,184903 #2030
184967,184969 #2031
184997,184999 #2032
185069,185071 #2033
185369,185371 #2034
185531,185533 #2035
185567,185569 #2036
185681,185683 #2037
185747,185749 #2038
185819,185821 #2039
185831,185833 #2040
185957,185959 #2041
186161,186163 #2042
186227,186229 #2043
186299,186301 #2044
186377,186379 #2045
186479,186481 #2046
186581,186583 #2047
186647,186649 #2048
186707,186709 #2049
186761,186763 #2050
186869,186871 #2051
187067,187069 #2052
187127,187129 #2053
187139,187141 #2054
187217,187219 #2055
187337,187339 #2056
187469,187471 #2057
187631,187633 #2058
187637,187639 #2059
187907,187909 #2060
188831,188833 #2061
188861,188863 #2062
188939,188941 #2063
189017,189019 #2064
189041,189043 #2065
189149,189151 #2066
189251,189253 #2067
189347,189349 #2068
189389,189391 #2069
189437,189439 #2070
189491,189493 #2071
189617,189619 #2072
189797,189799 #2073
189851,189853 #2074
189947,189949 #2075
190367,190369 #2076
190577,190579 #2077
190667,190669 #2078
190709,190711 #2079
190889,190891 #2080
191141,191143 #2081
191249,191251 #2082
191297,191299 #2083
191339,191341 #2084
191447,191449 #2085
191459,191461 #2086
191507,191509 #2087
191531,191533 #2088
191561,191563 #2089
191669,191671 #2090
191747,191749 #2091
191801,191803 #2092
191831,191833 #2093
192191,192193 #2094
192317,192319 #2095
192341,192343 #2096
192461,192463 #2097
192497,192499 #2098
192581,192583 #2099
192611,192613 #2100
192629,192631 #2101
192887,192889 #2102
192977,192979 #2103
193181,193183 #2104
193379,193381 #2105
193601,193603 #2106
193811,193813 #2107
193859,193861 #2108
193871,193873 #2109
193937,193939 #2110
194069,194071 #2111
194267,194269 #2112
194681,194683 #2113
194861,194863 #2114
194867,194869 #2115
195047,195049 #2116
195161,195163 #2117
195341,195343 #2118
195539,195541 #2119
195731,195733 #2120
195737,195739 #2121
195929,195931 #2122
195971,195973 #2123
196169,196171 #2124
196277,196279 #2125
196499,196501 #2126
196541,196543 #2127
196661,196663 #2128
196769,196771 #2129
196871,196873 #2130
196991,196993 #2131
197159,197161 #2132
197297,197299 #2133
197339,197341 #2134
197369,197371 #2135
197381,197383 #2136
197567,197569 #2137
197597,197599 #2138
197711,197713 #2139
197891,197893 #2140
197957,197959 #2141
197969,197971 #2142
198221,198223 #2143
198257,198259 #2144
198347,198349 #2145
198437,198439 #2146
198461,198463 #2147
198827,198829 #2148
198839,198841 #2149
198899,198901 #2150
198941,198943 #2151
199037,199039 #2152
199151,199153 #2153
199487,199489 #2154
199499,199501 #2155
199601,199603 #2156
199739,199741 #2157
199751,199753 #2158
199811,199813 #2159
199931,199933 #2160
200381,200383 #2161
200867,200869 #2162
200927,200929 #2163
200987,200989 #2164
201119,201121 #2165
201209,201211 #2166
201401,201403 #2167
201449,201451 #2168
201491,201493 #2169
201497,201499 #2170
201767,201769 #2171
201821,201823 #2172
201827,201829 #2173
202061,202063 #2174
202127,202129 #2175
202289,202291 #2176
202637,202639 #2177
202751,202753 #2178
202877,202879 #2179
202931,202933 #2180
203207,203209 #2181
203309,203311 #2182
203321,203323 #2183
203339,203341 #2184
203351,203353 #2185
203381,203383 #2186
203417,203419 #2187
203429,203431 #2188
203459,203461 #2189
203657,203659 #2190
203771,203773 #2191
203807,203809 #2192
203909,203911 #2193
203969,203971 #2194
204161,204163 #2195
204299,204301 #2196
204329,204331 #2197
204359,204361 #2198
204437,204439 #2199
204509,204511 #2200
204599,204601 #2201
204749,204751 #2202
204791,204793 #2203
204857,204859 #2204
205031,205033 #2205
205211,205213 #2206
205397,205399 #2207
205421,205423 #2208
205661,205663 #2209
205949,205951 #2210
205991,205993 #2211
206081,206083 #2212
206177,206179 #2213
206249,206251 #2214
206279,206281 #2215
206411,206413 #2216
206639,206641 #2217
206819,206821 #2218
206909,206911 #2219
206951,206953 #2220
207197,207199 #2221
207239,207241 #2222
207329,207331 #2223
207341,207343 #2224
207479,207481 #2225
207509,207511 #2226
207521,207523 #2227
207671,207673 #2228
207719,207721 #2229
207797,207799 #2230
207971,207973 #2231
208001,208003 #2232
208139,208141 #2233
208277,208279 #2234
208391,208393 #2235
208457,208459 #2236
208499,208501 #2237
208511,208513 #2238
208589,208591 #2239
208697,208699 #2240
208889,208891 #2241
208931,208933 #2242
208961,208963 #2243
208991,208993 #2244
209201,209203 #2245
209267,209269 #2246
209357,209359 #2247
209567,209569 #2248
209579,209581 #2249
209621,209623 #2250
209717,209719 #2251
209819,209821 #2252
209927,209929 #2253
210191,210193 #2254
210317,210319 #2255
210359,210361 #2256
210401,210403 #2257
210599,210601 #2258
210809,210811 #2259
210911,210913 #2260
211049,211051 #2261
211061,211063 #2262
211151,211153 #2263
211217,211219 #2264
211229,211231 #2265
211499,211501 #2266
211571,211573 #2267
211661,211663 #2268
211691,211693 #2269
211877,211879 #2270
211889,211891 #2271
211931,211933 #2272
212207,212209 #2273
212669,212671 #2274
212867,212869 #2275
213131,213133 #2276
213287,213289 #2277
213359,213361 #2278
213611,213613 #2279
213947,213949 #2280
214007,214009 #2281
214031,214033 #2282
214211,214213 #2283
214481,214483 #2284
214517,214519 #2285
214559,214561 #2286
214787,214789 #2287
215141,215143 #2288
215351,215353 #2289
215459,215461 #2290
215687,215689 #2291
215981,215983 #2292
216317,216319 #2293
216371,216373 #2294
216551,216553 #2295
216569,216571 #2296
216647,216649 #2297
216779,216781 #2298
216899,216901 #2299
216917,216919 #2300
217001,217003 #2301
217199,217201 #2302
217307,217309 #2303
217337,217339 #2304
217361,217363 #2305
217367,217369 #2306
217409,217411 #2307
217517,217519 #2308
217559,217561 #2309
217577,217579 #2310
217907,217909 #2311
217979,217981 #2312
218081,218083 #2313
218417,218419 #2314
218459,218461 #2315
218549,218551 #2316
218627,218629 #2317
218717,218719 #2318
218969,218971 #2319
218987,218989 #2320
219017,219019 #2321
219311,219313 #2322
219407,219409 #2323
219647,219649 #2324
219677,219679 #2325
219761,219763 #2326
219797,219799 #2327
219941,219943 #2328
219977,219979 #2329
220019,220021 #2330
220469,220471 #2331
220511,220513 #2332
220859,220861 #2333
220877,220879 #2334
220901,220903 #2335
220931,220933 #2336
221069,221071 #2337
221171,221173 #2338
221201,221203 #2339
221399,221401 #2340
221411,221413 #2341
221537,221539 #2342
221621,221623 #2343
221657,221659 #2344
221717,221719 #2345
221951,221953 #2346
221987,221989 #2347
222041,222043 #2348
222107,222109 #2349
222149,222151 #2350
222161,222163 #2351
222197,222199 #2352
222347,222349 #2353
222791,222793 #2354
222839,222841 #2355
222977,222979 #2356
223007,223009 #2357
223049,223051 #2358
223061,223063 #2359
223217,223219 #2360
223241,223243 #2361
223337,223339 #2362
223439,223441 #2363
223547,223549 #2364
223679,223681 #2365
223757,223759 #2366
223829,223831 #2367
223841,223843 #2368
223919,223921 #2369
224069,224071 #2370
224129,224131 #2371
224909,224911 #2372
225077,225079 #2373
225161,225163 #2374
225221,225223 #2375
225287,225289 #2376
225341,225343 #2377
225347,225349 #2378
225371,225373 #2379
225527,225529 #2380
225581,225583 #2381
225611,225613 #2382
225749,225751 #2383
225767,225769 #2384
225779,225781 #2385
225941,225943 #2386
226199,226201 #2387
226379,226381 #2388
226451,226453 #2389
226547,226549 #2390
226817,226819 #2391
226901,226903 #2392
227111,227113 #2393
227189,227191 #2394
227231,227233 #2395
227471,227473 #2396
227531,227533 #2397
227567,227569 #2398
227609,227611 #2399
227627,227629 #2400
227651,227653 #2401
228197,228199 #2402
228299,228301 #2403
228419,228421 #2404
228509,228511 #2405
228521,228523 #2406
228617,228619 #2407
228731,228733 #2408
228797,228799 #2409
228881,228883 #2410
228911,228913 #2411
228959,228961 #2412
229247,229249 #2413
229547,229549 #2414
229589,229591 #2415
229637,229639 #2416
229751,229753 #2417
229769,229771 #2418
229847,229849 #2419
229937,229939 #2420
229961,229963 #2421
229979,229981 #2422
230309,230311 #2423
230339,230341 #2424
230387,230389 #2425
230561,230563 #2426
230771,230773 #2427
230861,230863 #2428
230939,230941 #2429
230999,231001 #2430
231017,231019 #2431
231107,231109 #2432
231269,231271 #2433
231347,231349 #2434
231431,231433 #2435
231461,231463 #2436
231479,231481 #2437
231611,231613 #2438
231821,231823 #2439
231839,231841 #2440
232049,232051 #2441
232079,232081 #2442
232187,232189 #2443
232409,232411 #2444
232457,232459 #2445
232709,232711 #2446
232751,232753 #2447
232961,232963 #2448
233069,233071 #2449
233141,233143 #2450
233159,233161 #2451
233327,233329 #2452
233417,233419 #2453
233549,233551 #2454
233687,233689 #2455
233879,233881 #2456
233921,233923 #2457
233939,233941 #2458
234191,234193 #2459
234317,234319 #2460
234341,234343 #2461
234461,234463 #2462
234527,234529 #2463
234539,234541 #2464
234587,234589 #2465
234809,234811 #2466
234959,234961 #2467
234977,234979 #2468
235007,235009 #2469
235241,235243 #2470
235307,235309 #2471
235439,235441 #2472
235661,235663 #2473
235787,235789 #2474
235811,235813 #2475
235889,235891 #2476
236207,236209 #2477
236477,236479 #2478
236699,236701 #2479
236771,236773 #2480
236867,236869 #2481
236879,236881 #2482
236891,236893 #2483
236981,236983 #2484
237071,237073 #2485
237089,237091 #2486
237161,237163 #2487
237689,237691 #2488
237857,237859 #2489
237971,237973 #2490
238037,238039 #2491
238079,238081 #2492
238157,238159 #2493
238361,238363 #2494
238529,238531 #2495
238727,238729 #2496
238877,238879 #2497
238919,238921 #2498
239231,239233 #2499
239387,239389 #2500
239429,239431 #2501
239711,239713 #2502
239849,239851 #2503
240041,240043 #2504
240047,240049 #2505
240257,240259 #2506
240347,240349 #2507
240587,240589 #2508
240881,240883 #2509
241049,241051 #2510
241067,241069 #2511
241259,241261 #2512
241361,241363 #2513
241391,241393 #2514
241511,241513 #2515
241559,241561 #2516
241601,241603 #2517
241781,241783 #2518
241919,241921 #2519
241979,241981 #2520
242057,242059 #2521
242171,242173 #2522
242447,242449 #2523
242519,242521 #2524
242729,242731 #2525
243119,243121 #2526
243401,243403 #2527
243431,243433 #2528
243587,243589 #2529
243671,243673 #2530
243701,243703 #2531
243707,243709 #2532
244157,244159 #2533
244217,244219 #2534
244301,244303 #2535
244379,244381 #2536
244637,244639 #2537
244667,244669 #2538
244841,244843 #2539
244859,244861 #2540
245129,245131 #2541
245171,245173 #2542
245417,245419 #2543
245471,245473 #2544
245519,245521 #2545
245561,245563 #2546
245591,245593 #2547
245627,245629 #2548
245681,245683 #2549
245849,245851 #2550
245897,245899 #2551
245909,245911 #2552
245981,245983 #2553
246119,246121 #2554
246131,246133 #2555
246317,246319 #2556
246509,246511 #2557
246611,246613 #2558
246641,246643 #2559
246707,246709 #2560
246809,246811 #2561
246929,246931 #2562
247067,247069 #2563
247337,247339 #2564
247391,247393 #2565
247529,247531 #2566
247601,247603 #2567
247607,247609 #2568
247649,247651 #2569
247691,247693 #2570
247769,247771 #2571
247811,247813 #2572
247991,247993 #2573
247997,247999 #2574
248117,248119 #2575
248177,248179 #2576
248201,248203 #2577
248291,248293 #2578
248639,248641 #2579
248867,248869 #2580
248891,248893 #2581
249131,249133 #2582
249419,249421 #2583
249437,249439 #2584
249497,249499 #2585
249539,249541 #2586
249857,249859 #2587
249971,249973 #2588
250049,250051 #2589
250499,250501 #2590
250739,250741 #2591
250751,250753 #2592
250949,250951 #2593
250967,250969 #2594
251057,251059 #2595
251177,251179 #2596
251201,251203 #2597
251219,251221 #2598
251231,251233 #2599
251261,251263 #2600
251429,251431 #2601
251609,251611 #2602
251621,251623 #2603
251789,251791 #2604
251831,251833 #2605
251939,251941 #2606
251969,251971 #2607
252827,252829 #2608
252911,252913 #2609
253157,253159 #2610
253367,253369 #2611
253607,253609 #2612
253637,253639 #2613
253679,253681 #2614
253787,253789 #2615
253907,253909 #2616
253949,253951 #2617
254039,254041 #2618
254207,254209 #2619
254279,254281 #2620
254489,254491 #2621
254729,254731 #2622
254831,254833 #2623
254927,254929 #2624
255179,255181 #2625
255191,255193 #2626
255251,255253 #2627
255467,255469 #2628
255587,255589 #2629
255839,255841 #2630
255917,255919 #2631
255971,255973 #2632
256019,256021 #2633
256031,256033 #2634
256187,256189 #2635
256391,256393 #2636
256469,256471 #2637
256577,256579 #2638
256721,256723 #2639
256799,256801 #2640
256901,256903 #2641
257219,257221 #2642
257351,257353 #2643
257399,257401 #2644
257501,257503 #2645
257687,257689 #2646
257711,257713 #2647
257861,257863 #2648
257867,257869 #2649
257987,257989 #2650
258107,258109 #2651
258317,258319 #2652
258329,258331 #2653
258611,258613 #2654
258917,258919 #2655
259121,259123 #2656
259157,259159 #2657
259211,259213 #2658
259379,259381 #2659
259451,259453 #2660
259619,259621 #2661
259781,259783 #2662
259991,259993 #2663
260009,260011 #2664
260189,260191 #2665
260207,260209 #2666
260411,260413 #2667
260417,260419 #2668
260549,260551 #2669
260807,260809 #2670
260861,260863 #2671
261011,261013 #2672
261059,261061 #2673
261167,261169 #2674
261431,261433 #2675
261641,261643 #2676
261971,261973 #2677
262049,262051 #2678
262109,262111 #2679
262151,262153 #2680
262349,262351 #2681
262511,262513 #2682
262541,262543 #2683
262649,262651 #2684
262739,262741 #2685
262781,262783 #2686
263267,263269 #2687
263399,263401 #2688
263489,263491 #2689
263519,263521 #2690
263609,263611 #2691
263759,263761 #2692
263819,263821 #2693
263867,263869 #2694
263909,263911 #2695
263951,263953 #2696
264029,264031 #2697
264137,264139 #2698
264167,264169 #2699
264527,264529 #2700
264599,264601 #2701
264791,264793 #2702
264827,264829 #2703
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265247,265249 #2706
265271,265273 #2707
265337,265339 #2708
265511,265513 #2709
265541,265543 #2710
265619,265621 #2711
265709,265711 #2712
265871,265873 #2713
266027,266029 #2714
266051,266053 #2715
266081,266083 #2716
266291,266293 #2717
266351,266353 #2718
266447,266449 #2719
266477,266479 #2720
266489,266491 #2721
266681,266683 #2722
266687,266689 #2723
266837,266839 #2724
266897,266899 #2725
267131,267133 #2726
267227,267229 #2727
267299,267301 #2728
267389,267391 #2729
267401,267403 #2730
267431,267433 #2731
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267521,267523 #2733
267611,267613 #2734
267647,267649 #2735
267677,267679 #2736
267719,267721 #2737
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267899,267901 #2739
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268529,268531 #2742
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268781,268783 #2744
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268997,268999 #2748
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269177,269179 #2750
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270029,270031 #2754
270071,270073 #2755
270131,270133 #2756
270239,270241 #2757
270269,270271 #2758
270461,270463 #2759
270551,270553 #2760
270761,270763 #2761
270797,270799 #2762
271127,271129 #2763
271277,271279 #2764
271499,271501 #2765
271571,271573 #2766
271637,271639 #2767
271769,271771 #2768
271967,271969 #2769
272009,272011 #2770
272189,272191 #2771
272201,272203 #2772
272267,272269 #2773
272351,272353 #2774
272381,272383 #2775
272537,272539 #2776
272717,272719 #2777
272759,272761 #2778
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272981,272983 #2780
272999,273001 #2781
273059,273061 #2782
273269,273271 #2783
273281,273283 #2784
273311,273313 #2785
273641,273643 #2786
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273941,273943 #2788
274121,274123 #2789
274199,274201 #2790
274451,274453 #2791
274709,274711 #2792
274829,274831 #2793
275129,275131 #2794
275159,275161 #2795
275321,275323 #2796
275447,275449 #2797
275459,275461 #2798
275489,275491 #2799
275579,275581 #2800
275591,275593 #2801
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276047,276049 #2805
276371,276373 #2806
276587,276589 #2807
276671,276673 #2808
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277259,277261 #2813
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278147,278149 #2821
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282767,282769 #2860
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284129,284131 #2872
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285641,285643 #2890
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286367,286369 #2895
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290531,290533 #2926
290621,290623 #2927
290657,290659 #2928
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291041,291043 #2931
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292181,292183 #2941
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293147,293149 #2944
293177,293179 #2945
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294167,294169 #2949
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294311,294313 #2951
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294647,294649 #2953
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295199,295201 #2958
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296771,296773 #2970
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297809,297811 #2976
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300491,300493 #2999
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300647,300649 #3002
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305111,305113 #3040
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305759,305761 #3046
306167,306169 #3047
306191,306193 #3048
306329,306331 #3049
306347,306349 #3050
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306701,306703 #3053
306827,306829 #3054
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307169,307171 #3058
307187,307189 #3059
307259,307261 #3060
307337,307339 #3061
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307871,307873 #3065
308291,308293 #3066
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308519,308521 #3070
308639,308641 #3071
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309107,309109 #3075
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310019,310021 #3084
310127,310129 #3085
310229,310231 #3086
310361,310363 #3087
310727,310729 #3088
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311291,311293 #3090
311537,311539 #3091
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311681,311683 #3093
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312029,312031 #3097
312071,312073 #3098
312197,312199 #3099
312209,312211 #3100
312251,312253 #3101
312281,312283 #3102
312311,312313 #3103
312551,312553 #3104
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312941,312943 #3111
313127,313129 #3112
313151,313153 #3113
313331,313333 #3114
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313991,313993 #3119
314159,314161 #3120
314261,314263 #3121
314357,314359 #3122
314399,314401 #3123
314597,314599 #3124
314777,314779 #3125
315011,315013 #3126
315179,315181 #3127
315407,315409 #3128
315449,315451 #3129
315527,315529 #3130
315701,315703 #3131
315881,315883 #3132
316031,316033 #3133
316241,316243 #3134
316469,316471 #3135
316499,316501 #3136
316661,316663 #3137
316697,316699 #3138
316817,316819 #3139
316859,316861 #3140
317087,317089 #3141
317267,317269 #3142
317321,317323 #3143
317351,317353 #3144
317489,317491 #3145
317591,317593 #3146
317729,317731 #3147
317741,317743 #3148
317771,317773 #3149
317921,317923 #3150
317957,317959 #3151
317969,317971 #3152
318179,318181 #3153
318209,318211 #3154
318287,318289 #3155
318299,318301 #3156
318347,318349 #3157
318557,318559 #3158
318677,318679 #3159
318749,318751 #3160
318809,318811 #3161
318881,318883 #3162
318917,318919 #3163
319127,319129 #3164
319439,319441 #3165
319589,319591 #3166
319679,319681 #3167
319727,319729 #3168
319817,319819 #3169
319829,319831 #3170
320009,320011 #3171
320039,320041 #3172
320081,320083 #3173
320141,320143 #3174
320237,320239 #3175
320267,320269 #3176
320291,320293 #3177
320387,320389 #3178
320561,320563 #3179
320609,320611 #3180
320657,320659 #3181
320939,320941 #3182
321311,321313 #3183
321329,321331 #3184
321467,321469 #3185
321569,321571 #3186
321617,321619 #3187
321707,321709 #3188
321821,321823 #3189
321947,321949 #3190
322037,322039 #3191
322109,322111 #3192
322169,322171 #3193
322247,322249 #3194
322349,322351 #3195
322571,322573 #3196
322589,322591 #3197
322631,322633 #3198
322769,322771 #3199
322781,322783 #3200
322919,322921 #3201
322997,322999 #3202
323249,323251 #3203
323339,323341 #3204
323369,323371 #3205
323381,323383 #3206
323441,323443 #3207
323471,323473 #3208
323507,323509 #3209
323579,323581 #3210
323597,323599 #3211
323801,323803 #3212
324209,324211 #3213
324299,324301 #3214
324437,324439 #3215
324449,324451 #3216
324587,324589 #3217
324617,324619 #3218
324809,324811 #3219
324869,324871 #3220
324977,324979 #3221
325019,325021 #3222
325079,325081 #3223
325187,325189 #3224
325217,325219 #3225
325229,325231 #3226
325307,325309 #3227
325541,325543 #3228
325691,325693 #3229
325751,325753 #3230
325781,325783 #3231
325889,325891 #3232
326099,326101 #3233
326141,326143 #3234
326147,326149 #3235
326351,326353 #3236
326537,326539 #3237
326561,326563 #3238
326609,326611 #3239
326657,326659 #3240
326867,326869 #3241
326939,326941 #3242
326999,327001 #3243
327209,327211 #3244
327317,327319 #3245
327407,327409 #3246
327419,327421 #3247
327491,327493 #3248
327557,327559 #3249
327581,327583 #3250
327737,327739 #3251
327797,327799 #3252
327827,327829 #3253
327851,327853 #3254
327869,327871 #3255
328061,328063 #3256
328127,328129 #3257
328331,328333 #3258
328379,328381 #3259
328511,328513 #3260
328589,328591 #3261
328619,328621 #3262
328637,328639 #3263
328787,328789 #3264
328847,328849 #3265
328919,328921 #3266
329081,329083 #3267
329207,329209 #3268
329267,329269 #3269
329297,329299 #3270
329471,329473 #3271
329627,329629 #3272
329801,329803 #3273
330017,330019 #3274
330131,330133 #3275
330227,330229 #3276
330287,330289 #3277
330311,330313 #3278
330329,330331 #3279
330431,330433 #3280
330641,330643 #3281
330719,330721 #3282
330791,330793 #3283
330821,330823 #3284
330857,330859 #3285
331337,331339 #3286
331367,331369 #3287
331547,331549 #3288
331577,331579 #3289
331691,331693 #3290
331841,331843 #3291
331907,331909 #3292
331997,331999 #3293
332009,332011 #3294
332159,332161 #3295
332201,332203 #3296
332219,332221 #3297
332471,332473 #3298
332567,332569 #3299
332987,332989 #3300
333029,333031 #3301
333101,333103 #3302
333269,333271 #3303
333449,333451 #3304
333491,333493 #3305
333719,333721 #3306
333791,333793 #3307
334331,334333 #3308
334421,334423 #3309
334427,334429 #3310
334511,334513 #3311
334547,334549 #3312
334751,334753 #3313
334889,334891 #3314
334991,334993 #3315
335171,335173 #3316
335381,335383 #3317
335807,335809 #3318
336029,336031 #3319
336101,336103 #3320
336221,336223 #3321
336251,336253 #3322
336527,336529 #3323
336767,336769 #3324
336827,336829 #3325
336899,336901 #3326
337217,337219 #3327
337277,337279 #3328
337367,337369 #3329
337487,337489 #3330
337541,337543 #3331
337607,337609 #3332
337859,337861 #3333
337871,337873 #3334
337901,337903 #3335
338159,338161 #3336
338267,338269 #3337
338321,338323 #3338
338339,338341 #3339
338411,338413 #3340
338579,338581 #3341
339137,339139 #3342
339671,339673 #3343
339749,339751 #3344
339839,339841 #3345
340061,340063 #3346
340127,340129 #3347
340337,340339 #3348
340451,340453 #3349
340577,340579 #3350
340787,340789 #3351
340931,340933 #3352
340937,340939 #3353
341057,341059 #3354
341321,341323 #3355
341459,341461 #3356
341771,341773 #3357
341951,341953 #3358
342047,342049 #3359
342059,342061 #3360
342071,342073 #3361
342239,342241 #3362
342281,342283 #3363
342341,342343 #3364
342371,342373 #3365
342449,342451 #3366
342467,342469 #3367
342869,342871 #3368
343307,343309 #3369
343379,343381 #3370
343391,343393 #3371
343529,343531 #3372
343559,343561 #3373
343589,343591 #3374
343769,343771 #3375
343799,343801 #3376
343829,343831 #3377
344171,344173 #3378
344207,344209 #3379
344249,344251 #3380
344291,344293 #3381
344681,344683 #3382
344819,344821 #3383
344957,344959 #3384
345017,345019 #3385
345227,345229 #3386
345461,345463 #3387
345599,345601 #3388
345731,345733 #3389
345887,345889 #3390
346139,346141 #3391
346259,346261 #3392
346391,346393 #3393
346397,346399 #3394
346439,346441 #3395
346559,346561 #3396
346649,346651 #3397
346667,346669 #3398
346961,346963 #3399
347057,347059 #3400
347069,347071 #3401
347129,347131 #3402
347141,347143 #3403
347297,347299 #3404
347561,347563 #3405
347729,347731 #3406
347771,347773 #3407
347957,347959 #3408
347981,347983 #3409
347987,347989 #3410
348239,348241 #3411
348419,348421 #3412
348431,348433 #3413
348461,348463 #3414
348917,348919 #3415
348989,348991 #3416
349079,349081 #3417
349379,349381 #3418
349397,349399 #3419
349409,349411 #3420
349829,349831 #3421
349931,349933 #3422
350087,350089 #3423
350429,350431 #3424
350561,350563 #3425
350729,350731 #3426
350981,350983 #3427
351059,351061 #3428
351077,351079 #3429
351257,351259 #3430
351287,351289 #3431
351341,351343 #3432
351359,351361 #3433
351731,351733 #3434
351749,351751 #3435
351929,351931 #3436
352109,352111 #3437
352271,352273 #3438
352367,352369 #3439
352409,352411 #3440
352421,352423 #3441
352481,352483 #3442
352739,352741 #3443
352817,352819 #3444
352907,352909 #3445
352949,352951 #3446
353147,353149 #3447
353201,353203 #3448
353471,353473 #3449
353627,353629 #3450
353867,353869 #3451
354041,354043 #3452
354251,354253 #3453
354257,354259 #3454
354371,354373 #3455
354461,354463 #3456
354551,354553 #3457
354701,354703 #3458
354881,354883 #3459
354971,354973 #3460
355007,355009 #3461
355037,355039 #3462
355109,355111 #3463
355361,355363 #3464
355499,355501 #3465
355517,355519 #3466
355571,355573 #3467
355721,355723 #3468
355937,355939 #3469
355967,355969 #3470
356141,356143 #3471
356171,356173 #3472
356261,356263 #3473
356441,356443 #3474
356561,356563 #3475
356819,356821 #3476
356927,356929 #3477
357107,357109 #3478
357197,357199 #3479
357239,357241 #3480
357281,357283 #3481
357347,357349 #3482
357569,357571 #3483
357611,357613 #3484
357659,357661 #3485
357737,357739 #3486
357779,357781 #3487
358157,358159 #3488
358277,358279 #3489
358289,358291 #3490
358427,358429 #3491
358571,358573 #3492
358667,358669 #3493
358859,358861 #3494
358877,358879 #3495
358901,358903 #3496
358907,358909 #3497
359207,359209 #3498
359297,359299 #3499
359417,359419 #3500
359477,359479 #3501
359561,359563 #3502
360089,360091 #3503
360167,360169 #3504
360287,360289 #3505
360509,360511 #3506
360779,360781 #3507
360821,360823 #3508
360851,360853 #3509
360947,360949 #3510
360977,360979 #3511
361001,361003 #3512
361091,361093 #3513
361211,361213 #3514
361217,361219 #3515
361349,361351 #3516
361409,361411 #3517
361649,361651 #3518
361787,361789 #3519
361871,361873 #3520
361901,361903 #3521
362051,362053 #3522
362291,362293 #3523
362429,362431 #3524
362741,362743 #3525
362951,362953 #3526
363017,363019 #3527
363059,363061 #3528
363149,363151 #3529
363269,363271 #3530
363359,363361 #3531
363371,363373 #3532
363401,363403 #3533
363437,363439 #3534
363887,363889 #3535
363947,363949 #3536
364127,364129 #3537
364289,364291 #3538
364541,364543 #3539
364751,364753 #3540
364919,364921 #3541
365249,365251 #3542
365291,365293 #3543
365411,365413 #3544
365471,365473 #3545
365507,365509 #3546
365567,365569 #3547
365639,365641 #3548
365747,365749 #3549
365837,365839 #3550
366029,366031 #3551
366167,366169 #3552
366341,366343 #3553
366437,366439 #3554
366461,366463 #3555
366701,366703 #3556
366851,366853 #3557
367019,367021 #3558
367121,367123 #3559
367229,367231 #3560
367259,367261 #3561
367307,367309 #3562
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368231,368233 #3565
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369407,369409 #3570
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369827,369829 #3573
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371027,371029 #3579
371069,371071 #3580
371141,371143 #3581
371177,371179 #3582
371249,371251 #3583
371339,371341 #3584
371387,371389 #3585
371927,371929 #3586
371939,371941 #3587
372059,372061 #3588
372269,372271 #3589
372311,372313 #3590
372611,372613 #3591
372707,372709 #3592
372941,372943 #3593
372971,372973 #3594
373181,373183 #3595
373211,373213 #3596
373229,373231 #3597
373361,373363 #3598
373487,373489 #3599
373859,373861 #3600
374039,374041 #3601
374291,374293 #3602
374441,374443 #3603
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374837,374839 #3606
374987,374989 #3607
375017,375019 #3608
375101,375103 #3609
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375251,375253 #3611
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376529,376531 #3622
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376931,376933 #3626
377171,377173 #3627
377327,377329 #3628
377369,377371 #3629
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380879,380881 #3653
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381221,381223 #3655
381287,381289 #3656
381371,381373 #3657
381527,381529 #3658
381629,381631 #3659
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381791,381793 #3661
381989,381991 #3662
382001,382003 #3663
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383519,383521 #3673
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383837,383839 #3677
384257,384259 #3678
384287,384289 #3679
384299,384301 #3680
384479,384481 #3681
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385127,385129 #3683
385139,385141 #3684
385289,385291 #3685
385391,385393 #3686
385571,385573 #3687
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385661,385663 #3689
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386117,386119 #3692
386129,386131 #3693
386159,386161 #3694
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386369,386371 #3696
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405497,405499 #3851
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405677,405679 #3853
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1501427,1501429 #11608
1501481,1501483 #11609
1501499,1501501 #11610
1501679,1501681 #11611
1501781,1501783 #11612
1501847,1501849 #11613
1502021,1502023 #11614
1502099,1502101 #11615
1502141,1502143 #11616
1502201,1502203 #11617
1502327,1502329 #11618
1502687,1502689 #11619
1502717,1502719 #11620
1502861,1502863 #11621
1503317,1503319 #11622
1503371,1503373 #11623
1503611,1503613 #11624
1503659,1503661 #11625
1503881,1503883 #11626
1503959,1503961 #11627
1504409,1504411 #11628
1504469,1504471 #11629
1504661,1504663 #11630
1504691,1504693 #11631
1504859,1504861 #11632
1504967,1504969 #11633
1505087,1505089 #11634
1505291,1505293 #11635
1505519,1505521 #11636
1505657,1505659 #11637
1505681,1505683 #11638
1505849,1505851 #11639
1506077,1506079 #11640
1506389,1506391 #11641
1506497,1506499 #11642
1506509,1506511 #11643
1506551,1506553 #11644
1506611,1506613 #11645
1506731,1506733 #11646
1506779,1506781 #11647
1506887,1506889 #11648
1506977,1506979 #11649
1507139,1507141 #11650
1507421,1507423 #11651
1507481,1507483 #11652
1507607,1507609 #11653
1507697,1507699 #11654
1507769,1507771 #11655
1508249,1508251 #11656
1508279,1508281 #11657
1508321,1508323 #11658
1508471,1508473 #11659
1508621,1508623 #11660
1508627,1508629 #11661
1508909,1508911 #11662
1508951,1508953 #11663
1509059,1509061 #11664
1509437,1509439 #11665
1509551,1509553 #11666
1509587,1509589 #11667
1510217,1510219 #11668
1510307,1510309 #11669
1510319,1510321 #11670
1510337,1510339 #11671
1510361,1510363 #11672
1510391,1510393 #11673
1510427,1510429 #11674
1510679,1510681 #11675
1510757,1510759 #11676
1510961,1510963 #11677
1511099,1511101 #11678
1511231,1511233 #11679
1511327,1511329 #11680
1511441,1511443 #11681
1511597,1511599 #11682
1511687,1511689 #11683
1511819,1511821 #11684
1512221,1512223 #11685
1512281,1512283 #11686
1512479,1512481 #11687
1512557,1512559 #11688
1512689,1512691 #11689
1512827,1512829 #11690
1513019,1513021 #11691
1513067,1513069 #11692
1513091,1513093 #11693
1513121,1513123 #11694
1513271,1513273 #11695
1513319,1513321 #11696
1513397,1513399 #11697
1513427,1513429 #11698
1513487,1513489 #11699
1513529,1513531 #11700
1513619,1513621 #11701
1513667,1513669 #11702
1513739,1513741 #11703
1514099,1514101 #11704
1514321,1514323 #11705
1514327,1514329 #11706
1514549,1514551 #11707
1514561,1514563 #11708
1514657,1514659 #11709
1515719,1515721 #11710
1515821,1515823 #11711
1515971,1515973 #11712
1516127,1516129 #11713
1516187,1516189 #11714
1516259,1516261 #11715
1516391,1516393 #11716
1516589,1516591 #11717
1516607,1516609 #11718
1516661,1516663 #11719
1516817,1516819 #11720
1517051,1517053 #11721
1517099,1517101 #11722
1517141,1517143 #11723
1517519,1517521 #11724
1517567,1517569 #11725
1517651,1517653 #11726
1517687,1517689 #11727
1517939,1517941 #11728
1518089,1518091 #11729
1518311,1518313 #11730
1518551,1518553 #11731
1518677,1518679 #11732
1518707,1518709 #11733
1518731,1518733 #11734
1518947,1518949 #11735
1518971,1518973 #11736
1519097,1519099 #11737
1519121,1519123 #11738
1519421,1519423 #11739
1519517,1519519 #11740
1519547,1519549 #11741
1519709,1519711 #11742
1520009,1520011 #11743
1520339,1520341 #11744
1520357,1520359 #11745
1520501,1520503 #11746
1520537,1520539 #11747
1520681,1520683 #11748
1521029,1521031 #11749
1521227,1521229 #11750
1521671,1521673 #11751
1522019,1522021 #11752
1522049,1522051 #11753
1522361,1522363 #11754
1522457,1522459 #11755
1522691,1522693 #11756
1522769,1522771 #11757
1523087,1523089 #11758
1523099,1523101 #11759
1523441,1523443 #11760
1523567,1523569 #11761
1523651,1523653 #11762
1523939,1523941 #11763
1523981,1523983 #11764
1524071,1524073 #11765
1524077,1524079 #11766
1524137,1524139 #11767
1524179,1524181 #11768
1524359,1524361 #11769
1524377,1524379 #11770
1524401,1524403 #11771
1524431,1524433 #11772
1524569,1524571 #11773
1524629,1524631 #11774
1524701,1524703 #11775
1524827,1524829 #11776
1524839,1524841 #11777
1525031,1525033 #11778
1525217,1525219 #11779
1525331,1525333 #11780
1525421,1525423 #11781
1525607,1525609 #11782
1525637,1525639 #11783
1525961,1525963 #11784
1525967,1525969 #11785
1526069,1526071 #11786
1526087,1526089 #11787
1526267,1526269 #11788
1526339,1526341 #11789
1526639,1526641 #11790
1527107,1527109 #11791
1527287,1527289 #11792
1527311,1527313 #11793
1527347,1527349 #11794
1527521,1527523 #11795
1527551,1527553 #11796
1527677,1527679 #11797
1527791,1527793 #11798
1527857,1527859 #11799
1527899,1527901 #11800
1527971,1527973 #11801
1528139,1528141 #11802
1528937,1528939 #11803
1529027,1529029 #11804
1529069,1529071 #11805
1529189,1529191 #11806
1529387,1529389 #11807
1529501,1529503 #11808
1529531,1529533 #11809
1529849,1529851 #11810
1530071,1530073 #11811
1530227,1530229 #11812
1530311,1530313 #11813
1530521,1530523 #11814
1530539,1530541 #11815
1530827,1530829 #11816
1530869,1530871 #11817
1530911,1530913 #11818
1531091,1531093 #11819
1531331,1531333 #11820
1531631,1531633 #11821
1531811,1531813 #11822
1532351,1532353 #11823
1532579,1532581 #11824
1533107,1533109 #11825
1533137,1533139 #11826
1533197,1533199 #11827
1533437,1533439 #11828
1533461,1533463 #11829
1533797,1533799 #11830
1533899,1533901 #11831
1534019,1534021 #11832
1534067,1534069 #11833
1534151,1534153 #11834
1534217,1534219 #11835
1534451,1534453 #11836
1534787,1534789 #11837
1534961,1534963 #11838
1535069,1535071 #11839
1535291,1535293 #11840
1535351,1535353 #11841
1535669,1535671 #11842
1535717,1535719 #11843
1535969,1535971 #11844
1536011,1536013 #11845
1536047,1536049 #11846
1536581,1536583 #11847
1536641,1536643 #11848
1536677,1536679 #11849
1536809,1536811 #11850
1536959,1536961 #11851
1536989,1536991 #11852
1537397,1537399 #11853
1537439,1537441 #11854
1537559,1537561 #11855
1537799,1537801 #11856
1537967,1537969 #11857
1537997,1537999 #11858
1538027,1538029 #11859
1538057,1538059 #11860
1538081,1538083 #11861
1538501,1538503 #11862
1538597,1538599 #11863
1538609,1538611 #11864
1538627,1538629 #11865
1538837,1538839 #11866
1539257,1539259 #11867
1539449,1539451 #11868
1539719,1539721 #11869
1539971,1539973 #11870
1540151,1540153 #11871
1540169,1540171 #11872
1540541,1540543 #11873
1540619,1540621 #11874
1540697,1540699 #11875
1540709,1540711 #11876
1540751,1540753 #11877
1540787,1540789 #11878
1540871,1540873 #11879
1540961,1540963 #11880
1540967,1540969 #11881
1541117,1541119 #11882
1541357,1541359 #11883
1541429,1541431 #11884
1541819,1541821 #11885
1541921,1541923 #11886
1542029,1542031 #11887
1542041,1542043 #11888
1542089,1542091 #11889
1542347,1542349 #11890
1542509,1542511 #11891
1542521,1542523 #11892
1542689,1542691 #11893
1543391,1543393 #11894
1543511,1543513 #11895
1543637,1543639 #11896
1543811,1543813 #11897
1543979,1543981 #11898
1544129,1544131 #11899
1544507,1544509 #11900
1545041,1545043 #11901
1545239,1545241 #11902
1545389,1545391 #11903
1545431,1545433 #11904
1545617,1545619 #11905
1545701,1545703 #11906
1545809,1545811 #11907
1545911,1545913 #11908
1546217,1546219 #11909
1546229,1546231 #11910
1546271,1546273 #11911
1546547,1546549 #11912
1546757,1546759 #11913
1546901,1546903 #11914
1546967,1546969 #11915
1547129,1547131 #11916
1547477,1547479 #11917
1547519,1547521 #11918
1547591,1547593 #11919
1547657,1547659 #11920
1547717,1547719 #11921
1547771,1547773 #11922
1547837,1547839 #11923
1547879,1547881 #11924
1547927,1547929 #11925
1547939,1547941 #11926
1548179,1548181 #11927
1548539,1548541 #11928
1548719,1548721 #11929
1548761,1548763 #11930
1548947,1548949 #11931
1549319,1549321 #11932
1549367,1549369 #11933
1549529,1549531 #11934
1549547,1549549 #11935
1549739,1549741 #11936
1550051,1550053 #11937
1550207,1550209 #11938
1550231,1550233 #11939
1550441,1550443 #11940
1550777,1550779 #11941
1550999,1551001 #11942
1551497,1551499 #11943
1551617,1551619 #11944
1551659,1551661 #11945
1551731,1551733 #11946
1551791,1551793 #11947
1551887,1551889 #11948
1551917,1551919 #11949
1551959,1551961 #11950
1552121,1552123 #11951
1552379,1552381 #11952
1552541,1552543 #11953
1553009,1553011 #11954
1553309,1553311 #11955
1553507,1553509 #11956
1553807,1553809 #11957
1554101,1554103 #11958
1554281,1554283 #11959
1554347,1554349 #11960
1554611,1554613 #11961
1554737,1554739 #11962
1554779,1554781 #11963
1555157,1555159 #11964
1555187,1555189 #11965
1555247,1555249 #11966
1555259,1555261 #11967
1555289,1555291 #11968
1555469,1555471 #11969
1555571,1555573 #11970
1555637,1555639 #11971
1555691,1555693 #11972
1555817,1555819 #11973
1555997,1555999 #11974
1556327,1556329 #11975
1556369,1556371 #11976
1556561,1556563 #11977
1556669,1556671 #11978
1556717,1556719 #11979
1556759,1556761 #11980
1556771,1556773 #11981
1556837,1556839 #11982
1557041,1557043 #11983
1557089,1557091 #11984
1557287,1557289 #11985
1557341,1557343 #11986
1557707,1557709 #11987
1557947,1557949 #11988
1558307,1558309 #11989
1558559,1558561 #11990
1558727,1558729 #11991
1558757,1558759 #11992
1558787,1558789 #11993
1558811,1558813 #11994
1558829,1558831 #11995
1558937,1558939 #11996
1558979,1558981 #11997
1559057,1559059 #11998
1559447,1559449 #11999
1559477,1559479 #12000
1559609,1559611 #12001
1559849,1559851 #12002
1559891,1559893 #12003
1560047,1560049 #12004
1560131,1560133 #12005
1560239,1560241 #12006
1560407,1560409 #12007
1560707,1560709 #12008
1561037,1561039 #12009
1561121,1561123 #12010
1561421,1561423 #12011
1561577,1561579 #12012
1561757,1561759 #12013
1562051,1562053 #12014
1562087,1562089 #12015
1562129,1562131 #12016
1562291,1562293 #12017
1562357,1562359 #12018
1562591,1562593 #12019
1563017,1563019 #12020
1563227,1563229 #12021
1563257,1563259 #12022
1563281,1563283 #12023
1563407,1563409 #12024
1563431,1563433 #12025
1563467,1563469 #12026
1563629,1563631 #12027
1563971,1563973 #12028
1564307,1564309 #12029
1564361,1564363 #12030
1564499,1564501 #12031
1564571,1564573 #12032
1564907,1564909 #12033
1564991,1564993 #12034
1565189,1565191 #12035
1565381,1565383 #12036
1565489,1565491 #12037
1565519,1565521 #12038
1565561,1565563 #12039
1565609,1565611 #12040
1565741,1565743 #12041
1565789,1565791 #12042
1565867,1565869 #12043
1566197,1566199 #12044
1566209,1566211 #12045
1566281,1566283 #12046
1566401,1566403 #12047
1566449,1566451 #12048
1566749,1566751 #12049
1566767,1566769 #12050
1566821,1566823 #12051
1566881,1566883 #12052
1567001,1567003 #12053
1567037,1567039 #12054
1567169,1567171 #12055
1567259,1567261 #12056
1567301,1567303 #12057
1567409,1567411 #12058
1567901,1567903 #12059
1568141,1568143 #12060
1568351,1568353 #12061
1568519,1568521 #12062
1568561,1568563 #12063
1568921,1568923 #12064
1568969,1568971 #12065
1569011,1569013 #12066
1569257,1569259 #12067
1569317,1569319 #12068
1569329,1569331 #12069
1569551,1569553 #12070
1570097,1570099 #12071
1570451,1570453 #12072
1570631,1570633 #12073
1570769,1570771 #12074
1570871,1570873 #12075
1571027,1571029 #12076
1571237,1571239 #12077
1571417,1571419 #12078
1571681,1571683 #12079
1571741,1571743 #12080
1571747,1571749 #12081
1571957,1571959 #12082
1572251,1572253 #12083
1572281,1572283 #12084
1572377,1572379 #12085
1572509,1572511 #12086
1572587,1572589 #12087
1572677,1572679 #12088
1572749,1572751 #12089
1572869,1572871 #12090
1573079,1573081 #12091
1573109,1573111 #12092
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1573151,1573153 #12094
1573301,1573303 #12095
1573541,1573543 #12096
1573547,1573549 #12097
1573667,1573669 #12098
1573811,1573813 #12099
1573907,1573909 #12100
1573931,1573933 #12101
1573937,1573939 #12102
1574009,1574011 #12103
1574057,1574059 #12104
1574159,1574161 #12105
1574369,1574371 #12106
1575029,1575031 #12107
1575137,1575139 #12108
1575281,1575283 #12109
1575479,1575481 #12110
1575641,1575643 #12111
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1575989,1575991 #12113
1576037,1576039 #12114
1576241,1576243 #12115
1576499,1576501 #12116
1576649,1576651 #12117
1576721,1576723 #12118
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1577117,1577119 #12121
1577201,1577203 #12122
1577291,1577293 #12123
1577297,1577299 #12124
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1577531,1577533 #12126
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1577999,1578001 #12129
1578821,1578823 #12130
1578851,1578853 #12131
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1579931,1579933 #12137
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1580429,1580431 #12139
1580459,1580461 #12140
1580651,1580653 #12141
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1581707,1581709 #12147
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1581857,1581859 #12149
1582079,1582081 #12150
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1582577,1582579 #12153
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1582961,1582963 #12155
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1584437,1584439 #12167
1584641,1584643 #12168
1584701,1584703 #12169
1584827,1584829 #12170
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1585007,1585009 #12174
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1586537,1586539 #12180
1586621,1586623 #12181
1586771,1586773 #12182
1586789,1586791 #12183
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1588187,1588189 #12190
1588511,1588513 #12191
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1588877,1588879 #12196
1588901,1588903 #12197
1588931,1588933 #12198
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1589249,1589251 #12200
1589297,1589299 #12201
1589387,1589389 #12202
1589501,1589503 #12203
1589561,1589563 #12204
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1589849,1589851 #12206
1590047,1590049 #12207
1590077,1590079 #12208
1590131,1590133 #12209
1590539,1590541 #12210
1590551,1590553 #12211
1590791,1590793 #12212
1591097,1591099 #12213
1591871,1591873 #12214
1592111,1592113 #12215
1592321,1592323 #12216
1592429,1592431 #12217
1592579,1592581 #12218
1592621,1592623 #12219
1592777,1592779 #12220
1592861,1592863 #12221
1592867,1592869 #12222
1592879,1592881 #12223
1593197,1593199 #12224
1593269,1593271 #12225
1593377,1593379 #12226
1593497,1593499 #12227
1593539,1593541 #12228
1593797,1593799 #12229
1593857,1593859 #12230
1594127,1594129 #12231
1594259,1594261 #12232
1594631,1594633 #12233
1595051,1595053 #12234
1595309,1595311 #12235
1595729,1595731 #12236
1595831,1595833 #12237
1595861,1595863 #12238
1595927,1595929 #12239
1596059,1596061 #12240
1596311,1596313 #12241
1596347,1596349 #12242
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1597109,1597111 #12248
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1597721,1597723 #12250
1598237,1598239 #12251
1598447,1598449 #12252
1598501,1598503 #12253
1598789,1598791 #12254
1598897,1598899 #12255
1598951,1598953 #12256
1599461,1599463 #12257
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1599839,1599841 #12260
1600097,1600099 #12261
1600217,1600219 #12262
1600241,1600243 #12263
1600631,1600633 #12264
1600787,1600789 #12265
1600811,1600813 #12266
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1602077,1602079 #12276
1602101,1602103 #12277
1602119,1602121 #12278
1602281,1602283 #12279
1602527,1602529 #12280
1602551,1602553 #12281
1602719,1602721 #12282
1602749,1602751 #12283
1602827,1602829 #12284
1602899,1602901 #12285
1602941,1602943 #12286
1602959,1602961 #12287
1603079,1603081 #12288
1603331,1603333 #12289
1603337,1603339 #12290
1603361,1603363 #12291
1603517,1603519 #12292
1603529,1603531 #12293
1603697,1603699 #12294
1603709,1603711 #12295
1603799,1603801 #12296
1604129,1604131 #12297
1604147,1604149 #12298
1604177,1604179 #12299
1604297,1604299 #12300
1604609,1604611 #12301
1605029,1605031 #12302
1605419,1605421 #12303
1605431,1605433 #12304
1605509,1605511 #12305
1605551,1605553 #12306
1605629,1605631 #12307
1605887,1605889 #12308
1606151,1606153 #12309
1606247,1606249 #12310
1606259,1606261 #12311
1606289,1606291 #12312
1606541,1606543 #12313
1606739,1606741 #12314
1606751,1606753 #12315
1607141,1607143 #12316
1607477,1607479 #12317
1607699,1607701 #12318
1607831,1607833 #12319
1608107,1608109 #12320
1608239,1608241 #12321
1608461,1608463 #12322
1608569,1608571 #12323
1608821,1608823 #12324
1608911,1608913 #12325
1609061,1609063 #12326
1609247,1609249 #12327
1609667,1609669 #12328
1609691,1609693 #12329
1609871,1609873 #12330
1609901,1609903 #12331
1609997,1609999 #12332
1610177,1610179 #12333
1610237,1610239 #12334
1610309,1610311 #12335
1610429,1610431 #12336
1610471,1610473 #12337
1610657,1610659 #12338
1610771,1610773 #12339
1611689,1611691 #12340
1611761,1611763 #12341
1611851,1611853 #12342
1611899,1611901 #12343
1611947,1611949 #12344
1612181,1612183 #12345
1612211,1612213 #12346
1612307,1612309 #12347
1612361,1612363 #12348
1612619,1612621 #12349
1612997,1612999 #12350
1613321,1613323 #12351
1613411,1613413 #12352
1613639,1613641 #12353
1613669,1613671 #12354
1614329,1614331 #12355
1614461,1614463 #12356
1614491,1614493 #12357
1614629,1614631 #12358
1614647,1614649 #12359
1614659,1614661 #12360
1614719,1614721 #12361
1614911,1614913 #12362
1615181,1615183 #12363
1615331,1615333 #12364
1615499,1615501 #12365
1615631,1615633 #12366
1615841,1615843 #12367
1615847,1615849 #12368
1615919,1615921 #12369
1616609,1616611 #12370
1616621,1616623 #12371
1616687,1616689 #12372
1616801,1616803 #12373
1616807,1616809 #12374
1616897,1616899 #12375
1617137,1617139 #12376
1617347,1617349 #12377
1617437,1617439 #12378
1617689,1617691 #12379
1617767,1617769 #12380
1618049,1618051 #12381
1618079,1618081 #12382
1618091,1618093 #12383
1618187,1618189 #12384
1618367,1618369 #12385
1618457,1618459 #12386
1618679,1618681 #12387
1618739,1618741 #12388
1618829,1618831 #12389
1619069,1619071 #12390
1619207,1619209 #12391
1619327,1619329 #12392
1619339,1619341 #12393
1619381,1619383 #12394
1619417,1619419 #12395
1619549,1619551 #12396
1619669,1619671 #12397
1619687,1619689 #12398
1620329,1620331 #12399
1620467,1620469 #12400
1620569,1620571 #12401
1620611,1620613 #12402
1620629,1620631 #12403
1620677,1620679 #12404
1621031,1621033 #12405
1621349,1621351 #12406
1621421,1621423 #12407
1621469,1621471 #12408
1621619,1621621 #12409
1621637,1621639 #12410
1621721,1621723 #12411
1621727,1621729 #12412
1621769,1621771 #12413
1621931,1621933 #12414
1622039,1622041 #12415
1622141,1622143 #12416
1622207,1622209 #12417
1622471,1622473 #12418
1622639,1622641 #12419
1622669,1622671 #12420
1623161,1623163 #12421
1623287,1623289 #12422
1623827,1623829 #12423
1623929,1623931 #12424
1624169,1624171 #12425
1624199,1624201 #12426
1624277,1624279 #12427
1624349,1624351 #12428
1624589,1624591 #12429
1624661,1624663 #12430
1624811,1624813 #12431
1624967,1624969 #12432
1624991,1624993 #12433
1625177,1625179 #12434
1625207,1625209 #12435
1625417,1625419 #12436
1625717,1625719 #12437
1625747,1625749 #12438
1625807,1625809 #12439
1625837,1625839 #12440
1626071,1626073 #12441
1626089,1626091 #12442
1626281,1626283 #12443
1626377,1626379 #12444
1626431,1626433 #12445
1626479,1626481 #12446
1626617,1626619 #12447
1627061,1627063 #12448
1627487,1627489 #12449
1627601,1627603 #12450
1627607,1627609 #12451
1627649,1627651 #12452
1627727,1627729 #12453
1627781,1627783 #12454
1627859,1627861 #12455
1627979,1627981 #12456
1628057,1628059 #12457
1628171,1628173 #12458
1628381,1628383 #12459
1628489,1628491 #12460
1628591,1628593 #12461
1628987,1628989 #12462
1629011,1629013 #12463
1629107,1629109 #12464
1629209,1629211 #12465
1629317,1629319 #12466
1629359,1629361 #12467
1629449,1629451 #12468
1629557,1629559 #12469
1629581,1629583 #12470
1629599,1629601 #12471
1629851,1629853 #12472
1630019,1630021 #12473
1630049,1630051 #12474
1630091,1630093 #12475
1630127,1630129 #12476
1630379,1630381 #12477
1630427,1630429 #12478
1630457,1630459 #12479
1630547,1630549 #12480
1630619,1630621 #12481
1630841,1630843 #12482
1631027,1631029 #12483
1631051,1631053 #12484
1631057,1631059 #12485
1631261,1631263 #12486
1631297,1631299 #12487
1631489,1631491 #12488
1631519,1631521 #12489
1631657,1631659 #12490
1631897,1631899 #12491
1632311,1632313 #12492
1632467,1632469 #12493
1632479,1632481 #12494
1632569,1632571 #12495
1632647,1632649 #12496
1632749,1632751 #12497
1632767,1632769 #12498
1632779,1632781 #12499
1633127,1633129 #12500
1633169,1633171 #12501
1633319,1633321 #12502
1633337,1633339 #12503
1633361,1633363 #12504
1633559,1633561 #12505
1633691,1633693 #12506
1633787,1633789 #12507
1633991,1633993 #12508
1634051,1634053 #12509
1634069,1634071 #12510
1634201,1634203 #12511
1634231,1634233 #12512
1634291,1634293 #12513
1634441,1634443 #12514
1634681,1634683 #12515
1634879,1634881 #12516
1634951,1634953 #12517
1635371,1635373 #12518
1635497,1635499 #12519
1635971,1635973 #12520
1636007,1636009 #12521
1636067,1636069 #12522
1636331,1636333 #12523
1636541,1636543 #12524
1636667,1636669 #12525
1636697,1636699 #12526
1636757,1636759 #12527
1637549,1637551 #12528
1637639,1637641 #12529
1638059,1638061 #12530
1638209,1638211 #12531
1638347,1638349 #12532
1638797,1638799 #12533
1639151,1639153 #12534
1639199,1639201 #12535
1639241,1639243 #12536
1639511,1639513 #12537
1639577,1639579 #12538
1639607,1639609 #12539
1640057,1640059 #12540
1640621,1640623 #12541
1640927,1640929 #12542
1640939,1640941 #12543
1641089,1641091 #12544
1641359,1641361 #12545
1641377,1641379 #12546
1641587,1641589 #12547
1641797,1641799 #12548
1641929,1641931 #12549
1642031,1642033 #12550
1642049,1642051 #12551
1642481,1642483 #12552
1642517,1642519 #12553
1642631,1642633 #12554
1642811,1642813 #12555
1643219,1643221 #12556
1643231,1643233 #12557
1643597,1643599 #12558
1643639,1643641 #12559
1643819,1643821 #12560
1643867,1643869 #12561
1643891,1643893 #12562
1643987,1643989 #12563
1644197,1644199 #12564
1644371,1644373 #12565
1644437,1644439 #12566
1644491,1644493 #12567
1644689,1644691 #12568
1644899,1644901 #12569
1644947,1644949 #12570
1644989,1644991 #12571
1645559,1645561 #12572
1645601,1645603 #12573
1645667,1645669 #12574
1645727,1645729 #12575
1645769,1645771 #12576
1645907,1645909 #12577
1645937,1645939 #12578
1646147,1646149 #12579
1646171,1646173 #12580
1646219,1646221 #12581
1646717,1646719 #12582
1646921,1646923 #12583
1647251,1647253 #12584
1647377,1647379 #12585
1647551,1647553 #12586
1647599,1647601 #12587
1647857,1647859 #12588
1648067,1648069 #12589
1648259,1648261 #12590
1648289,1648291 #12591
1648481,1648483 #12592
1648529,1648531 #12593
1649099,1649101 #12594
1649147,1649149 #12595
1649171,1649173 #12596
1649309,1649311 #12597
1649771,1649773 #12598
1649801,1649803 #12599
1649861,1649863 #12600
1650107,1650109 #12601
1650611,1650613 #12602
1651211,1651213 #12603
1651409,1651411 #12604
1651511,1651513 #12605
1651589,1651591 #12606
1651691,1651693 #12607
1652351,1652353 #12608
1652489,1652491 #12609
1652771,1652773 #12610
1652837,1652839 #12611
1652879,1652881 #12612
1652897,1652899 #12613
1652921,1652923 #12614
1653059,1653061 #12615
1653101,1653103 #12616
1653107,1653109 #12617
1653191,1653193 #12618
1653329,1653331 #12619
1653341,1653343 #12620
1653497,1653499 #12621
1653917,1653919 #12622
1654019,1654021 #12623
1654031,1654033 #12624
1654199,1654201 #12625
1654649,1654651 #12626
1654787,1654789 #12627
1654979,1654981 #12628
1655021,1655023 #12629
1655177,1655179 #12630
1655207,1655209 #12631
1655279,1655281 #12632
1655321,1655323 #12633
1655807,1655809 #12634
1655891,1655893 #12635
1656047,1656049 #12636
1656119,1656121 #12637
1656167,1656169 #12638
1656227,1656229 #12639
1656311,1656313 #12640
1656647,1656649 #12641
1656791,1656793 #12642
1656827,1656829 #12643
1656839,1656841 #12644
1656899,1656901 #12645
1657037,1657039 #12646
1657571,1657573 #12647
1657697,1657699 #12648
1657937,1657939 #12649
1658051,1658053 #12650
1658201,1658203 #12651
1658309,1658311 #12652
1658387,1658389 #12653
1658411,1658413 #12654
1658441,1658443 #12655
1659101,1659103 #12656
1659107,1659109 #12657
1659347,1659349 #12658
1659569,1659571 #12659
1659809,1659811 #12660
1659881,1659883 #12661
1660037,1660039 #12662
1660229,1660231 #12663
1660259,1660261 #12664
1660409,1660411 #12665
1660469,1660471 #12666
1660661,1660663 #12667
1660697,1660699 #12668
1660721,1660723 #12669
1660739,1660741 #12670
1660871,1660873 #12671
1661159,1661161 #12672
1661249,1661251 #12673
1661831,1661833 #12674
1662119,1662121 #12675
1662161,1662163 #12676
1662629,1662631 #12677
1662641,1662643 #12678
1662779,1662781 #12679
1662839,1662841 #12680
1662959,1662961 #12681
1662977,1662979 #12682
1663217,1663219 #12683
1663301,1663303 #12684
1663349,1663351 #12685
1663379,1663381 #12686
1663547,1663549 #12687
1664459,1664461 #12688
1664561,1664563 #12689
1664651,1664653 #12690
1664711,1664713 #12691
1664867,1664869 #12692
1665071,1665073 #12693
1665107,1665109 #12694
1665527,1665529 #12695
1665569,1665571 #12696
1665581,1665583 #12697
1665647,1665649 #12698
1665929,1665931 #12699
1665941,1665943 #12700
1666037,1666039 #12701
1666211,1666213 #12702
1666307,1666309 #12703
1666727,1666729 #12704
1666781,1666783 #12705
1667051,1667053 #12706
1667249,1667251 #12707
1667357,1667359 #12708
1667441,1667443 #12709
1667507,1667509 #12710
1667597,1667599 #12711
1667639,1667641 #12712
1667747,1667749 #12713
1667777,1667779 #12714
1667789,1667791 #12715
1667957,1667959 #12716
1668131,1668133 #12717
1668299,1668301 #12718
1668479,1668481 #12719
1668551,1668553 #12720
1668617,1668619 #12721
1668647,1668649 #12722
1668911,1668913 #12723
1669097,1669099 #12724
1669469,1669471 #12725
1669541,1669543 #12726
1669649,1669651 #12727
1669781,1669783 #12728
1669931,1669933 #12729
1670057,1670059 #12730
1670411,1670413 #12731
1670489,1670491 #12732
1670531,1670533 #12733
1670561,1670563 #12734
1670567,1670569 #12735
1670657,1670659 #12736
1670831,1670833 #12737
1671209,1671211 #12738
1671347,1671349 #12739
1671641,1671643 #12740
1672079,1672081 #12741
1672337,1672339 #12742
1672379,1672381 #12743
1672421,1672423 #12744
1672469,1672471 #12745
1672499,1672501 #12746
1672607,1672609 #12747
1672637,1672639 #12748
1672751,1672753 #12749
1672961,1672963 #12750
1673069,1673071 #12751
1673207,1673209 #12752
1673279,1673281 #12753
1673627,1673629 #12754
1673807,1673809 #12755
1673951,1673953 #12756
1673981,1673983 #12757
1674161,1674163 #12758
1674269,1674271 #12759
1674557,1674559 #12760
1674599,1674601 #12761
1674767,1674769 #12762
1674887,1674889 #12763
1674917,1674919 #12764
1674947,1674949 #12765
1674989,1674991 #12766
1675109,1675111 #12767
1675181,1675183 #12768
1675577,1675579 #12769
1675769,1675771 #12770
1675787,1675789 #12771
1675799,1675801 #12772
1676027,1676029 #12773
1676069,1676071 #12774
1676471,1676473 #12775
1676627,1676629 #12776
1676711,1676713 #12777
1676891,1676893 #12778
1677197,1677199 #12779
1677251,1677253 #12780
1677281,1677283 #12781
1677461,1677463 #12782
1677521,1677523 #12783
1678067,1678069 #12784
1678091,1678093 #12785
1678151,1678153 #12786
1678217,1678219 #12787
1678319,1678321 #12788
1678361,1678363 #12789
1678421,1678423 #12790
1678601,1678603 #12791
1678751,1678753 #12792
1678757,1678759 #12793
1678769,1678771 #12794
1678877,1678879 #12795
1678889,1678891 #12796
1679057,1679059 #12797
1679099,1679101 #12798
1679471,1679473 #12799
1679681,1679683 #12800
1679831,1679833 #12801
1680101,1680103 #12802
1680179,1680181 #12803
1680269,1680271 #12804
1680317,1680319 #12805
1680359,1680361 #12806
1680527,1680529 #12807
1680821,1680823 #12808
1681259,1681261 #12809
1681571,1681573 #12810
1681619,1681621 #12811
1681649,1681651 #12812
1681721,1681723 #12813
1681871,1681873 #12814
1681877,1681879 #12815
1682249,1682251 #12816
1682477,1682479 #12817
1682537,1682539 #12818
1682669,1682671 #12819
1682831,1682833 #12820
1683041,1683043 #12821
1683467,1683469 #12822
1683839,1683841 #12823
1684097,1684099 #12824
1684169,1684171 #12825
1684229,1684231 #12826
1684301,1684303 #12827
1684607,1684609 #12828
1684691,1684693 #12829
1685111,1685113 #12830
1685207,1685209 #12831
1685267,1685269 #12832
1685441,1685443 #12833
1685447,1685449 #12834
1685477,1685479 #12835
1685711,1685713 #12836
1685777,1685779 #12837
1685819,1685821 #12838
1685861,1685863 #12839
1685951,1685953 #12840
1686257,1686259 #12841
1686341,1686343 #12842
1686701,1686703 #12843
1687451,1687453 #12844
1687667,1687669 #12845
1687757,1687759 #12846
1687781,1687783 #12847
1687799,1687801 #12848
1688261,1688263 #12849
1688327,1688329 #12850
1688369,1688371 #12851
1688411,1688413 #12852
1688969,1688971 #12853
1689197,1689199 #12854
1689377,1689379 #12855
1689551,1689553 #12856
1689659,1689661 #12857
1689911,1689913 #12858
1689929,1689931 #12859
1690079,1690081 #12860
1690097,1690099 #12861
1690187,1690189 #12862
1690217,1690219 #12863
1690229,1690231 #12864
1690571,1690573 #12865
1690691,1690693 #12866
1690781,1690783 #12867
1690847,1690849 #12868
1691099,1691101 #12869
1691411,1691413 #12870
1691531,1691533 #12871
1691861,1691863 #12872
1691867,1691869 #12873
1692137,1692139 #12874
1692239,1692241 #12875
1692827,1692829 #12876
1692947,1692949 #12877
1693091,1693093 #12878
1693169,1693171 #12879
1693271,1693273 #12880
1693331,1693333 #12881
1693427,1693429 #12882
1693577,1693579 #12883
1693631,1693633 #12884
1693661,1693663 #12885
1693889,1693891 #12886
1694027,1694029 #12887
1694081,1694083 #12888
1694309,1694311 #12889
1694351,1694353 #12890
1694447,1694449 #12891
1695347,1695349 #12892
1695401,1695403 #12893
1695437,1695439 #12894
1695509,1695511 #12895
1695641,1695643 #12896
1695761,1695763 #12897
1695779,1695781 #12898
1696421,1696423 #12899
1696577,1696579 #12900
1696691,1696693 #12901
1696859,1696861 #12902
1697039,1697041 #12903
1697411,1697413 #12904
1697459,1697461 #12905
1697621,1697623 #12906
1697741,1697743 #12907
1697867,1697869 #12908
1697957,1697959 #12909
1697987,1697989 #12910
1698119,1698121 #12911
1698131,1698133 #12912
1698311,1698313 #12913
1698377,1698379 #12914
1698509,1698511 #12915
1698797,1698799 #12916
1698857,1698859 #12917
1698869,1698871 #12918
1698881,1698883 #12919
1699067,1699069 #12920
1699109,1699111 #12921
1699331,1699333 #12922
1699391,1699393 #12923
1699469,1699471 #12924
1699499,1699501 #12925
1699679,1699681 #12926
1699739,1699741 #12927
1699781,1699783 #12928
1699799,1699801 #12929
1699877,1699879 #12930
1699937,1699939 #12931
1700141,1700143 #12932
1700267,1700269 #12933
1700339,1700341 #12934
1700591,1700593 #12935
1700759,1700761 #12936
1700849,1700851 #12937
1700981,1700983 #12938
1701017,1701019 #12939
1701041,1701043 #12940
1701059,1701061 #12941
1701179,1701181 #12942
1701389,1701391 #12943
1701437,1701439 #12944
1701521,1701523 #12945
1701641,1701643 #12946
1701827,1701829 #12947
1701857,1701859 #12948
1701899,1701901 #12949
1701911,1701913 #12950
1702319,1702321 #12951
1702637,1702639 #12952
1702661,1702663 #12953
1702709,1702711 #12954
1702739,1702741 #12955
1702781,1702783 #12956
1702817,1702819 #12957
1702901,1702903 #12958
1702931,1702933 #12959
1702991,1702993 #12960
1703267,1703269 #12961
1703717,1703719 #12962
1704119,1704121 #12963
1704587,1704589 #12964
1704611,1704613 #12965
1704671,1704673 #12966
1704929,1704931 #12967
1705127,1705129 #12968
1705139,1705141 #12969
1705271,1705273 #12970
1705397,1705399 #12971
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2472929,2472931 #17784
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2810579,2810581 #19831
2810711,2810713 #19832
2810909,2810911 #19833
2810957,2810959 #19834
2811089,2811091 #19835
2811161,2811163 #19836
2811227,2811229 #19837
2811617,2811619 #19838
2811629,2811631 #19839
2811659,2811661 #19840
2811707,2811709 #19841
2812421,2812423 #19842
2812751,2812753 #19843
2812811,2812813 #19844
2813339,2813341 #19845
2813411,2813413 #19846
2813477,2813479 #19847
2813507,2813509 #19848
2813579,2813581 #19849
2813807,2813809 #19850
2813819,2813821 #19851
2813849,2813851 #19852
2814167,2814169 #19853
2814431,2814433 #19854
2814839,2814841 #19855
2815739,2815741 #19856
2816057,2816059 #19857
2816087,2816089 #19858
2816171,2816173 #19859
2816291,2816293 #19860
2816531,2816533 #19861
2817077,2817079 #19862
2817167,2817169 #19863
2817251,2817253 #19864
2817467,2817469 #19865
2817671,2817673 #19866
2818157,2818159 #19867
2818391,2818393 #19868
2818469,2818471 #19869
2818997,2818999 #19870
2819021,2819023 #19871
2819051,2819053 #19872
2819099,2819101 #19873
2819147,2819149 #19874
2819471,2819473 #19875
2819489,2819491 #19876
2819519,2819521 #19877
2819627,2819629 #19878
2819681,2819683 #19879
2819741,2819743 #19880
2820017,2820019 #19881
2820359,2820361 #19882
2820401,2820403 #19883
2820479,2820481 #19884
2820707,2820709 #19885
2820749,2820751 #19886
2820887,2820889 #19887
2820941,2820943 #19888
2821151,2821153 #19889
2821769,2821771 #19890
2821829,2821831 #19891
2821979,2821981 #19892
2821997,2821999 #19893
2822009,2822011 #19894
2822189,2822191 #19895
2822297,2822299 #19896
2822711,2822713 #19897
2822717,2822719 #19898
2822879,2822881 #19899
2823437,2823439 #19900
2823521,2823523 #19901
2823671,2823673 #19902
2823809,2823811 #19903
2823971,2823973 #19904
2824187,2824189 #19905
2824649,2824651 #19906
2825099,2825101 #19907
2825411,2825413 #19908
2825477,2825479 #19909
2825489,2825491 #19910
2825819,2825821 #19911
2825861,2825863 #19912
2825957,2825959 #19913
2825981,2825983 #19914
2826071,2826073 #19915
2826149,2826151 #19916
2826179,2826181 #19917
2826737,2826739 #19918
2826851,2826853 #19919
2826917,2826919 #19920
2827211,2827213 #19921
2827547,2827549 #19922
2827631,2827633 #19923
2827679,2827681 #19924
2828297,2828299 #19925
2828429,2828431 #19926
2828597,2828599 #19927
2828627,2828629 #19928
2828741,2828743 #19929
2828867,2828869 #19930
2829569,2829571 #19931
2829677,2829679 #19932
2829707,2829709 #19933
2829887,2829889 #19934
2830097,2830099 #19935
2830151,2830153 #19936
2830349,2830351 #19937
2830871,2830873 #19938
2830937,2830939 #19939
2830967,2830969 #19940
2831657,2831659 #19941
2831669,2831671 #19942
2831789,2831791 #19943
2831861,2831863 #19944
2831951,2831953 #19945
2831999,2832001 #19946
2832131,2832133 #19947
2832257,2832259 #19948
2832329,2832331 #19949
2832629,2832631 #19950
2833319,2833321 #19951
2833331,2833333 #19952
2833799,2833801 #19953
2833811,2833813 #19954
2834261,2834263 #19955
2834411,2834413 #19956
2834651,2834653 #19957
2834717,2834719 #19958
2834747,2834749 #19959
2835137,2835139 #19960
2835221,2835223 #19961
2835269,2835271 #19962
2835587,2835589 #19963
2835671,2835673 #19964
2835689,2835691 #19965
2836079,2836081 #19966
2836241,2836243 #19967
2836259,2836261 #19968
2836367,2836369 #19969
2836487,2836489 #19970
2836607,2836609 #19971
2836619,2836621 #19972
2836961,2836963 #19973
2836991,2836993 #19974
2837057,2837059 #19975
2837069,2837071 #19976
2837279,2837281 #19977
2837501,2837503 #19978
2837711,2837713 #19979
2837801,2837803 #19980
2837951,2837953 #19981
2837981,2837983 #19982
2838137,2838139 #19983
2838149,2838151 #19984
2838287,2838289 #19985
2838461,2838463 #19986
2838629,2838631 #19987
2838767,2838769 #19988
2838851,2838853 #19989
2838917,2838919 #19990
2839469,2839471 #19991
2839547,2839549 #19992
2839841,2839843 #19993
2839931,2839933 #19994
2839937,2839939 #19995
2840039,2840041 #19996
2840237,2840239 #19997
2840261,2840263 #19998
2840267,2840269 #19999
2840417,2840419 #20000

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Introduction to twin primes and Brun's constant computation

Source : http://numbers.computation.free.fr/Constants/Primes/twin.html

 

Introduction to twin primes and Brun's constant computation

 

(Click here for a Postscript version of this page and here for a pdf version)

 

1  Introduction

It's a very old fact (Euclid 325-265 B.C., in Book IX of the Elements) that the set of primes is infinite and a much more recent and famous result (by Jacques Hadamard (1865-1963) and Charles-Jean de la Vallee Poussin (1866-1962)) that the density of primes is ruled by the law

p(n) ~  n

log(n)
 

where the prime counting function p(n) is the number of prime numbers less than a given integer n. This result proved in 1896 is the celebrated prime numbers theorem and was conjectured earlier, in 1792, by young Carl Friedrich Gauss (1777-1855) and by Adrien-Marie Legendre (1752-1833) who studied the repartition of those numbers in published tables of primes.

This approximation may be usefully replaced by the more accurate logarithmic integral Li(n):

p(n) ~ Li(n)= ó
õ
n

2 
   dt

log(t)
.

However among the deeply studied set of primes there is a famous and fascinating subset for which very little is known and has generated some famous conjectures: the twin primes (the term prime pairs was used before [5]).

Definition 1 A couple of primes (p,q) are said to be twins if q=p+2. Except for the couple (2,3), this is clearly the smallest possible distance between two primes.

Example 2 (3,5),(5,7),(11,13),(17,19),(29,31),...,(419,421),... are twin primes.

 

2  Counting twin primes

As for the set of primes the most natural question is wether the set of twin primes is finite or not. But unlike prime numbers for which numerous and elementary proofs exist [10], the answer to this natural question is still unknown for twin primes ! Today, this problem remains one of the greatest challenge in mathematics and has occupied numbers of mathematicians. Of course, as we will see, there are some empirical and numerical results suggesting an answer and most mathematicians believe that there are infinitely many twin primes.

In 1849, Alphonse de Polignac (1817-1890) made the general conjecture that there are infinitely many primes distant from 2k. The case for which k=1 is the twin primes case.

It's now natural to introduce the twin prime counting function p2(n) which is the number of twin primes smaller than a given n.

 

2.1  Numerical results

Using huge table of primes (Glaisher 1878 [5], before computer age, enumerated p2(105)) and with intensive computations during modern period (Shanks and Wrench 1974 [12], Brent 1976 [1], Nicely 1996-2002 [9], Sebah 2001-2002 [11], see also [10]) it's possible to compute the exact values of p2(n) for large n and its conjectured approximation 2C2Li2(n) (see next section for the definition).

The following array includes the relative error e (in %) between the approximation and the real value.

 

n p2(n) 2C2Li2(n) e
10 2 5 150.00
102 8 14 75.00
103 35 46 31.43
104 205 214 4.39
105 1224 1249 2.04
106 8169 8248 0.97
107 58980 58754 -0.38
108 440312 440368 0.013
109 3424506 3425308 0.023
1010 27412679 27411417 -0.0046
1011 224376048 224368865 -0.0032
1012 1870585220 1870559867 -0.0013
1013 15834664872 15834598305 -0.00042
1014 135780321665 135780264894 -0.000042
1015 1177209242304 1177208491861 -0.000064
1016 10304195697298 10304192554496 -0.000031

At present time (2002), Pascal Sebah has reached p2(1016) and his values are confirmed by Thomas Nicely up to p2(4.1015) who used an independent approach and implementation.

 

2.1.1  Sieving twin primes

Because the most convenient (in fact the only available) way to compute p2(n) is to find all twin primes and just count them, it's of great importance to improve as much as possible such an algorithm. All known methods use variations on the historical Eratosthenes sieve.

In order to accelerate the sieve, a possible idea is to represent integers modulo a base m, so that any integer has the form mk+r with 0 £ r < m. Primes numbers are such as m and r are relatively primes and for any value of m, there are f(m) numbers r which are prime with m (this is the definition of Euler's f totient function).

Modulo 6  

For example modulo 6 all integers have one of the form

6k,6k+1,6k+2,6k+3,6k+4,6k+5

but 6k may not be prime, 6k+2 and 6k+4 are divisible by 2, 6k+3 is divisible by 3, therefore primes (except 2 and 3) must be of the form

6k+1,6k+5

or which is more convenient to sieve twin primes

( 6k+5,6k+7) .

This allows to sieve a proportion of only f(m)/m=2/6 or 33.3% of all the numbers.

Modulo 30  

The same kind of approach modulo 30 gives for candidates

30k+1,30k+7,30k+11,30k+13,30k+17,30k+19,30k+23,30k+29

and here we need to sieve only f(m)/m=8/30 or 26.66% of the integers to find all primes (except 2,3 and 5).

But remember that we are only trying to sieve twin primes hence are left only the candidate couples

( 30k+11,30k+13) ,( 30k+17,30k+19) ,(30k+29,30k+31)

and the proportion drops to 6/30 or 20% of all integers.

This suggest to introduce the function f2(m) which is the number of pairs of integer 0 £ r < m such as r and r+2 are relatively prime with m. We observe from the last two examples that

f2(6)=1,f2(30)=3.


   Example  

Let's illustrate this on a numerical example. The enumeration of twin primes modulo 30 up to 1010 gives, respectively, for each of the 3 previous couples:

9137078,9138015,9137584

twin primes. So that (don't forget to count the two couples (3,5) and (5,7)!)

p2(1010)=9137078+9138015+9137584+2=27412679.


And, for the same couples, up to 1012:

623517830,623557143,623510245

which produces

p2(1012)=623517830+623557143+623510245+2=1870585220.

It's interesting to observe that the contribution to the enumeration of the twin primes of each couple is almost equivalent. This was also observed during all numerical estimations for other modulo like 210, 2310, 30030, ... [11].

This result is well known when enumerating just prime numbers but may be conjectured for twin primes.

Other modulo  

In this table we show the proportion 2f2(m)/m of integer to sieve in order to count twin primes as a function of the modulo m:

 

m f2(m) %
2 . 50.0
6 1 33.3
30 3 20.0
210 15 14.3
2310 135 11.7
30030 1485 9.9
510510 22275 8.7

The smallest ratios are obtained for values of m which are the product of the first primes (2#=2,3#=2×3,5#=2×3×5,7#=2×3×5×7,...), that is the first values of the primorial # function. For example in [11], the sieves were made modulo 30030 and 510510, therefore less than 10% of the set of integers were considered by the algorithm. In some others implementations sieves modulo 6 or 30 are used.

 

2.2  Twin prime conjecture

Based on heuristic considerations, a law (the twin prime conjecture) was developed, in 1922, by Godfrey Harold Hardy (1877-1947) and John Edensor Littlewood (1885-1977) to estimate the density of twin primes.

According to the prime number theorem the probability that a number n is prime is about 1/log(n), therefore, if the probability that n+2 is also prime was independent of the probability for n, we should have the approximation

p2(n) ~  n

log2(n)
,

but a more careful analysis shows that this model is too simplified (an argument is given in [6]). In fact we have the following and more accurate conjecture (called conjecture B in[7]).

Conjecture 3 [Twin prime conjecture]For large values of n, the two following equivalent approximations are conjectured

p2(n) ~ 2C2Li2(n)=2C2 ó
õ
n

2 
   dt

log2(t)
 
(1)

or

p2(n) ~ 2C2  n

log2(n)
 
(2)

Note that C2 is the twin prime constant and is defined by

C2=
Õ
p ³ 3 
  æ
è
 p(p-2)

(p-1)2
ö
ø
=0.6601618158468695739278121100145...

This last constant occurs in some asymptotic estimations involving primes and it's interesting to observe that it may be estimated using properties of the Riemann Zeta function to thousand of digits (Sebah computed it to more than 5000 digits).

 

Figure 1: p2(n) and 2C2Li2(n)

Remark 4 The function Li2(n) occuring in (1) may be related to the logarithmic integral Li(n) by the trivial relation

Li2(n)=Li(n)+  2

log(2)
-  n

log(n)
.

 

2.2.1  Generalizations

In fact, Hardy and Littlewood made a more general conjecture on the primes separated by a gap of d. A natural generalization of the twin primes is to search for primes distant of d=2k (which should be infinite for any d according to Polignac's conjecture). The case d=2 is the twin primes set, d=4 forms the cousin primes set, d=6 is the sexy primes set, ...

If we denote pd(n) the number of primes p £ n such as p+d is also prime (observe that here p and p+d may not be consecutive), Hardy-Littlewood's conjecture states (in [7]) that for d ³ 2:

pd(n) ~ 2C2Rd ó
õ
n

2 
   dt

log2(t)
,

with Rd ³ 1 being the rational number

Rd=
Õ
p|d,p > 2 
   p-1

p-2
,       p is prime.

The first values of the function Rd are

 

d 2 4 6 8 10 12 14 16 18 20
Rd 1 1 2 1 4/3 2 6/5 1 2 4/3

According to this conjecture the density of twin primes is equivalent to the density of cousin primes. For example, the exact computed values up to 1012 are: 

p2(1012) =1870585220
p4(1012) =1870585458, 

which can be compared to the predicted value 1870559867 by the conjecture. Marek Wolf has studied the function

W(n)=p2(n)-p4(n)

and its fractal properties and approximate dimension of 1.48 ([13]).

 

3  Brun's constant

 

3.1  From Euler's constant to Brun's constant

Euler's constant  

It's very natural to understand the nature of the harmonic numbers

H(n)=1+  1

2
+  1

3
+  1

4
+...+  1

n
 

when n becomes large and the sum takes all integers in account. We know since Euler, for instance, that

  ê
ê
ê
H(n) ~ log(n)
H(n)-log(n) ~ g
 

so that the harmonic numbers tends to infinite like log(n). Note that g is Euler's constant and may be evaluated to million of digits.

Mertens' constant  

The next step is to take in account only the primes numbers in the sum that is

P(p)=1+  1

2
+  1

3
+  1

5
+...+  1

p
 

and we have the beautiful results that

  ê
ê
ê
P(p) ~ log( log(p))
P(p)-log( log(p)) ~ M
 

Therefore the sum diverges (this was also observed by Euler) but at the very low rate log(log(p)) and M is the interesting Mertens' constant which may be evaluated to much less digits than g, say a few thousands.

Brun's Constant  

In the last step we only take in account the twin primes less than p in the sum

B2(p)= æ
è
 1

3
+  1

5
ö
ø
+ æ
è
 1

5
+  1

7
ö
ø
+ æ
è
 1

11
+  1

13
ö
ø
+...

and here comes the remarkable result due to Norwegian Mathematician Viggo Brun (1885-1978) in 1919 [2].

Theorem 5 The sum of the inverse of the twin primes converges to a finite constant B2.

We write this result as

 
lim
p®¥ 
B2(p)=B2.

Note that this theorem doesn't answer to the question of the infinitude of twin primes, it just says that the limit exists (and may or may not contains a finite number of terms !). The proof is rather complex and based on a majoration of the density of twin primes ; a more modern one may also be found in [8].

Unlike Euler's constant or Mertens' constant, Brun's constant is one of the hardest to evaluate and we are not even sure to know 9 digits of it. By mean of very intensive computations, we only have guaranteed minorations !

 

3.2  Estimation of Brun's constant

 

3.2.1  Direct estimation

In the following table we have try to estimate this constant by computing the partial sums B2(p) up to different values of p.

 

p B2(p) 
102 1.330990365719...
104 1.616893557432...
106 1.710776930804...
108 1.758815621067...
1010 1.787478502719...
1012 1.806592419175...
1014 1.820244968130...
1015 1.825706013240...
1016 1.830484424658...

From this, we observe that the convergence is extremely slow and irregular. If we expect to find even just a few digits, we have to make some assumptions.

 

3.2.2  Extrapolation

An easy consequence of the twin prime conjecture is that we may write the numbers B2(p) as (see [4] and [9])

B2(p)=B2-  4C2

log(p)
+O æ
è
 1

Öplog(p)
ö
ø
 

and thanks to this relation, the extrapolated value

B2*(p)=B2(p)+  4C2

log(p)
 

converges much faster to Brun's constant B2.

Let's take a look to numerical values:

 

p B2*(p) 
102 1.904399633290...
104 1.903598191217...
106 1.901913353327...
108 1.902167937960...
1010 1.902160356233...
1012 1.902160630437...
1014 1.902160577783...
1015 1.902160582249...
1016 1.902160583104...

which suggest that the value of B2 should be around 1.902160583... (a similar value was first proposed by Nicely after intensive computations and checked later by Sebah, see [9] and [11]).

The relation

B2(p) ~ B2-  4C2

log(p)
 

invites us to draw the function B2(p)=f( 1/log(p)) (see Figure 2) which should be a line with a negative slop with a value near -4C2 » -2.64064726.

 

Figure 2: B2(p)=f( [ 1/log(p)])

The intersection of the line with the vertical axis (that is p=¥) is Brun's constant if the twin prime conjecture is valid. And according to this line the direct estimation B2(p) should reach 1.9 not before the value p ~ 10530 which is far beyond any computational project !

 

4  Twin prime characterization

There is a result from Clement (1949, [3]) which permits to see if a couple (p,p+2) is a twin primes pair. This theorem extends Wilson's famous theorem on prime numbers.

Theorem 6 Let p ³ 3, the integers (p,p+2) form a twin primes pair if and only if

4( (p-1)!+1) º -p mod p(p+2)

Example 7 For p=17, 4( (p-1)!+1) = 83691159552004 º 306 mod 323 and -p º 306 mod 323, therefore (17,19) is a twin prime pair.

The huge value of the factorial makes this theorem of no practical use to find large twin primes.

 

4.1  Large twin primes

Today, thanks to modern computers, a lot of huge twin primes are known. Many of those primes are of the form k×2n±1 because there are efficient primality testing algorithms for such numbers when k is not too large.

The following theorem due to the French farmer François Proth (1852-1879) may be used.

Theorem 8 [Proth's theorem - 1878]Let N=k.2n+1 with k < 2n, if there is an integer a such as

a(N-1)/2 º -1 mod N

then N is prime.

To help finding large pairs, an idea is to take a value for n and then to start a sieve in order to reduce the set of possible values for the k. It should take a few hours to find twin primes with a few thousands digits.

For example the following numbers are twin prime pairs (some are given from [10]): 

 
 
459.28529±1       Dubner, 1993
 
 
594501.29999±1
 
 
6797727.215328±1       Forbes, 1995
 
 
697053813.216352±1       Indlekofer & Janai, 1994
 
 
318032361.2107001±1       Underbakke & Carmody & ..., 2001

The last one is a twin primes pair of more than 32000 digits !

 

References

[1]
R.P. Brent, Tables Concerning Irregularities in the Distribution of Primes and Twin Primes Up to 1011, Math. Comput., (1976), vol. 30, p. 379

 

[2]
V. Brun, La série 1/5+1/7+1/11+1/13+1/17+1/19+1/29+1/31+1/41+1/43+1/59+1/61+..., où les dénominateurs sont nombres premiers jumeaux est convergente ou finie, Bulletin des sciences mathématiques, (1919), vol. 43, p. 100-104 and p. 124-128

 

[3]
P.A. Clement, Congruences for sets of primes, American Mathematical Monthly, (1949), vol. 56, p. 23-25

 

[4]
C.E. Fröberg, On the sum of inverses of primes and twin primes, Nordisk Tidskr. Informationsbehandling (BIT), (1961), vol. 1, p. 15-20.

 

[5]
J.W.L. Glaisher, An enumeration of prime-pairs, Messenger of Mathematics, (1878), vol. 8, p. 28-33

 

[6]
G.H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford Science Publications, (1979)

 

[7]
G.H. Hardy and J.E. Littlewood, Some problems of 'Partitio Numerorum' III : On the expression of a number as a sum of primes, Acta Mathematica, (1922), vol. 44, p. 1-70

 

[8]
W.J. LeVeque, Fundamentals of Number Theory, New York, Dover, (1996)

 

[9]
T. Nicely, Enumeration to 1014 of the Twin Primes and Brun's Constant, Virginia J. Sci., (1996), vol. 46, p. 195-204

 

[10]
P. Ribenboim, The new Book of Prime Number Records, Springer, (1996)

 

[11]
P. Sebah, Counting Twin Primes and estimation of Brun's Constant up to 1016, Computational project at http://numbers.computation.free.fr/Constants/constants.html, (2002)

 

[12]
D. Shanks and J.W. Wrench, Brun's Constant, (1974), Math. Comput., vol. 28, p. 293-299

 

[13]
M. Wolf, On the Twin and Cousin primes, (1996), See http://www.ift.uni.wroc.pl/~mwolf/




File translated from TEX by TTH, version 3.01.
On 29 Jul 2002, 14:41.

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L'union fait la force des mathématiciens LE MONDE SCIENCE ET TECHNO | 24.06.2013 à 15h40 | En savoir plus sur http://www.lemonde.fr/sciences/article/2013/06/24/l-union-fait-la-force-des-mathematiciens_3435624_1650684.html#64byvQVhFuDDXMoA.99

L'union fait la force des mathématiciens

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Le résultat en question, accepté pour publication en mai 2013 par les prestigieuses Annals of Mathematics, annonce..."


En savoir plus sur http://www.lemonde.fr/sciences/article/2013/06/24/l-union-fait-la-force-des-mathematiciens_3435624_1650684.html#64byvQVhFuDDXMoA.99


En savoir plus sur http://www.lemonde.fr/sciences/article/2013/06/24/l-union-fait-la-force-des-mathematiciens_3435624_1650684.html#64byvQVhFuDDXMoA.99

 

 

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Nombres premiers jumeaux

Nombres premiers jumeaux

 
 

En mathématiques, deux nombres premiers jumeaux sont deux nombres premiers qui ne diffèrent que de 2. Hormis pour le couple (2, 3), cet écart entre nombres premiers de 2 est le plus petit possible. Les plus petits nombres premiers jumeaux sont 3 et 5, 5 et 7, 11 et 13.

Au 25 décembre 2011, les plus grands nombres premiers jumeaux connus, découverts dans le cadre du projet de calcul distribué PrimeGrid, sont 3 756 801 695 685 × 2666 669 ± 1 ; ils possèdent 200 700 chiffres en écriture décimale.

Selon la conjecture des nombres premiers jumeaux, il existe une infinité de nombres premiers jumeaux ; les observations numériques et des raisonnements heuristiquesjustifient la conjecture, mais aucune démonstration n'en a encore été faite.

 

 

Définition[modifier | modifier le code]

Soient p et q deux nombres premiers. On dit que (p, q) forme un couple de nombres premiers jumeaux si q = p + 2.

Liste des premiers nombres premiers jumeaux[modifier | modifier le code]

L'ensemble des nombres premiers jumeaux jusqu'à 1000 :

(3, 5) (5, 7) (11, 13) (17, 19) (29, 31)
(41, 43) (59, 61) (71, 73) (101, 103) (107, 109)
(137, 139) (149, 151) (179, 181) (191, 193) (197, 199)
(227, 229) (239, 241) (269, 271) (281, 283) (311, 313)
(347, 349) (419 , 421) (431 , 433) (461 , 463) (521 , 523)
(569 , 571) (599 , 601) (617 , 619) (641 , 643) (659 , 661)
(809 , 811) (821 , 823) (827 , 829) (857 , 859) (881 , 883)

Quelques propriétés[modifier | modifier le code]

  • Le couple (2, 3) est le seul couple de nombres premiers consécutifs.
  • Si l'on omet le couple (2, 3), la plus petite distance possible entre deux nombres premiers est 2 ; deux nombres premiers jumeaux sont ainsi deux nombres impairsconsécutifs.
  • Tout couple de nombres premiers jumeaux, à l'exception du couple (3, 5), est de la forme (6n – 1, 6n + 1) pour un certain entier n. En effet, tout triplet d'entiers consécutifs comporte au moins un multiple de 2 (éventuellement deux) et un seul multiple de 3 ; l'entier qui se trouve entre les deux nombres premiers jumeaux est à la fois ce multiple de 2 et ce multiple de 3, car cela ne peut pas être l'un des nombres premiers.
  • Pour tout entier m ≥ 2, le couple (m, m + 2) est constitué de nombres premiers jumeaux si et seulement si 4[(m - 1)! + 1] + m est divisible par m(m + 2). Cette caractérisation des nombres premiers jumeaux, remarquée par P. A. Clement en 19491, résulte du théorème de Wilson.
  • Alors que la série des inverses des nombres premiers est divergente, la série des inverses de nombres premiers jumeaux est convergente (vers un nombre appelé constante de Brun). Cette propriété fut démontrée par Viggo Brun en 19192.

Records[modifier | modifier le code]

Le 15 janvier 2007, les deux projets de calcul distribué Twin Prime Search et PrimeGrid ont découvert le plus grand couple de nombres premiers jumeaux connu à l'époque, de 58 711 chiffres en écriture décimale. Le découvreur était le Français Éric Vautier3.

Le 25 décembre 2011, le couple record3 est 3 756 801 695 685 × 2666 669 ± 1 ; les deux nombres possèdent 200 700 chiffres.

Conjecture des nombres premiers jumeaux[modifier | modifier le code]

La conjecture des nombres premiers jumeaux affirme qu'il existe une infinité de nombres premiers jumeaux:

Il existe une infinité de nombres premiers p tels que p + 2 soit aussi premier.

Cette conjecture partage avec l'hypothèse de Riemann et la conjecture de Goldbach le numéro 8 des problèmes de Hilbert, énoncés par ce dernier en 1900. Bien que la plupart des chercheurs en théorie des nombres pensent que cette conjecture est vraie, elle n'a jamais été démontrée. Ils se basent sur des observations numériques et des raisonnements heuristiques utilisant la distribution probabiliste des nombres premiers.

En 1849, Alphonse de Polignac émit une conjecture plus générale : la conjecture de Polignac :

Tout nombre pair est égal à la différence de deux nombres premiers consécutifs d'une infinité de manières.

dont le cas n = 2 correspond à la conjecture des nombres premiers jumeaux.

Il existe également une version plus forte de cette conjecture : la première conjecture de Hardy-Littlewood (cf. infra), qui fournit une loi de distribution des nombres premiers jumeaux et qui s'inspire du théorème des nombres premiers.

La conjecture des nombres premiers jumeaux est un cas particulier de la conjecture de Schinzel.

Résultats partiels[modifier | modifier le code]

En 1940, Paul Erdős démontra l'existence d'une constante positive c < 1 pour laquelle l'ensemble des nombres premiers p tels que p'p < c ln(p) est infini, où p' désigne le nombre premier suivant immédiatement p.

Ce résultat fut plusieurs fois amélioré ; en 1986, Helmut Maier montra que c peut être choisi inférieur à 1/4. En 2005, Daniel Goldston, János Pintz et Cem Yıldırım démontrèrent que c peut être choisi arbitrairement petit.

Par ailleurs, en 1966, Chen Jingrun démontra l'existence d'une infinité de « nombres premiers de Chen », c'est-à-dire de nombres premiers p tels que p + 2 soit premier ou semi-premier (un nombre semi-premier est le produit de deux nombres premiers). Son approche est celle de la théorie des cribles, qu'il a utilisée pour traiter de façon similaire la conjecture des nombres premiers jumeaux et la conjecture de Goldbach (voir Théorème de Chen).

À partir de 2009, à la suite de la découverte d'une optimisation du crible d'Eratosthène, Zhang Yitang établit qu'il existe une infinité de nombres premiers consécutifs dont l'écart est inférieur à 70 000 000, résultat qui constitue une forme faible de la conjecture des nombres premiers jumeaux. Début 2013, le projet Polymath, un projet de mathématiques collaboratives mené par Tim Gowers et Terence Tao, a proposé de réduire progressivement cet écart N = 70 millions pour le faire tendre vers 2 : en septembre 2013 l'écart a été réduit à N = 4 6804,5. En novembre 2013, une amélioration significative de ces résultats est annoncée indépendamment par James Maynard (en) et Terence Tao6 : non seulement l'écart entre deux nombres premiers consécutifs est inférieur ou égal à 600 infiniment souvent, mais un résultat équivalent est valable pour m nombres premiers consécutifs, quel que soit m ≥ 2. Une nouvelle amélioration est annoncée par le projet Polymath (section Polymath8) début 2014 : d'une part, l'écart serait inférieur à 270 infiniment souvent, d'autre part, en admettant une version généralisée de la conjecture d'Elliott-Halberstam, l'écart serait alors inférieur ou égal à 67.

Le résultat de Zhang a été publié dans les Annals of Mathematics8. Dans un premier temps il a été difficile de trouver des relecteurs acceptant d'évaluer le travail9.

La conjecture de Hardy-Littlewood[modifier | modifier le code]

Il existe aussi une généralisation de la conjecture des nombres premiers jumeaux, connue sous le nom de première10 conjecture de Hardy-Littlewood, en rapport avec la distribution des premiers jumeaux, par analogie avec le théorème des nombres premiers. Soit π2(x) le nombre de nombres premiers px tels que p + 2 soit aussi premier.

On note C2 le nombre obtenu de la façon suivante :

C_2 = prod_{pge 3} frac{p(p-2)}{(p-1)^2} approx 0,66016ldots11

(ici le produit s'étend à l'ensemble des nombres premiers p ≥ 3). C2 est appelé constante des nombres premiers jumeaux12 ou constante de Shah et Wilson13.

Alors la conjecture de Hardy-Littlewood s'énonce de la façon suivante :

pi_2(x) sim 2 C_2 int_2^x {{rm d}t over (ln t)^2}

(ce qui signifie que le quotient des deux expressions tend vers 1 quand x tend vers l'infini).

Comme le second membre a une limite infinie quand x tend vers l'infini, cette conjecture démontrerait que le nombre de nombres premiers jumeaux est bien infini.

Cette conjecture peut être justifiée (mais pas démontrée) en supposant que 1/ln(t) est la fonction de densité de la distribution des nombres premiers, une hypothèse suggérée par le théorème des nombres premiers. Cette conjecture est un cas particulier d'une conjecture plus générale appelée conjecture des n-uplets premiers de Hardy-Littlewood14utilisée dans les recherches sur la conjecture de Goldbach.

Notes et références[modifier | modifier le code]

  1. (en) P. A. Clement, « Congruences for sets of primes », American Mathematical Monthly, vol. 56,‎ , p. 23-25 (lire en ligne [archive])
  2. Viggo Brun, « La série 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + 1/19 + 1/29 + 1/31 + 1/41 + 1/43 + 1/59 + 1/61 + ... où les dénominateurs sont « nombres premiers jumeaux » est convergente ou finie », Bulletin des Sciences Mathématiques, vol. 43,‎ , p. 100-104 et 124-128
  3. a et b (en) « Twin Primes » [archive], sur Top Twenty
  4. L'union fait la force des mathématiciens [archive], Le Monde, 24/06/2013.
  5. (en) Bounded gaps between primes [archive].
  6. (en) Polymath8b: Bounded intervals with many primes, after Maynard [archive], sur le blog de Terence Tao.
  7. (en) Annonce de ce résultat [archive] sur le blog de Gil Kalai (en)
  8. Y. Zhang, « Bounded gaps between primes », Annals of Mathematics, 179 (2014), p. 1121–1174
  9. John Friedlander, « Prime Numbers: A Much Needed Gap Is Finally Found », Notices of the American Mathematical Society, vol. 62, no 6,‎ (lire en ligne [archive]).
  10. Il existe une seconde conjecture de Hardy-Littlewood
  11. suite A005597 de l'OEIS des décimales de cette constante.
  12. (en) Eric W. Weisstein, « Twin Primes Constant [archive] », MathWorld
  13. François Le Lionnais, Les nombres remarquables, Hermann, 1983, p. 30 [archive]
  14. (en) Eric W. Weisstein, « Twin Prime Conjecture [archive] », MathWorld

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A Short Biography of A.N. Kolmogorov

Source : http://homepages.cwi.nl/~paulv/KOLMOGOROV.BIOGRAPHY.html

 

A Short Biography of A.N. Kolmogorov


(``Andrei Nikolaevich Kolmogorov,'' CWI Quarterly, 1(1988), pp. 3-18.)
by Paul M.B. Vitanyi, CWI and University of Amsterdam

 



Andrei Nikolaevich Kolmogorov, born 25 April 1903 in Tambov, Russia, died 20 October 1987 in Moscow. He was perhaps the foremost contemporary Soviet mathematician and counts as one of the great mathematicians of this century. His many creative and fundamental contributions to a vast variety of mathematical fields are so wide-ranging that I cannot even attempt to treat them either completely or in any detail.
For now let me mention a non-exhaustive list of areas he enriched by his fundamental research: The theory of trigonometric series, measure theory, set theory, the theory of integration, constructive logic (intuitionism), topology, approximation theory, probability theory, the theory of random processes, information theory, mathematical statistics, dynamical systems, automata theory, theory of algorithms, mathematical linguistics, turbulence theory, celestial mechanics, differential equations, Hilbert's 13th problem, ballistics, and applications of mathematics to problems of biology, geology, and the crystallization of metals.
In over 300 research papers, textbooks and monographs, Kolmogorov covered almost every area of mathematics except number theory. In all of these areas even his short contributions did not just study an isolated question, but in contrast exposed fundamental insights and deep relations, and started whole new fields of investigations.
Apart from his penetrating work in Mathematics and the Sciences, he devoted much of his time to improving the teaching of mathematics in secondary schools in the Soviet Union, and in providing special schools for the mathematically gifted - which were very successful. Famous are also his efforts to capture in quantitative form some aspects of Russian poetry, especially that of Pushkin. It is told that ``it was fascinating to hear him lecture on this, whether one understood Russian or not.'' In 1942 Kolmogorov married Anna Dmitriyevna Egorov. He did not have children of his own.
Apart from being acknowledged without question in science, Kolmogorov was also blessed with social recognition. The USSR conferred to him seven orders of Lenin, the Order of the October Revolution, and also the high title of Hero of Socialist Labour; he gained Lenin prizes and State prizes. He occupies the first place among all Soviet mathematicians in the number of foreign academies and scientific societies that have elected him as member. These number over twenty, among them the Royal Netherlands Academy of Sciences (1963), the London Royal Society (1964), the USA National Society (1967), the Paris Academy of Sciences (1968), the Polish Academy of Sciences, the Rumanian Academy of Sciences (1956), the German Academy of Sciences Leopoldina (1959), the American Academy of Sciences and Arts in Boston (1959).
He was presented with honorary doctorates from the universities of Paris, Berlin, Warsaw, Stockholm, etc. He was elected a honorary member of the Moscow, London, Indian, and Calcutta Mathematical Societies, of the London Royal Statistical Society, the International Statistical Institute, and the American Meteorological Society. In 1963 he was awarded the International Bolzano prize.
Let me state here that I do not claim any personal relations with Kolmogorov. These remarks are based on second hand information, and primarily on sources in the Russian Mathematical Surveys, other references, and to a much lesser extent on personal communications. My credentials for writing about Kolmogorov's achievements are founded solely on my interests in that excellent notion we call``Kolmogorov complexity''. Since Kolmogorov was a man of many aspects, it is a pleasure to share some of these with the reader. This writeup was originally published as an obituary: P.M.B. Vitanyi, Andrei Nikolaevich Kolmogorov, CWI Quarterly, 1(1988), pp. 3-18. See also references in Section 1.6 of M. Li and P.M.B. Vitányi, An Introduction to Kolmogorov Complexity and its Applications, Springer-Verlag, New York, 1993 (xx + 546 pp). (This is Section 1.13 in the Second Edition of 1997.)

Early Years: 1903-1933

Kolmogorov was born on 25 April 1903 in the town of Tambov, where his mother Mariya Yakovlevna Kolmogorova had been delayed on her way from the Crimea. She died in childbed, and the responsibility to bring up the child was taken over by her sister Vera Yakovlevna Kolmogorova, ``an independent woman who held high social ideals. She passed this over to her nephew, raising him in the sense of responsibility, independence of opinion, intolerance towards idleness and poorly performed tasks, and the desire to understand and not just to memorize.''
K. treated her as his mother until her death in 1950 at Komarovka (his dacha) at the age of 87. From his mother's side K. was of aristocratic stock, his grandfather Yakov Stephanovitch Kolmogorov was a district head of the nobles in Uglich.
He spent his early years (before the revolution of 1917) at the family estate. The sources are less clear about his father. Apparently, K.'s father was the son of a clergyman, and was himself an agronomist with highly specialized training, what they called at the time ``a learned agronomist.''
K. started to work already at an early age (but presumably after the revolution); and before he became a student at Moscow University, he worked for some time as a railway conductor. He arrived at the University in autumn 1920, with already a fair knowledge of mathematics, gleaned from a book called ``New Ideas in Mathematics.'' Students at the time received grants that had little material value, but at the second course received, in addition, a ration of 16 kilos of baked bread and a kilo of fat. Hence, K. lost little time to check the minimum requirements for moving to the second course (lecture attendance being noncompulsory).
Conditions were generally harsh, and lecture rooms cold and unheated in the winter of 1920/1921. The following (unattributed) lines describe it:
``That grim year, nineteen twenty-one,
the scientific march began
Of Moscow University.
Though I was not then very old,
Though sheepskin coats enveloped me,
I still recall that beastly cold.''
For some time K. was interested in Russian history as well as mathematics. He did serious scientific research on XV-XVI century manuscripts concerning agrarian relations in ancient Novgorod. In the twenties he made a hypothesis on the way the upper Pinega was settled, and this conjecture was later confirmed by an expedition to that area.
In this early post-October revolution period the mathematical life in Moscow was dominated by ``young Luzitania'' (1920-1923) and ``post-Luzitania'' (1923-1927), a nickname for the school of real function theory headed by N.N. Luzin. This legendary personality apparently created either enthusiastic admiration or, in their struggle for independence, one-sided negation in his pupils. Among the first subjects in mathematics K. took were set theory, projective geometry, and theory of analytic functions. In 1921-1922 he obtained his first independent mathematical result (the existence of Fourier-Lebesgue series with arbitrarily slowly decreasing Fourier coefficients), and he became a pupil of N.N. Luzin. During this time he was also approached by P.S. Urysohn, who tried to interest him in topological problems. Since K. had obtained some results on the descriptive theory of functions, work that did not fit into Luzin's plans, Urysohn brought him into contact with P.S. Alexandrov whose research interests were better related to this topic. However, at about this time K. constructed a Fourier series divergent everywhere, a result that attracted international attention, and brought him for the time being in Luzin's orbit again. For this reason K.'s initial contacts with Aleksandrov stayed very limited at the time.
K. got interested in mathematical logic, and in 1925 published a paper in Mathematicheskii Sbornik on the law of the excluded middle, which has been a continuous source for later work in mathematical logic. This was the first Soviet publication on mathematical logic containing (very substantial) new results, and the first systematic research in the world on intuitionistic logic. K. anticipated to a large extentA. Heyting 's formalization of intuitionistic reasoning, and made a more definite correlation between classical and intuitionistic mathematics. K. defined an operation for `embedding' one logical theory in another. Using this - historically the first such operation, now called the `Kolmogorov operation' - to embed classical logic in intuitionistic logic, he proved that application of the law of the excluded middle in itself cannot lead to a contradiction. In 1932 K. published a second paper on intuitionistic logic, in which for the first time a semantics was proposed (for this logic), free from the philosophical aims of intuitionism. This paper made it possible to treat intuitionistic logic as constructive logic.
His interest in probability theory originated in 1924. His first steps in this area were performed jointly with A.Y. Khinchin. In 1928 he succeeded in finding necessary and sufficient conditions for the strong law of large numbers to hold, and proved the law of the iterated logarithm for sums of independent random variables, under very general conditions on the summands. In "A general theory of measure and the calculus of probabilities", 1929, he put forward a first draft of an axiom system for probability theory based on the theory of measure and the theory of functions of a real variable. Such a theory had been first suggested by E. Borel in 1909, was further developed by Lomnicki in 1923, and received its so successful final form with K.'s classic treatment of 1933. Much important work on probability theory had already been done without benefit of foundations, but this little book ``Foundations of the Calculus of Probabilities,'' published in German in 1933, immediately became the definitive formulation of the subject. This determined not only a new stage in the development of probability theory as a branch of mathematics, but also gave the necessary basis for the creation of the theory of random processes - the subject of his 1931 paper below. It was here that the basic theorems on infinite-dimensional distributions, now the logical foundations for the rigorous construction of the theory of random functions and sequences of random variables, were first formulated. The involved ideas lie at the heart of the modern theory of random processes; they form essential concepts in the very idea of control theory, and play a vital role in K.'s later synthesis of information theory and ergodic theory. K.'s many contributions in the theory of probability and statistics made him generally acknowledged as the foremost representative of this discipline.
In 1931 K.'s paper ``Analytical methods in probability theory'' appeared, in which he laid the foundations for the modern theory of Markov processes. According to Gnedenko: "In the history of probability theory it is difficult to find other works that changed the established points of view and basic trends in research work in such a decisive way. In fact, this work could be considered as the beginning of a new stage in the development of the whole theory".
The theory had a few forerunners: A.A. Markov, Poincaré and Bashelier, Fokker, Planck, Smolukhovski and Chapman. Their particular equations for individual problems in physics, informally obtained, followed as special cases in K.'s theory. A long series of subsequent publications followed, by K. and his followers, among which a paper by K. dealing with one of the basic problems of mathematical statistics, where he introduces his famous criterion (Kolmogorov's test) for using the empirical distribution function of observed random variables to test the validity of an hypothesis about their true distribution. In general K.'s ideas on probability and statistics have led to numerous theoretic developments, and to numerous applications in present-day physical sciences.
After graduation in 1925, K. stretched his stay at the University for four more years as a research student, but finally in 1928-1929 stricter control on the number of years a student had for research was enforced. An unprecedented number of 70 students finished in 1929, including K. This raised the problem of where to continue his research. Aleksandrov was instrumental in securing for K. the single available vacancy in 1929 at the Institute of Mathematics and Mechanics of Moscow University, against heavy competition.

Youth: 1929-1940

From 1930-1940 K. published more than sixty papers on probability theory, projective geometry, mathematical statistics, the theory of functions of a real variable, topology, mathematical logic, mathematical biology, philosophy and the history of mathematics. In 1931 K. was made professor at Moscow University, and from 1937 held the chair of theory of probability. From this time dates the life long friendship between K. and Aleksandrov. Says Aleksandrov: ``in 1979 this friendship [with K.] celebrated its fiftieth anniversary and over the whole of this half century there was not only never any breach in it, there was also never any quarrel, in all this time there was never any misunderstanding between us on any question, no matter how important for our lives and our philosophy; even when our opinions on one of these questions differed, we showed complete understanding and sympathy for the views of each other.''
Says K.: ``for me these 53 years of close and indissoluble friendship were the reason why all my life was on the whole full of happiness, and the basis of that happiness was the unceasing thoughtfulness on the part of Aleksandrov.''
K. describes how this friendship started in 1929 during a sailing trip on the Volga. At that time, the ``Society for Proletarian Tourism and Excursions'' offered active vacations: one obtained a boat and camping equipment at one city on the Volga which could be handed in at other cities downstream. K., already experienced in boating, decided to organize such a trip, and asked (besides two others) Aleksandrov to join. The young men bought the then popular ``Jungsturm'' suits for all of the crew. By way of books they took along only a steamboat timetable and a copy of the Odyssey (and also manuscripts to work on and a folding writing desk). They started out at June 16, and covered 1300 kilometers before handing in the boat at Samar downstream. K. and Aleksandrov then proceeded together to the Caucasus by steamer. After some more wandering, they set up residence in an unused cell of a monastery on a small peninsula in Lake Savan. Whiling their time away at secluded bays, in between swimming and sun bathing they also managed to get some work done: K. in the shadows on integration theory and analytic description of Markov processes in continuous time, and Aleksandrov dressed only in dark glasses and white panama hat in the burning sun on his Topology book with Hopf. They stayed in these idyllic surroundings for about three weeks, then set off partly on foot, partly by other means of transport, and eventually climbed the Alagez mountain (4100m). They wound up at Tiflis, from where Aleksandrov proceeded alone to a prearranged appointment with a group of mathematicians. K. continued hiking and mountain climbing. (By this time it was August.) Later, they joined up again at Gagra, on the Black Sea, and spent some more time there, sunbathing and swimming and doing mathematics. At about this time they decided to share a house together.
After returning to Moscow they forthwith rented the first in a series of houses in the nearby vacation village of Klyaz'm, and moved in together with K.'s aunt Vera Yakovlevna. A short time later, Masha Barbanova, who had been K.'s nanny at the family estate near Jaroslavl' before the revolution, joined as housekeeper. In 1935 they acquired (initially part of) an old manor house at Komarovka, with room for a large library and several guests. This `house at Komarovka' became a meeting place for mathematicians. One of them said ``It is just like Oberwolfach (a mathematical institute in the Black Forest), except that here Kolmogorov buys all the drinks.''
It is perhaps instructive to see a glimpse of the mathematicians' country life:
``As a rule,'' says K., ``of the seven days a week, four were spent in Komarovka, one of which was devoted entirely to physical recreation - skiing, rowing, long excursions on foot (these long walks covered on average about 30 kilometers, rising to 50; on sunny March days we went out on skis wearing nothing but shorts, for as much as four hours on a stretch. On the other days, morning exercise was compulsory, supplemented in the winter by a 10 kilometer ski run ... Especially did we love swimming in the river just as it began to melt ... I swam only short distances in icy water but Aleksandrov swam much further. It was I however who skied naked for considerably longer distances.'' As P. Halmos, visiting K. in Moscow in 1965 tells it:
``Kolmogorov [had] five rooms [apartment in the University]. ... stacks of reprints in one corner, a collection of theatrical masks somewhere, and a couple of skis somewhere else. "Is this where you work? I asked. "No, no", he said: "I work out at the dacha; I am here only three days a week."'' (At the celebration meeting of K's seventieth birthday, a skiing trip was organized where K. clad only in shorts outskied every other participant.)
In 1930-1931 K. and Aleksandrov were mainly abroad. The year 1930 they spent both at Gottingen. Here K. had contacts with R. Courant on limit theorems, with H. Weyl on intuitionistic logic, and with E. Landau on function theory. K. relates the story that he solved a problem Landau much liked to be solved, and wrote it up in detail. Landau being very pleased told everybody about the success and invited a paper on the subject, but, to his embarrassment, K. discovered a few weeks later exactly the same result with the same proof by Besicovitch in Fundamenta Mathematicae. The summer both K. and Aleksandrov visited Caratheodory at Munich (measure theory), and were invited to stay with Frechet on the Mediterranean (to work on probability theory in K.'s case). The journey there involved hiking through Bavaria, staying with Frechet for about a month, visiting P.S. Urysohn's grave in Normandy, and continuing on to Paris. Aleksandrov left Paris by the end of September for Goettingen, and K. stayed on until December, and had some meetings with Borel and P. Lévy, especially the last. While K. returned to Goettingen, Aleksandrov spent the spring 1931 semester in the U.S.A.
As another highlight of this period the article "Mathematics" for the second edition of the Great Soviet Encyclopaedia is often mentioned. Another area he turned to at the time was topology. Simultaneously with the U.S. topologist J.W. Alexander and independently of him, K. discovered the notion of cohomology and founded the theory of cohomological operations. The work of K. and his school on the deep connections between topology, the theory of ordinary differential equations, celestial mechanics and the theory of dynamical systems, determined to a considerable extent its present state.
At the end of the thirties, K.'s attention was drawn to the mechanics of turbulence. In the hands of K. and his school the theory of turbulence obtained an accurate mathematical form as an applied chapter in the theory of measure of function spaces. With great physical intuition, in two short papers in 1941, K. posited in concise mathematical form ideas about the structure of the small-scale components of turbulent motion of fluids and gasses, latent in earlier experimental work, particularly by G.I. Taylor. These hypotheses imply many qualitative results that are widely applicable - what goes on, for instance, within the turbulence that occurs in the wake of a jet aircraft. Some of the quantitative relations arising have the character of new laws of nature - like K.'s law of "2/3": in each developed turbulent flow the mean square difference of the velocities at two points is proportional to the 2/3rd power of their distance (if the distance is not too small or not too large). K. made also quantitative predictions on the basis of his theories, that were later confirmed by experiments, e.g., the stratified structure of the ocean, an effect known as "pancakes". K.'s 1941 contributions to the theory of turbulence are perhaps the most important ones in the long and unfinished history of the theory of turbulence.

Middle Years: 1940-1960

He was interested in every branch of science, he and his pupils wrote about crystal growth, about geometry of the interaction of plants, and also made significant contributions to ``birth and death'' processes and to genetics. One of these papers brought him to a head-on confrontation with Lysenko. In a courageous stand in emphasizing scientific truth, in a paper published in 1940 in the "Genetics" section of Dokl. Akad. Nauk SSSR , K. showed that the material gathered by followers of Stalin's proteg e Academician Lysenko, contrary to opinion, supported Mendel's laws. Another joint work (with Piscounov andPetrovsky) treated the rate of advance of an advantageous gene in a linear environment, (a topic studied independently by R.A. Fisher, for whom K. had high regard). This was later adapted to describe spreading of epidemics of innovations, and rumours.
The theory of smoothing and prediction of stationary time-series is usually associated with the name of Norbert Wiener but in fact it was developed simultaneously by Wiener and K. during the second world war.
In the post-war period K. turned again to turbulence, and made small improvements on laws he discovered before, that were experimentally verified as well. Topics in the vast range of classical mechanics, ergodic theory, function theory, information theory and the theory of algorithms belong to this period. He managed to find links between totally unconnected fields, and published a small number of papers, but quite fundamental ones, on each topic. In his work on dynamical systems one can distinguish two periods. In 1953-1954 he made a seminal contribution to the fundamental problem of classical mechanics, identified fifty years earlier by H. Poincare in his study of the motion of planets around the sun. Neglecting all but one planet one deals with an ``integrable'' problem that is well understood. However, the small effects associated with gravitational interaction between the planets introduces a profound qualitative change related to the fact that the equations are now ``nonintegrable.'' In attacking this problem, K.'s great achievement was to develop a general theory of Hamiltonian systems under small perturbations, which has several practical applications, among others in the study of magnetic fields and plasma physics. This work also spawned, together with improvements of K.'s pupil Arnol'd and by Moser, what is now known as the study of ``KAM-tori.'' Subsequent computational studies aptly confirm K.'s insights and have opened up the enormously fruitful field of ``chaos in dynamical systems,'' which is currently attracting much attention. These studies lead, for example, to better weather forecasting.
At this time he also started to work on the theory of automata and the theory of algorithms. Together with his pupil Uspenskii he formulated the important notion of Kolmogorov-Uspenskii machine. He supported the up and coming field of cybernetics (theory of computation) against heavy initial antagonism (in the USSR). Many USSR computer scientists are K.'s pupils or pupils of K.'s pupils.
The second period from 1955-1959 consisted in applications from information theory to the ergodic theory of dynamical systems. He introduced the fruitful idea of informational (entropic) characteristics in the study of metric spaces and of dynamical systems. Together with Arnol'd, K. settled in 1956-1957 Hilbert's 13th problem, disproving the conjectured outcome, by showing that a continuous function in any number of variables can be represented as a composition of continuous functions of a single variable and addition. The ideas of introducing entropic characteristics in the theory of dynamical systems opened up a large new area. Another important concept, that of a quasi-regular system (now called K-system), plays a very important role in the analysis of classical dynamic systems with strong stochastic properties, such as in physics, biology and chemistry. In the years 1958-1959 K. applied ergodic theory to phenomena of the type of turbulence, which had a great influence on subsequent work.

Later Years: 1960-1987

While in previous years K. used concepts of information theory in mathematical sciences, now it was the turn of information theory to be reconstructed using the theory of algorithms, incidentally closing the circle of his research by giving logico-algorithmic foundations to the theory of probability. Algorithmic information theory, or " Kolmogorov complexity theory", originated with the discovery of universal descriptions of finite objects, and a recursively invariant approach to the concepts of complexity of description, randomness and a priori probability. Historically, it is firmly rooted in R. von Mises' notion of random infinite sequences ( Kollektivs ), proposed from 1919 onwards as foundation for the theory of probability in the spirit of a physical theory (according to the program outlined in D. Hilbert's 6th problem), using the frequency interpretation of probability. In 1940 A. Church proposed an algorithmic version of von Mises random sequences, but the results were not yet satisfactory.
In his 1933 booklet K. had in some sense executed Hilbert's suggestion in his 6th problem: "To treat (in the same manner as geometry) by means of axioms, those physical sciences in which mathematics plays an important part; in the first rank are the theory of probability ..", in 1963 K. observes: "This theory [K's 1933 set theoretic axiomatic approach] was so successful, that the problem of finding the basis of real applications of the results of the mathematical theory of probability became rather secondary to many investigators. ..[However] the basis for the applicability of the results of the mathematical theory of probability to real 'random' phenomena must depend in some form on the frequency concept of probability , the unavoidable nature of which has been established by von Mises in a spirited manner."
However, von Mises based his approach on axiomatically postulated infinite random sequences, representing repetitious independent trials with a limiting frequency. To this K. objects: "The frequency concept based on the notion of limiting frequency as the number of trials increases to infinity, does not contribute anything to substantiate the application of the results of probability theory to real practical problems where we always have to deal with a finite number of trials."
Following a four decades long controversy on von Mises' intended notion of an infinite random sequence, in a 1965 paper K. used the theory of algorithms to describe the complexity of a finite object as the length of the smallest description (algorithm to reconstruct it). This would seem to make the definition depend on the algorithmic method used. However, it turns out that there are optimal and universal methods for which the complexities of the objects described are asymptotically optimal. Although there are many optimal methods, the corresponding complexities differ by no more than an additive constant. It is natural to call a finite object random if it has no description of complexity less than it has itself. It is seductive to define an random infinite sequence as one of which the growth of complexity if the initial segments with the length is sufficiently fast, thus relating to von Mises' earlier approach. Due to unavoidable oscillations of the complexity of prefixes as function of their length this did not work out. However, P. Martin-Loef, a Swedish mathematician visiting K. in Moscow in 1964-1965, was able to show that under appropriate axiomatic definitions of randomness, one can prove once and for all that the thus defined sequences satisfy all effective tests for randomness, and have measure one in the set of all such infinite sequences. This rigorously defined an appropriate class, intuitively satisfactory as well, to qualify as von Mises' Kollektivs. Later it was shown by L.A. Levin, P. Gacs and G.J. Chaitin that one can refine the notion of complexity by defining it relative to a set of admissible descriptions. If admissible descriptions are restricted such that no description is a proper prefix of any other description, then an infinite sequence is Martin-L of random if and only if each of its finite initial sequences has a complexity that equals (up to a fixed constant) its length.
With the advent of electronic computers in the 1950's, a new emphasis on computer algorithms, and a maturing general recursive function theory, ideas tantamount to Kolmogorov complexity came to many people's minds, because ``when the time is ripe for certain things, these things appear in different places in the manner of violets coming to light in early spring,'' in the phrase of Wolfgang Bolyai in another famous context. Thus, R. Solomonoff in Cambridge, Massachusetts, had formulated the same ideas already in 1960. and had published his truly innovative work on the subject already in 1964 in `Information and Control'.
According to Solomonoff his work got far more attention after K. started to refer to it from 1968 onward, even though the attribution ``Kolmogorov'' complexity seems to have stuck. Says K.: ``I came to similar conclusions before becoming aware of Solomonoff's work, in 1963-1964.''
Yet a third independent inventor entered slightly later, Gregory Chaitin who was an 18 year old undergraduate in New York when he submitted a very similar set of inventions for publication end 1965 for publication in `J. Assoc. Comp. Mach.' (published in 1966 and 1969, the last paper containing the definition of Kolmogorov complexity and results thereof, while the 1966 paper extends C.E. Shannon's non-invariant notion of state-symbol measure for the complexity of Turing machines). Says Chaitin: ``this definition [of Kolmogorov complexity] was independently proposed about 1965 by A.N. Kolmogorov and me ... Both Kolmogorov and I were then unaware of related proposals made in 1960 by Ray Solomonoff.''
One of the last papers of K. was on the topic of algorithmic information theory - a paper together with Uspenskii published in 1987. For a comprehensive introduction and a survey of the astonishing range of applications of Kolmogorov complexity, see M. Li and P.M.B. Vitányi, An Introduction to Kolmogorov Complexity and its Applications, Springer-Verlag, New York, 1993 (xx + 546 pp).

As a Teacher

K.'s pedagogical activities began in 1922, when he became teacher at the experimental model school of the People's Commissariat for Education. He taught there until 1925. From 1925 till 1929 he was instructor at the University. Passing on knowledge and scientific ideas was very important for K. His interests in this subject ranged over the full scale from earliest education to higher education, and occupied much of his time. He actively took part in organizing mathematical Olympiads in schools and gave talks to school children. Thus he wrote a booklet on the topic ``Mathematics as a Profession'', which circulated in tens of thousands of copies. He put special emphasis on selection of mathematically gifted adolescents, since even the nonmathematicians will need such training in their later career
According to K., by 14-15 years about half of the pupils have come to the conclusion that mathematics and physics will be of little use to them. In recognition of that fact a special simplified program should be followed by such pupils. ``The mechanically understood principles of uniformity of schools providing general education, which excludes schools with a more detailed study of individual subjects, has outlived itself. As applied to mathematics it has already been destroyed by the creation of schools giving special training to computer operators and computer programmers.'' And: ``At 14-16 everything changes. At this age interest in mathematics usually becomes apparent, which quickly and painlessly leads the student to concentrated work and then to the real research work of the young scientist (at 18-20 years). ... For the beginners, the young people entering science for the first time, it is important to be convinced as soon as possible that they are capable of doing something original, their very own. When offering a subject for research to a graduate or a research student, the supervisor must not think only about the objective importance, or urgency of the subject, but also whether the work on the subject will stimulate the development of the young scientist, and whether it is within his powers to carry out, and at the same time demand the maximal effort of which he is capable.'' The ability to offer the students exactly what is most important and ripe in the development of science, and avoid pursuing dead-ends, and what is at the same time in their powers to accomplish is very characteristic for K.
The number of Kolmogorov's research students who have obtained their Ph.D. exceeds sixty. He was instrumental in substantial transformation (in the Soviet Union) of the very character of university education in mathematics, in particular the organization of practical work in mathematics, and updating the contents of mathematics. He also engaged in the search for new contents of mathematics in secondary schools, the founding of mathematical boarding schools, gave cycles of lectures for teachers on the structure of modern mathematics, and so on. Finally, he created an author's collective, and took part himself in writing textbooks on geometry, algebra and analysis for 6th through 10th grades. At the mathematical boarding school No. 18 at the University of Moscow, otherwise known as the ``Kolmogorov school'', he gave for years lessons up to 26 hours a week, and wrote accompanying syllabi. He also gave lectures to the students on music, art and literature. He felt that intellectual development must be evenly balanced. The former pupils of this school are very successful and systematically take the first places in All-Union and International Mathematical Olympiads.
In 1964 K. became head of the mathematical section of a joint syllabus committee of the USSR Academy of Sciences and that of Pedagogical Sciences. K. also organized a Statistical Laboratory at the University of Moscow, and succeeded in upgrading the budding library by obtaining large funds, and also international literature through partial use of money he received as part of the international Bolzano prize. In 1972 on K.'s initiative a compulsory course in mathematical logic was introduced for the first time in the Department of Mechanics and Mathematics at Moscow State University. He wrote the syllabus (which was still followed in 1983) and was the first to teach it.
According to V.I. Arnol'd, ``K. never explained anything, just posed problems, and didn't chew them over. He gave the student complete independence and never forced one to do anything, always waiting to hear from the student something remarkable. He stood out from the other professors I met by his complete respect for the personality of the student. I remember only one case where he interfered with my work: in 1959 he asked me to omit from the paper on self-maps of the circle the section on applications to heartbeats, adding "That is not one of the classical problems one ought to work on". The application to the theory of heartbeats was published by L. Glass 25 years later, while I had to concentrate my efforts on the celestial-mechanical applications of the same theory.''
L.S. Pontryagin relates: ``Kolmogorov gave me an interesting task..: to study [some problems in] locally compact algebraic fields in which multiplication is not necessarily commutative... A week later I reported to Aleksandrov that I had solved it in the case of commutative fields. Directly afterwards the three of us, Aleksandrov, Kolmogorov and I, met in Aleksandrov's flat. With a shade of ironical doubt, Kolmogorov said: "Well now, Lev Semenovich, I hear you have already solved my problem, let's hear you." Kolmogorov declared my very first statement to be false, but I immediately refuted him. Then he said: "Yes, it seems that the problem turned out to be much easier than I supposed." None of the rest of my answer aroused doubt. For the case of the noncommutative field the problem was immeasurably more difficult. It took me a whole year to work it out.'' It is also said that K. was one of the very few non-political mathematicians in the Soviet Union with yet real power. He quietly helped talented people with otherwise unfashionable views.
K.'s pupils included in the early years: Millionshchikov (later Vice-President of the USSR Academy of Sciences), Mal'tsev, Nikol'skii, Gnedenko, Gel'fand, Bavli and Verchenko. The subjects ranged from theoretical geophysics, mathematical logic, functional analysis, probability theory, function theory. During and after the war: Shilov, Fage, Sevast'yanov, Sirazhdinov, Pinsker, Prikhorov, Barenblatt, Bol'shev, Dobrushin, Medvedev, Mikhalevich, Uspenskii, Borovkov, Zolotarev, Alekseev, Belyaev, Mehhalkin, Epokhin, Rozanov, Sinai, Tikhomirov, Shiryaev, Arnol'd, Bassalygo, and Ofman. Later also Prokhorov, L.A. Levin, Kozlov, Zhurbenko, Abramov, and Bulinskii. His pupils include a number of well-known foreign mathematicians, among who the Swede P. Martin-L of. Pupils who became member of the USSR Academy of Sciences: A.I. Mal'tsev (algebra, mathematical logic), S.M. Nikol'skii (function theory), A.M. Obukhov (physics of the atmosphere), I.M. Gel'fand (functional analysis), Yu.V. Prokhorov (probability theory); and corresponding member: L.N. Bol'shev (mathematical statistics), A.A. Borovokov (probability theory, mathematical statistics), A.S. Monin (oceanology), and V.I. Arnol'd. The Ukrainian Academy of Sciences: B.V. Gnedenko (probability theory, history of mathematics), V.M. Mikhalevich (cybernetics), etc.

Scientific Career

K. entered Moscow University in 1920, graduated in 1925, and got his (equivalent of) Ph.D. in 1929, when he also got a position on the faculty. In 1931 K. became professor at Moscow University, and from 1933-1939 he also became Director of the Scientific Research Institute of Mathematics at the Moscow State University. Apparently, he was involved with the scientific research of all graduate students at the institute, not only his own. Most of them mention the unforgettable hikes on Sundays when K. invited all his own students (graduates and undergraduates) as well as students from other supervisors. These 40 km walks in the environment of Bolshevo, Klyaz'm, later Komarovka, are remembered as intellectually stimulating and culturally wide ranging experiences, ending when he and Aleksandrov treated the whole company to dinner in their dacha. In 1939 K. was elected as an Academician of the All-Union Academy of Sciences and as Academician-Secretary of the Physics-Mathematical Section. He also did enormous work as head of the mathematics editorial board of the Publishing House of Foreign Literature and as editor of the mathematics section of the Great Soviet Encyclopaedia. During the second world war K. engaged in the war effort by solving problems in ballistics and began research on problems of quality control of mass industrial production. From 1964 to 1966, and from 1976 till at least 1983 K. has been President of the Moscow Mathematical Society; from 1946 to 1954 and from 1983 on Editor-in-chief of Uspekhi Math. Nauk (Russian Mathematical Surveys). At the University of Moscow, K. held from 1938 to 1966 the chair of probability theory. From 1966 till 1976 he was the head of the Interdepartmental Laboratory of Statistical Methods, and from 1976 to 1980 he held the chair of mathematical statistics, which he organized. From 1980 on K. held the chair of mathematical logic. From 1951 to 1953 he was Director of the Institute of Mathematics and Mechanics of the Moscow State University; from 1954 to 1956 and from 1978 to at least 1983 the head of the mathematics section of the Faculty of Mechanics and Mathematics. From 1954 to 1958 he was Dean of the Faculty of Mechanics and Mathematics of the University.



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Mathematics Genealogy Project

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2003 Steele Prizes

Voir le pdf : http://www.ams.org/notices/200304/comm-steele.pdf

 

2003 Steele Prizes 462 NOTICES OF THE AMS VOLUME 50, NUMBER 4 The 2003 Leroy P. Steele Prizes were awarded at the 109th Annual Meeting of the AMS in Baltimore in January 2003. The Steele Prizes were established in 1970 in honor of George David Birkhoff, William Fogg Osgood, and William Caspar Graustein. Osgood was president of the AMS during 1905–06, and Birkhoff served in that capacity during 1925–26. The prizes are endowed under the terms of a bequest from Leroy P. Steele. Up to three prizes are awarded each year in the following categories: (1) Mathematical Exposition: for a book or substantial survey or expository-research paper; (2) Seminal Contribution to Research (limited for 2003 to the field of logic): for a paper, whether recent or not, that has proved to be of fundamental or lasting importance in its field or a model of important research; and (3) Lifetime Achievement: for the cumulative influence of the total mathematical work of the recipient, high level of research over a period of time, particular influence on the development of a field, and influence on mathematics through Ph.D. students. Each Steele Prize carries a cash award of $5,000. The Steele Prizes are awarded by the AMS Council acting on the recommendation of a selection committee. For the 2003 prizes, the members of the selection committee were: M. S. Baouendi, Andreas R. Blass, Sun-Yung Alice Chang, Michael G. Crandall, Constantine M. Dafermos, Daniel J. Kleitman, Barry Simon, Lou P. van den Dries, and Herbert S. Wilf (chair). The list of previous recipients of the Steele Prize may be found in the November 2001 issue of the Notices, pages 1216–20, or on the World Wide Web, http://www.ams.org/prizes-awards. The 2003 Steele Prizes were awarded to JOHN B. GARNETT for Mathematical Exposition, to RONALD JENSEN and to MICHAEL D. MORLEY for a Seminal Contribution to Research, and to RONALD GRAHAM and to VICTOR GUILLEMIN for Lifetime Achievement. The text that follows presents, for each awardee, the selection committee’s citation, a brief biographical sketch, and the awardee’s response upon receiving the prize. Mathematical Exposition: John B. Garnett Citation An important development in harmonic analysis was the discovery, by C. Fefferman and E. Stein, in the early seventies, that the space of functions of bounded mean oscillation (BMO) can be realized as the limit of the Hardy spaces Hp as p tends to infinity. A crucial link in their proof is the use of “Carleson measure”—a quadratic norm condition introduced by Carleson in his famous proof of the “Corona” problem in complex analysis. In his book Bounded Analytic Functions (Pure and Applied Mathematics, 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981, xvi + 467 pp.), Garnett brings together these far-reaching ideas by adopting the techniques of singular integrals of the Calderón-Zygmund school and combining them with techniques in complex analysis. The book, which covers a wide range of beautiful topics in analysis, is extremely well organized and well written, with elegant, detailed proofs. The book has educated a whole generation of mathematicians with backgrounds in complex analysis and function algebras. It has had a great impact on the early careers of many leading analysts and has been widely adopted as a textbook for graduate courses and learning seminars in both the U.S. and abroad. Biographical Sketch John B. Garnett was born in Seattle in 1940. He received a B.A. degree from the University of Notre Dame in 1962 and a Ph.D. degree in mathematics APRIL 2003 NOTICES OF THE AMS 463 from the University of Washington in 1966. His thesis advisor at Washington was Irving Glicksberg. In 1968, following a two-year appointment as C.L.E. Moore Instructor at the Massachusetts Institute of Technology, Garnett became assistant professor at the University of California, Los Angeles, where he has worked ever since. At UCLA, Garnett was promoted to tenure in 1970 and to professor in 1974. In 1989 he received the UCLA Distinguished Teaching Award primarily for his work with Ph.D. students, and from 1995 to 1997 he served as department chairman. Garnett’s research focuses on complex analysis and harmonic analysis. He has held visiting positions at Institut Mittag-Leffler; Université de Paris-Sud; Eidgenössische Technische Hochschule, Zurich; Yale University; Institut des Hautes Études Scientifiques; and Centre de Recerca Matemática, Barcelona. He gave invited lectures to the AMS in 1979 and to the International Congress of Mathematicians in 1986. Response I am honored to receive the Steele Prize for the book Bounded Analytic Functions. It is especially satisfying because the prize had previously been awarded for some of the classic books in analysis by L. Ahlfors, Y. Katznelson, W. Rudin, and E. M. Stein, from which I first learned much mathematics and to which I still return frequently. I wrote Bounded Analytic Functions around 1980 to explain an intricate subject that was rapidly growing in surprising ways, to teach students techniques in their simplest cases, and to argue that the subject, which had become an offshoot of abstract mathematics, was better understood using the concrete methods of harmonic analysis and geometric function theory. I want to thank several mathematicians: L. Carleson, C. Fefferman, K. Hoffman, and D. Sarason, whose ideas prompted the development of the subject; and S.-Y. A. Chang, P. Jones, D. Marshall, and the late T. Wolff, whose exciting new results at the time were some of the book’s highlights. Encouragement is critical to the younger mathematician, and from that time I owe much to my mentors I. Glicksberg, K. Hoffman, and L. Carleson, and to my contemporaries T. W. Gamelin, P. Koosis, and N. Varopoulos. I also want to thank the young mathematicians who over the years have told me that they learned from the book. Seminal Contribution to Research: Ronald Jensen Citation Ronald Jensen’s paper “The fine structure of the constructible hierarchy” (Annals of Mathematical Logic 4 (1972) 229–308) has been of seminal importance for two different directions of research in contemporary set theory: the inner model program and the use of combinatorial principles of the sort that Jensen established for the constructible universe. The inner model program, one of the most active parts of set theory nowadays, has as its goals the understanding of very large cardinals and their use to measure the consistency strength of assertions about John B. Garnett Ronald Jensen Michael D. Morley Ronald Graham Victor Guillemin 464 NOTICES OF THE AMS VOLUME 50, NUMBER 4 much smaller sets. A central ingredient of this program is to build, for a given large cardinal axiom, a model of set theory that either is just barely large enough to contain that type of cardinal or is just barely too small to contain it. The fine structure techniques introduced in Jensen’s paper are the foundation of the more recent work of Mitchell, Steel, Jensen himself, and others constructing such models. The paradigm, initiated by Jensen, for relating large cardinals to combinatorial properties of smaller sets is first to show that the desired properties hold in these inner models and then to show that, if they failed to hold in the universe of all sets, then that universe and the inner model would differ so strongly that a large cardinal that is barely missing from the inner model would be present in the universe. The paper cited here contains the first steps in this direction, establishing for the first time combinatorial properties of an inner model, in this case Gödel’s constructible sets, that go far beyond Gödel’s proof of the generalized continuum hypothesis in this model. The second direction initiated by Jensen’s paper involves applying these combinatorial principles to problems arising in other parts of mathematics. The principle , which Jensen proved to hold in the constructible universe, has been particularly useful in such applications. A good example is Shelah’s solution of the Whitehead problem in abelian group theory; half of the solution was to show that a positive answer to the problem follows from . By now, has become part of the standard tool kit of several branches of mathematics, ranging from general topology to module theory. Biographical Sketch Ronald Jensen received his Ph.D. in 1964 from the University of Bonn. He continued his research at Bonn as a scientific assistant (1964–69). From 1969 until 1973 Jensen was a professor of mathematics at the University of Oslo. During this period he held concurrent positions at Rockefeller University (1969–71) and the University of California, Berkeley (1971–73). At the University of Bonn he was awarded the Humboldt Prize (1974–75) and served as a professor of mathematics (1976–78). He was a visiting fellow at Oxford University’s Wolfson College (1978–79), a professor of mathematics at the University of Freiburg (1979–81), and a senior research fellow at Oxford University’s All Souls College (1981–94). He moved to Humboldt University of Berlin, where he was a professor of mathematics (1994–2001). His areas of research interest include set theory. Response I feel deeply honored that on the basis of my paper “The fine structure of the constructible hierarchy”, I was chosen to share the Steele Prize for seminal research with Michael Morley. I came to set theory in the wake of Cohen’s discovery of the forcing method, together with a group of other young mathematicians such as Bob Solovay, Tony Martin, and Jack Silver, all of whom influenced my work. It was an exciting time. Much of the work centered on independence proofs using Cohen’s method, but the research on the consequences of strong existence axioms, such as large cardinals and determinacy, was also beginning. The theory of inner models—in particular Gödel’s model L—was comparatively underdeveloped. After discovering that the axiom V = L settles Souslin’s problem, I began developing a body of methods, now known as “fine structure theory”, for investigating the structure L. Much of this work was done in 1969–71 at Rockefeller University and the University of Oslo. The above-mentioned paper was subsequently written at Berkeley. In the ensuing years it became apparent that these methods were also applicable to larger inner models in which strong existence axioms are realized. The most important breakthrough in this direction was made by John Steel. He and Hugh Woodin have applied the methods widely. This work is being extended by a very capable group of younger mathematicians, such as Itay Neeman, Ernest Schimmerling, and Martin Zeman. I feel privileged to have worked in such gifted company. Seminal Contribution to Research: Michael D. Morley Citation Michael Morley’s paper “Categoricity in power” (Transactions of the AMS 114 (1965) 514–538) set in motion an extensive development of pure model theory by proving the first deep theorem in this subject and introducing in the process completely new tools to analyze theories (sets of first-order axioms) and their models. When does a theory have (up to isomorphism) a unique model? An early result in mathematical logic is that, for basic cardinality reasons, a theory never has a unique infinite model. The next question is: when does a theory have exactly one model of some specified infinite cardinality? An important example is the theory of algebraically closed fields of any given characteristic, which has a unique model in every uncountable cardinality. Answering a question of L os´, Morley proved that a countable theory which is categorical (has a unique model) in one uncountable cardinality is categorical in every uncountable cardinality. Morley used most of the then-existing model theory, but what makes his paper seminal are its new techniques, which involve a systematic study of Stone spaces of Boolean algebras of definable sets, called type spaces. For the theories under consideration, these type spaces admit a CantorBendixson analysis, yielding the key notions of Morley rank and ω-stability. This property of ω-stability of a theory was the first of many to APRIL 2003 NOTICES OF THE AMS 465 follow that are of an intrinsic nature, that is, invariant under biinterpretability. Morley’s work set the stage for studying the difficult problem of the possible isomorphism types of models of a given theory. This was pursued with great success by Shelah, who vastly generalized Morley’s methods. Also, the recognition grew that categoricity properties and notions like Morley rank and ω-stability are intimately tied to underlying combinatorial geometries (Baldwin-Lachlan, Zil ber). In combination with the fact that an infinite field with uncountably categorical theory has to be algebraically closed (Macintyre), this led to the geometric orientation of current model theory. In the last ten years, the development started by Morley enabled remarkable applications by Hrushovski and others to questions of diophantine character, with impact on areas such as differential and difference algebra. Biographical Sketch Michael Morley was born in Youngstown, Ohio, in 1930. In 1951 he received a B.S. degree in mathematics from Case Institute of Technology and began graduate work at the University of Chicago. There was a five-and-one-half year hiatus (1955–61) in his graduate education, during which he worked as a mathematician at the Laboratories for Applied Sciences of the University of Chicago. After returning to graduate school, he received his Ph.D. from the University of Chicago in 1962, though the last year of his graduate work was done at the University of California, Berkeley. He was an instructor for one year at Berkeley, an assistant professor for three years at the University of Wisconsin, and joined the Cornell faculty in 1966. He was associate chairman and director of undergraduate studies for the mathematics department at Cornell from 1984–95. He achieved emeritus status at the end of 2002. He served as president of the Association for Symbolic Logic in 1986–89. Response I am grateful for this award. By definition, a paper is judged seminal because of work that follows it. Therefore, I am aware that I am being honored in large part for the work of other people. This paper was written just over forty years ago. At that time most mathematicians considered mathematical logic as philosophically very interesting but mathematically not very deep. (After all, some of the work was done by professors of philosophy.) There was some justification for this attitude. However, in the early 1960s several papers appeared that obtained spectacular results by applying nontrivial mathematics to logic. This attracted many of the best young mathematicians to mathematical logic. Today there is a large body of mathematically deep and lovely work in logic. One worries that we may have lost some of the philosophical significance. The paper was my doctoral dissertation written under the supervision of Professor Robert Vaught. Bob Vaught died last spring. I must express the gratitude that I, and indeed many of his students, felt towards Robert Vaught, not just for his mathematical direction, but for his great personal kindness and generosity of spirit. He was a fine mathematician and a truly good man. Lifetime Achievement: Ronald Graham Citation Ron Graham has been one of the principal architects of the rapid development worldwide of discrete mathematics in recent years. He has made many important research contributions to this subject, including the development, with Fan Chung, of the theory of quasirandom combinatorial and graphical families, Ramsey theory, the theory of packing and covering, etc., as well as to the theory of numbers, and seminal contributions to approximation algorithms and computational geometry (the “Graham scan”). Furthermore, his talks and his writings have done much to shape the positive public image of mathematical research in the USA, as well as to inspire young people to enter the subject. He was chief scientist at Bell Labs for many years and built it into a world-class center for research in discrete mathematics and theoretical computer science. He served as president of the AMS in 1993–94. Biographical Sketch Ronald Graham’s undergraduate training included three years at the University of Chicago (in Robert Maynard Hutchins’ Great Books program); a year at Berkeley as an electrical engineering major; and four years in the U.S. Air Force, three of which were spent in Fairbanks, Alaska, where he concurrently received a B.S. in physics in 1959. He subsequently was awarded a Ph.D. in mathematics from the University of California, Berkeley, in 1962. He spent the next thirty-seven years at Bell Labs as a researcher, leaving from what is now AT&T Labs in 1999 as chief scientist. During that time he also held visiting positions at Princeton University, Stanford University, the California Institute of Technology, and the University of California, Los Angeles, and was a (part-time) University Professor at Rutgers for ten years. He currently holds the Irwin and Joan Jacobs Chair of Computer and Information Science at the University of California at San Diego. Graham has received the Pólya Prize in Combinatorics from the Society for Industrial and Applied Mathematics, the Euler Medal from the Institute of Combinatorics and Its Applications, the Lester R. Ford Award from the Mathematical Association of America (MAA), and the Carl Allendoerfer Award 466 NOTICES OF THE AMS VOLUME 50, NUMBER 4 from the MAA. He is currently treasurer of the National Academy of Sciences, a foreign member of the Hungarian Academy of Sciences, a fellow of the American Academy of Arts and Sciences, a fellow of the American Association for the Advancement of Science, and past president of the International Jugglers Association. He was an invited speaker at the International Congress of Mathematicians in Warsaw in 1983 and was the AMS Gibbs Lecturer in 2000. Response from Professor Graham I must say that it is a great honor and pleasure for me to receive this award in recognition of a life in mathematics, and I would like to express my deep appreciation to the American Mathematical Society and to the Steele Prize Committee for their selection. When I was first notified, my initial reaction was to recall the famous quote of Mark Twain, who, upon seeing his obituary printed in a local newspaper, wrote that “the reports of my death are greatly exaggerated.” I can’t remember a time when I didn’t love doing mathematics, and that desire has not dimmed over the years (yet!). But I also get great pleasure sharing mathematical discoveries and insights with others, even though this can present a special challenge for mathematicians talking to nonmathematicians. However, I really believe that this type of communication will become increasingly important in the future. As an undergraduate at Berkeley, a one-year course in number theory taught by D. H. Lehmer fired my imagination for the subject and formed the basis for my Ph.D. dissertation under him (after a slight detour of four years in the military and Alaska). Although I never took another course from Dick Lehmer, he taught me the value of independence of thought and an appreciation for the algorithmic issues in mathematics. I feel that I have been very lucky to have been at the right place and time in history for participating in the rapid and exciting current developments in combinatorics. No doubt, all mathematicians in every generation feel this way! In particular, I have had the good fortune to work with, and be inspired by, such giants as Paul Erdo˝s and Gian-Carlo Rota, who, though different in many ways, were both driven by grand visions which have helped guide the paths of many combinatorial researchers today. Number theory and combinatorics are especially rife with simple-looking problems which, like Socratic gadflies, constantly remind us how little we really know. (For example, are there infinitely many pairs of primes which differ by 2? The answer, of course, is yes! However, at present we don’t have a clue how to prove this.) I recall the story of a civilization so advanced that a prize was awarded to the first mathematician who realized that the Riemann Hypothesis actually needed a proof. Perhaps more imminent (and more likely?) is the related version in which the Great Computer a hundred years from now, when asked whether the Riemann Hypothesis is true, pauses for a moment and then says, “Yes, it is true. But you wouldn’t be able to understand the proof!” Still, I am a firm believer in Hilbert’s famous dictum “Wir müssen wissen, wir werden wissen” (“We must know, we shall know”). And with this thought in mind, I will happily continue to keep hammering pitons into the sides of the infinite mountain of mathematical truth, as we all slowly inch our way up its irresistible slopes. Lifetime Achievement: Victor Guillemin Citation Victor Guillemin has played a critical role in the development of a number of important areas in analysis and geometry. In particular, he has made fundamental contributions to microlocal analysis, symplectic group actions, and spectral theory of elliptic operators on manifolds. His work on generalizations of the Poisson and Selberg trace formulae has been particularly influential. Moreover, Guillemin has greatly advanced these areas, and mathematics in general, by mentoring many graduate students and postdoctoral fellows, some of whom have become leading mathematicians in their own right. Biographical Sketch Victor Guillemin was born in Cambridge, Massachusetts, on October 15, 1937. He received his B.A. from Harvard in 1959, his M.A. from the University of Chicago in 1960, and his Ph.D. from Harvard in 1962. He was an instructor at Columbia from 1963 to 1966 and an assistant professor at the Massachusetts Institute of Technology from 1966 to 1969. He was promoted to associate professor in 1969 and to full professor in 1973. He has held a Sloan fellowship (1969–70), a Guggenheim grant (1988–89), and an Alexander Humboldt fellowship (1998). He was elected to the American Academy of Arts and Sciences in 1984 and to the National Academy of Sciences in 1985. Response I want to thank the AMS Steele Prize Committee for the wonderful honor of being selected as corecipient, with Ron Graham, of this year’s Steele Lifetime Achievement award. For me personally, my main “lifetime achievement” has been to have had, over the course of my career, some remarkable mentors, collaborators, and students. In particular, as a graduate student I had the good fortune to have Raoul Bott and Shlomo Sternberg as teachers at a time when Morse theory, index theory, and K-theory were revolutionizing differential topology. It was also a time when Raoul Bott was, for Shlomo and me, not only a teacher and mentor but APRIL 2003 NOTICES OF THE AMS 467 a greater-than-life role model. I can’t speak for Shlomo, but “greater-than-life” remains my view of Raoul to this day. In the collaborations I’ve been involved in, I feel I have been extraordinarily lucky. I was Shlomo Sternberg’s Ph.D. student when we wrote our first paper together in 1962, neither of us imagining that this was going to be the first of thirty papers and six books that we would produce together or that we would still be actively working together four decades later. These four decades have tempered somewhat the awe I felt in his presence when I first started working with him, but not my awe for the range and depth of his understanding of mathematics. When I met Richard Melrose at a conference in Nice in 1973, he seemed, with his scruffy beard and ponytail, the embodiment of the 1970s counterculture Zeitgeist. He had, however, just settled an important special case of one of the main open problems in physical optics, the glancing ray problem; and two years later, together with Mike Taylor, he solved this problem in complete generality (a result for which he won the Bôcher Prize in 1979). Thirty years later the ponytail is gone and the beard marginally less scruffy, and when the occasion requires, he can pass himself off as a respectable middle-aged academic. However, he is still, with his many students and collaborators (of whom I am fortunate to be one), exploring the consequences of this result and the beautiful ideas to which it has led in microlocal analysis on manifolds-with-corners and singular spaces. One of the most rewarding collaborations of my life was working with Hans Duistermaat on the Poisson formula for elliptic operators; however, at the time it was also one of the most exasperating. I enjoy writing mathematical papers but find it hard to edit and revise and am often content with efforts that give one a glimpse of, without entirely embodying, the good, the true, and the beautiful. Hans is the opposite: With the fiercely competitive instincts of the accomplished chess player that he is, he is content with nothing short of perfection, and our paper went through many rewrites before he was completely happy with it. With each rewrite my exasperation mounted, and when we finally sent it off, I recalled his once warning me that Duistermaat is Dutch for “dark mate”. The early 1990s saw a curious blip in the demographics of the population of Generation-X mathematicians of that era. Jobs in theoretical physics became hard to come by, and as a consequence many would-be graduate students in physics gravitated to adjoining areas of mathematics. My own field of symplectic geometry was one of the beneficiaries of this development, and in the early and mid-1990s there were a large number of exceptionally talented postdocs in our department at MIT, some of whom became my collaborators and many of whom became cherished friends. Among them were Jiang-Hua Lu, Reyer Sjamaar, Sue Tolman, Yael Karshon, Jaap Kalkman, and Eckhard Meinrenken. I like to believe that they learned a little symplectic geometry from me, but I suspect I learned much, much more from them. (In particular, I learned from Eckhard Meinrenken that, as Shlomo and I had conjectured fifteen years before, “quantization and reduction commute”.) My first student, in 1968, was Marty Golubitsky, and my last student, in 2002, Tara Holm. To them and to the students in between I owe everything that has made my life in mathematics worthwhile.

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Préface : CONCRETE MATHEMATICS: A Foundation for Computer Science

Source : http://cs.ioc.ee/yik/lib/1/Graham1pre.html

 

BackCONCRETE MATHEMATICS: A Foundation for Computer Science
2nd ed

by Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik

Addison-Wesley Publishing Co. - Reading, Mass.
ISBN: 0-201-55802-5 * Hardcover * 657 p. * © 1994


This book is based on a course of the same name that has been taught annually at Stanford University since 1970. About fifty students have taken it each year juniors and seniors, but mostly graduate students - and alumni of these classes have begun to spawn similar courses elsewhere. Thus the time seems ripe to present the material to a wider audience (including sophomores).

It was dark and stormy decade when Concrete Mathematics was born. Long-held values were constantly being questioned during those turbulent years; college campuses were hotbeds of controversy. The college curriculum itself was challenged, and mathematics did not escape scrutiny. John Hammersley had just written a thought-provoking article "On the enfeeblement of mathematical skills by 'Modern Mathematics' and by similar soft intellectual trash in schools and universities" [176] ; other worried mathematicians [332] even asked, "Can mathematics be saved?" One of the present authors had embarked on a series of books called The Art of Computer Programming, and in writing the first volume he (DEK) had found that there were mathematical tools missing from his repertoire; the mathematics he needed for a thorough, well-grounded understanding of computer programs was quite different from what he'd learned as a mathematics major in college. So he introduced a new course, teaching what he wished somebody had taught him.

The course title "Concrete Mathematics" was originally intended as an antidote to "Abstract Mathematics," since concrete classical results were rapidly being swept out of the modern mathematical curriculum by a new wave of abstract ideas popularly called the "New Math." Abstract mathematics is a wonderful subject, and there's nothing wrong with it: It's beautiful, general, and useful. But its adherents had become deluded that the rest of mathematics was inferior and no longer worthy of attention. The goal of generalization had become so fashionable that a generation of mathematicians had become unable to relish beauty in the particular, to enjoy the challenge of solving quantitative problems, or to appreciate the value of technique. Abstract mathematics was becoming inbred and losing touch with reality; mathematical education needed a concrete counterweight in order to restore a healthy balance.

When DEK taught Concrete Mathematics at Stanford for the first time he explained the somewhat strange title by saying that it was his attempt to teach a math course that was hard instead of soft. He announced that, contrary to the expectations of some of his colleagues, he was not going to teach the Theory of Aggregates, not Stone's Embedding Theorem, nor even the Stone-Cech compactification. (Several students from the civil engineering department got up and quietly left the room.)

Although Concrete Mathematics began as a reaction against other trends, the main reasons for its existence were positive instead of negative. And as the course continued its popular place in the curriculum, its subject matter "solidified" and proved to be valuable in a variety of new applications. Meanwhile, independent confirmation for the appropriateness of the name came from another direction, when Z.A. Melzak published two volumes entitled Companion to Concrete Mathematics [267].

The material of concrete mathematics may seem at first to be a disparate bag of tricks, but practice makes it into a disciplined set of tools. Indeed, the techniques have an underlying unity and a strong appeal for many people. When another one of the authors (RLG) first taught the course in 1979, the students had such fun that they decided to hold a class reunion a year later.

But what exactly is Concrete Mathematics? It is a blend of continuous and discrete mathematics. More concretely, it is the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems. Once you, the reader, have learned the material in this book, all you will need is a cool head, a large sheet of paper, and fairly decent handwriting in order to evaluate horrendous-looking sums, to solve complex recurrence relations, and to discover subtle patterns in data. You will be so fluent in algebraic techniques that you will often find it easier to obtain exact results than to settle for approximate answers that are valid only in a limiting sense.

The major topics treated in this book include sums, recurrences, elementary number theory, binomial coefficients, generating functions, discrete probability, and asymptotic methods. The emphasis is on manipulative techniques rather than on existence theorems or combinatorial reasoning; the goal is for each reader to become as familiar with discrete operation (like the greatest integer function and finite summation) as a student of calculus is familiar with continuous operations (like the absolute-value function and infinite integration)

Notice that this list of topics is quite different from what is usually taught nowadays in undergraduate course entitled "Discrete Mathematics." Therefore the subject needs a distinctive name, and "Concrete Mathematics" has proved to be as suitable as another

The original textbook for Stanford's course on concrete mathematics was the "Mathematical Preliminaries" section in The Art of Computer Programming [207]. But the presentation in those 110 pages is quite terse, so another author (OP) was inspired to draft a lengthy set of supplementary notes. The present book is an outgrowth of those notes; it is an expansion of, and a more leisurely introduction to, the material if Mathematical Preliminaries. Some of the more advanced parts have been omitted; on the other hand, several topics not found there have been included here so that the story will be complete

The authors have enjoyed putting this book together because the subject began to jell and to take on a life of its own before our eyes; this book almost seemed to write itself. Moreover, the somewhat unconventional approaches we have adopted in several places have seemed to fit together so well, after these years of experience, that we can't help feeling that this book is a kind of manifesto about our favorite way to do mathematics. So we think the book has turned out to be a tale of mathematical beauty and surprise, and we hope that our readers will share at least of the pleasure we had while writing it

Since this book was born in a university setting, we have tried to capture the spirit of a contemporary classroom by adopting an informal style. Some people think that mathematics is a serious business that must always be cold and dry; but we think mathematics is fun, and we aren't ashamed to admit the fact. Why should a strict boundary line be drawn between work and play? Concrete mathematics is full of appealing patterns; the manipulations are not always easy, but the answers can be astonishingly attractive. The joy and sorrows of mathematical work are reflected explicitly in this book because they are part of our lives.

Students always know better than their teachers, so we have asked the first students of this material to contribute their frank opinions, as "graffiti" in the margins. Some of these marginal markings are merely corny, some are profound; some of them warn about ambiguities or obscurities, others are typical comments made by wise guys in the back row; some are positive, some are negative, some are zero. But they all are real indications of feelings that should make the text material easier to assimilate. (the inspiration for such marginal notes comes from a student handbook entitled Approaching Stanford, where the official university line is counterbalanced by the remarks of outgoing students. For example, Stanford says, "There are a few things you cannot miss in this amorphous .. what the h*** does that mean? Typical of the pseudo-intellectualism around her." Stanford: There is no end to the potential of a group of students living together." Graffito: "Stanford dorms are like zoos without a keeper."

The margins also include direct quotations from famous mathematicians of past generations, giving the actual words in which they announced some of their fundamental discoveries. Somehow it seems appropriate to mix the words of Leibniz, Euler, Gauss, and others with those of the people who will be continuing the work. Mathematics is an ongoing endeavor for people everywhere; many strands are being woven into one rich fabric.

This book contains more than 500 exercises, divided into six categories:

  1. Warmups are exercises that every reader should try to do when first reading the material.
  2. Basics are exercises to develop facts that are best learned by trying one's own derivation rather than by reading somebody else's.
  3. Homework exercises are problems intended to deepen an understanding of material in the current chapter.
  4. Exam problems typically involve ideas from two or more chapters simultaneously; they are generally intended for use in take-home exams (not for in-class exams under time pressure).
  5. Bonus problems go beyond what an average student of concrete mathematics is expected to handle while taking a course based on this book; they extend the text in interesting ways.
  6. Research problems may or may not be humanly solvable, but the ones presented here seen to be worth a try (without time pressure).

Answers to all the exercises appear in Appendix A, often with additional information about related results. (Of course the "answers" to research problems are incomplete; but even in these cases, partial results or hints are given that might prove to be helpful.) Readers are encouraged to look at the answers especially the answers to the warmup problems, but only after making a serious attempt to solve the problems without peeking.

We have tried in Appendix C to give proper credit to the sources of each exercise, since a great deal of creativity and/or luck often goes into the design of an instructive problem. Mathematicians have unfortunately developed a tradition of borrowing exercises without an acknowledgment; we believe that the opposite tradition, practiced for example books and magazines about chess (where names, dates, and location of original chess problems are routinely specified) is far superior. However, we have not been able to pin down the sources of many problems that have become part of the folklore. If any reader knows the origin of an exercise for which our citation is missing or inaccurate, we would be glad to learn the details so that we can correct the omission in subsequent editions of this book.

The typeface used for mathematics throughout this book is a new design by Hermann Zapf [227], commissioned by the American Mathematical Society and developed with the help of a committee that included B. Beeton, R.P. Boas. L.K. Durst, D. E. Knuth, P. Murdock, R.S. Palais, P Renz, E. Swanson, S.B. Whidden and W.B. Woolf. The underlying philosophy of Zapf's design is to capture the flavor of mathematics as it might be written by a mathematician with excellent handwriting. A handwritten rather than mechanical style is appropriate because people generally create mathematics with pen, pencil, or chalk. (For example, one of the trademarks of the new design is the symbol for zero, 'O', which is slightly pointed at the top because a handwritten zero rarely closes together smoothly when the curve returns to its starting point.) The letters are upright, not italic, so the subscripts, superscripts, and accents are more easily fitted with ordinary symbols. This new type of family has been named AMS Euler, after the great Swiss mathematician Leonhard Euler (1707-1783) who discovered so much of mathematics as we know it today. The alphabets include Euler Text, Euler Fraktur, and Euler Script Capitals, as well as Euler Greek and special symbols such as <p> and <N>. We are especially pleased to be able to inaugurate the Euler Family of typefaces in this book, because Leonhard Euler's spirit truly lives on every pare: Concrete mathematics is Eulerian mathematics.

The authors are extremely grateful to Andrei Broder, Ernst Mayr, Andrew Yao, and Frances Yao, who contributed greatly to this book during the years that they taught Concrete Mathematics at Stanford. Furthermore we offer 1024 thanks to the teaching assistants who creatively transcribed what took place in class each year and who helped to design the examination questions; their names are listed in Appendix C. This book, which is essentially a compendium of sixteen years' worth of lecture notes, would have been impossible without their first-rate work.

Many other people have helped to make this book a reality. For examples, we wish to commend the students at Brown, Columbia, CUNY, Princeton, Rice, and Stanford who contributed the choice of graffiti and helped to debug our first drafts. Our contacts at Addison-Wesley were especially efficient and helpful; in particular, we wish to thank our publisher (Peter Gordon), production supervisor (Bette Aaronson), designer (Roy Brown), and copy editor (Lyn Dupré). The National Science Foundation and the Office of Naval Research have given invaluable support. Cheryl Graham was tremendously helpful as we prepared the index. An above all, we wish to thank our wives (fan, Jill, and Amy) for their patience, support, encouragement, and ideas.

This second edition features a new Section 5.8, which describes some important ideas that Doron Zeilberger discovered shortly after the first edition went to press. Additional improvements to the first printing can also be found on almost every page.

We have tried to produce a perfect book, but we are imperfect authors. Therefore we solicit help in correcting any mistakes that we've made. A reward of $2.56 will gratefully be paid to the first finder if any error, whether it is mathematical, historical, or typographical.

Murray Hill, New Jersey RLG  
and Stanford California DEK Top
May 1988 and October 1993 OP Back

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Concrete Mathematics

Concrete Mathematics

 
 
Concrete Mathematics: A Foundation for Computer Science
Auteur Ronald Graham, Donald Knuth, etOren Patashnik
Pays États-Unis
Genre Mathématiques
Informatique
Éditeur Addison–Wesley
Nombre de pages 657 pp (seconde édition)
ISBN 0-201-55802-5

Concrete Mathematics, sous-titré A Foundation for Computer Science (Mathématiques concrètes : Fondations pour l'informatique) est un manuel de cours écrit par Ronald Graham, Donald Knuth et Oren Patashnik (en), fréquemment utilisé dans l'enseignement de l'informatique.

 

 

Historique et contenu[modifier | modifier le code]

Concrete Mathematics a pour objectif d'exposer les connaissances et les compétences mathématiques nécessaires en informatique (théorique), et plus particulièrement celles permettant l'analyse de l'efficacité des algorithmes. La préface précise que les sujets abordés « combinent des mathématiques CONtinues et disCRÈTES. » ; bien que les méthodes employées soit essentiellement celles de la combinatoire (dénombrements, raisonnement par récurrence, etc.) et de la théorie des nombres (arithmétique modulaire), les explications et les exercices utilisent fréquemment des outils provenant de l'analyse, comme les intégrales ou les développements asymptotiques. L'expression « concrete mathematics (mathématiques concrètes) » fait contraste avec abstract mathematics (mathématiques pures) et se rapproche de mathématiques constructives ; de plus, elle contient un jeu de mot intraduisible, concrete signifiant également béton en anglais, ce qui renvoie à l'idée de fondations (d'un bâtiment), et explique la couverture de l'ouvrage, représentant le symbole somme sum imprimé dans du béton.

Le livre est basé sur un cours donné par Donald Knuth à partir de 1970 à l'université Stanford. Il développe le matériel exposé dans la section Mathematical Preliminaries(Préliminaires mathématiques) du livre de Knuth, The Art of Computer Programming, et peut être utilisé comme une introduction à cette célèbre série d'ouvrages.

Concrete Mathematics est écrit dans un langage informel et souvent humoristique, les auteurs rejetant ce qu'ils voient comme le style aride de la plupart des manuels de mathématiques. Les marges contiennent des « graffitis mathématiques », commentaires proposés par les premiers lecteurs du manuscrit : les étudiants de Knuth et de Patashnik à Stanford.

Comme pour la plupart des livres de Knuth, les lecteurs se voient proposé une récompense (en) pour toute erreur qu'ils découvriraient dans le texte, que cela soit « techniquement, historiquement, typographiquement, ou politiquement incorrect »1.

Le livre est à l'origine de la popularité de nombreuses notations en combinatoire, par exemple les crochets de Iverson, les notations de la partie entière et de la partie fractionnaire, et celles des factorielles croissantes et décroissantes.

Typographie[modifier | modifier le code]

Donald Knuth utilisa la première édition de Concrete Mathematics comme un test en grandeur réelle de la police d'écriture AMS Euler (en) et de la fonte de caractères Concrete Roman (en)2.

Table des matières[modifier | modifier le code]

Éditions[modifier | modifier le code]

  • Première édition (septembre 1988) : (en) Ronald Graham, Donald Knuth et Oren Patashnik, Concrete Mathematics, Reading, MA, First, coll. « Advanced Book Program »,‎, xiv+625 p. (ISBN 0-201-14236-8)
  • Deuxième édition (février 1994) : (en) Ronald Graham, Donald Knuth et Oren Patashnik, Concrete Mathematics, Reading, MA, Second,‎ , xiv+657 p. (ISBN 0-201-55802-5)
  • Traduction en français de la deuxième édition (octobre 2003) : (en) Ronald Graham, Donald Knuth et Oren Patashnik (trad. Alain Denise), Mathématiques concrètes : Fondations pour l'informatique, Paris, deuxième,‎ , xiv+688 p. (ISBN 978-2711748242)

Notes[modifier | modifier le code]

  1. (en) Graham, Knuth and Patashnik : Concrete Mathematics [archive]
  2. Donald E. Knuth. Typesetting Concrete Mathematics [archive], TUGboat 10 (1989), 31–36, 342. Réimprimé comme le chapitre 18 du livre Digital Typography.

Liens externes[modifier | modifier le code]

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Beaux ordres et graphes Bastien Le Gloannec 21 avril 2009

Voir le pdf : http://perso.ens-lyon.fr/eric.thierry/Graphes2009/bastien...

 

Beaux ordres et graphes Bastien Le Gloannec 21 avril 2009 1 Introduction L’´etude des beaux ordres n’est pas sp´ecifique `a la th´eorie des graphes. Ces ordres apparaissent en effet `a de tr`es nombreuses occasions et l’on peut les voir comme une forme affaiblie des bons ordres. En th´eorie des graphes, l’embl´ematique th´eor`eme des mineurs de Robertson et Seymour a notamment contribu´e `a mettre en avant certaines notions comme les d´ecompositions arborescentes et les beaux ordres. Ces derniers ´etaient toutefois ´etudi´es depuis longtemps, y compris en th´eorie des graphes, ce que nous illustrerons notamment avec le th´eor`eme de Kruskal, qui a lui-mˆeme ´egalement publi´e un survey ayant pour objet la th´eorie des beaux pr´eordres en 1970 ([3]). Nous pr´esenterons dans ce rapport une approche tout d’abord g´en´erale, puis centr´ee sur la th´eorie des graphes, de la notion de bel ordre. Cela nous offrira l’occasion de nous int´eresser finalement au th´eor`eme des mineurs, dont les enjeux et cons´equences ont notamment ´et´e synth´etis´ees dans un survey de Lovasz sur la th´eorie des mineurs ([5]), et de le mettre en relation avec la notion de bel ordre. Enfin, il convient de faire remarquer au lecteur qu’un nombre non n´egligeable de preuves ou exemples de ce rapport sont issus d’exercices (non corrig´es) de [1]. Il n’est par cons´equent pas impossible que des erreurs ou impr´ecisions y figurent malencontreusement. Le lecteur est donc invit´e `a rester vigilant. 2 G´en´eralit´es 2.1 D´efinitions et caract´erisation Commen¸cons par rappeler qu’´etant donn´e un ensemble X quelconque, on appelle pr´eordre sur X toute relation binaire r´eflexive et transitive sur X. Si X est muni d’un pr´eordre, nous parlerons d’anti-chaˆıne pour d´esigner un sous-ensemble de X dans lequel tous les ´el´ements distincts sont deux `a deux incomparables. On appelle beau pr´eordre tout pr´eordre 6 sur X tel que pour toute suite infinie (xn)n∈N d’´el´ements de X, il existe deux indices i < j tels que xi 6 xj . Le couple (xi , xj ) est dans ce cas appel´e bonne paire et toute suite contenant une bonne paire sera appel´ee bonne suite. A l’inverse, une suite qui n’est pas bonne sera ` 1 Beaux ordres et graphes Bastien Le Gloannec dite mauvaise. Ainsi, tout pr´eordre sur X est un beau pr´eordre si et seulement si toute suite infinie de X est bonne. On donne le th´eor`eme de caract´erisation des beaux pr´eordres suivant. Th´eor`eme 1 (Caract´erisation des beaux pr´eordres) Les propositions suivantes sont ´equivalentes : (i) 6 est un beau pr´eordre sur X. (ii) De toute suite infinie d’´el´ements de X, on peut extraire une sous-suite croissante. (iii) X ne contient ni anti-chaˆıne infinie, ni suite infinie strictement d´ecroissante. Bien qu’il soit assez simple de donner une preuve directe de cette caract´erisation, il existe une jolie d´emonstration passant par un th´eor`eme de Ramsey ´enonc´e et d´emontr´e ciapr`es. Mais avant, d´efinissons quelques notations utiles. Etant donn´e un ensemble ´ X, nous noterons Pk(X) l’ensemble des parties finies de X `a exactement k ´el´ements. Pour c > 1, on appelle c-coloration de X toute fonction de X dans {0, . . . , c−1}, qui `a chaque ´el´ement de X associe une couleur parmi c couleurs possibles. Si X est muni d’une c-coloration, on dira qu’un ensemble Y ⊆ X est monocromatique si tous les ´el´ements de Y ont la mˆeme couleur. Th´eor`eme 2 (Ramsey) Soit X un ensemble infini, k > 1, c > 1 et l’on suppose donn´ee une c-coloration de Pk(X). Alors il existe une partie infinie Y ⊆ X telle que Pk(Y ) soit monochromatique. Preuve (Ramsey) On proc`ede par r´ecurrence sur k `a c fix´e. Pour k = 1, quel que soit X infini et une c-coloration de P1(X) (singletons) que l’on assimilera `a une coloration de X, il n’y a qu’un nombre fini de couleurs attribu´ees `a un nombre infini d’´el´ements donc au moins une couleur est affect´ee `a une infinit´e d’´el´ements. Pour k > 1 on suppose que pour tout ensemble Z infini et toute c-coloration de Pk−1(X), il existe une partie infinie Z 0 ⊆ Z telle que Pk−1(Z 0 ) soit monochromatique. Nous allons construire une suite (xn)n∈N d’´el´ements de X ainsi qu’une suite (Xn)n∈N strictement d´ecroissante de parties infinies de X v´erifiant les conditions suivantes (pour tout i) : (i) xi ∈ Xi (ii) Xi+1 ⊆ Xi{xi} (iii) L’ensemble {S∪{xi}, S ∈ Pk−1(Xi{xi})} (i.e. l’ensemble des parties `a k ´el´ements de X contenant xi et dont tous les autres ´el´ements sont pris dans Xi) est monochromatique, et l’on note ci sa couleur. On commence par poser X0 = X et x0 ∈ X quelconque. 2/16 Beaux ordres et graphes Bastien Le Gloannec Supposons construite la suite jusqu’au rang i. On consid`ere l’ensemble Pk−1(Xi{xi}) et l’on d´efinit une c-coloration sur cet ensemble ainsi : pour tout S ∈ Pk−1(Xi{xi}), on pose la couleur de S comme ´etant la couleur de S ∪ {xi} (ensemble `a k ´el´ements car xi ∈/ Xi) dans la c-coloration de Pk(X). Par hypoth`ese de r´ecurrence, il existe Xi+1 ⊆ Xi{xi} ((ii) est v´erifi´ee) tel que Pk−1(Xi+1) soit monochromatique ((iii) est v´erifi´ee) et l’on pose ci la couleur correspondante1 . On choisit alors xi+1 ∈ Xi+1 quelconque ((i) est v´erifi´ee). La suite ´etant maintenant construite, on remarque que la suite des couleurs associ´ees ne prend qu’un nombre fini de valeurs, il en existe donc une extraction ϕ telle que le suite infinie (cϕ(n) )n∈N soit constante, notons C sa valeur. Il ne reste plus qu’`a poser Y = {xϕ(n) , n ∈ N} ⊆ X. Y v´erifie (on a tout fait pour) la propri´et´e attendue : Pk(Y ) est monochromatique. En effet, quel que soit S ∈ Pk(Y ), S est constitu´e d’´el´ements de la suite (xϕ(n) ) et posons xi l’´el´ement de plus petit indice de S. Par construction, S{xi} ∈ Pk−1(Xi+1), et donc, par (iii), S est de couleur ci = C (car xi est issu de le suite extraite (xϕ(n) )) et ce pour tout S, donc Pk(Y ) est monochromatique. D’o`u le r´esultat. On peut maintenant d´emontrer le th´eor`eme de caract´erisation qui nous int´eresse. Preuve (Caract´erisation) Remarquons tout d’abord que les implications (ii) ⇒ (i) et (i) ⇒ (iii) sont ´evidentes. Pour la premi`ere, si pour toute suite il existe une soussuite croissante, alors il existe une infinit´e de bonnes paires et a fortiori la suite est bonne. Pour la seconde, toute suite d’´el´ements distincts d’une anti-chaˆıne infinie, ainsi que toute suite infinie strictement d´ecroissante est une mauvaise suite, ce qui n’existe pas par d´efinition mˆeme d’un beau pr´eordre. Consid´erons maintenant l’implication (iii) ⇒ (i). Soit (xn)n∈N une suite d’´el´ements de X. Consid´erons le graphe infini dont les sommets sont les indices de la suite et les arˆetes les couples (i, j) pour i < j. Pour tous i < j, on colorie l’arˆete (i, j) de la fa¸con suivante : • si xi et xj sont incomparables, on colorie l’arˆete (i, j) en gris. • sinon, si xi 6 xj , i.e. (xi , xj ) est une bonne paire, on colorie l’arˆete (i, j) en vert. • sinon, on a xi > xj et l’on colorie l’arˆete (i, j) en rouge. Par th´eor`eme de Ramsey (pour k = 2 et c = 3 sur les paires d’´el´ements de n, la paire {i, j} avec i < j se voyant attribu´ee la couleur de l’arˆete (i, j) de notre graphe), il existe un sous-graphe infini monochrome. S’il ´etait gris alors on aurait form´e une anti-chaˆıne infinie. S’il ´etait rouge, on aurait trouv´e une sous-suite infinie strictement d´ecroissante. Il ne peut donc qu’ˆetre vert : c’est une sous-suite infinie croissante et donc bonne a fortiori. On notera au passage que par cette mˆeme m´ethode on peut extraire de toute suite de X une sous-suite croissante, i.e. l’implication (i) ⇒ (ii) est d´emontr´ee du mˆeme coup. 3/16 Beaux ordres et graphes Bastien Le Gloannec Dans la suite, nous appellerons bel ordre tout beau pr´eordre qui est en plus un ordre (i.e. anti-sym´etrique). 2.2 Exemples et premi`eres propri´et´es Nous allons exposer ici quelques exemples de beaux pr´eordres et beaux ordres ainsi que quelques propri´et´es simples, naturelles et utiles. Voici tout d’abord quelques remarques imm´ediates sur les beaux pr´eordres. Proposition 1 (Beau pr´eordre induit) Soit 6 est un beau pr´eordre (resp. bel ordre) sur X et Y ⊆ X, le pr´eordre (resp. ordre) induit par 6 sur Y est un beau pr´eordre (resp. bel ordre). Preuve Tout mauvaise suite sur Y muni de l’ordre induit serait aussi une mauvaise suite sur X, or il n’en existe pas par hypoth`ese. Proposition 2 (Ordres et sous-ordres) Soient 61 et 62 sont deux pr´eordres sur X v´erifiant 61⊆62, i.e. ∀x, y ∈ X, x 61 y ⇒ x 62 y. Alors on a 61 beau pr´eordre ⇒ 62 beau pr´eordre mais la r´eciproque est fausse. Preuve Toute bonne suite pour 61 est encore bonne pour 62. Toute suite est donc bonne pour 62 qui est donc un beau pr´eordre. Pour la r´eciproque, comme nous le verrons, la relation de mineur est un bel ordre sur les graphes finis mais pas la relation de mineur topologique. Exemple 1 Les ordres sur N. 1. L’ordre usuel sur N est un bel ordre (car il est total et qu’il n’existe pas de chaˆıne infinie strictement d´ecroissante). 2. L’ordre produit usuel (composante par composante) sur N k est aussi un bel ordre. En effet, de toute suite de N k , on peut extraire une sous-suite croissante ainsi : on extrait une sous-suite croissante (pour l’ordre usuel, bon ordre sur N) suivant la premi`ere composante ; de cette suite on extrait une sous-suite croissante suivant la deuxi`eme composante, et on it`ere ainsi sur toutes les composantes. . . On arrive finalement `a une suite de N k simultan´ement croissante sur toutes les composantes, i.e. croissante pour l’ordre produit. Il est int´eressant de constater qu’`a travers de l’exemple de N k , nous avons donn´e une m´ethode de preuve qui assure imm´ediatement le r´esultat suivant. Proposition 3 (Bel ordre produit) Pour tout n > 1 et tous ensembles X1, . . . , Xn munis respectivement de beaux pr´eordres (resp. beaux ordres) 61, . . . , 6n, le pr´eordre (resp. l’ordre) produit sur X = Qn k=1 Xk, d´efinit par (x1, . . . , xn) 6 (y1, . . . , yn) si et seulement si ∀1 6 k 6 n, xk 6 yk, est un beau pr´eordre (resp. bel ordre) sur X. 4/16 Beaux ordres et graphes Bastien Le Gloannec La preuve est imm´ediate par la m´ethode que nous avons propos´e pour l’exemple 1. Dans la mˆeme veine, ce r´esultat sur le produit cart´esien est ´egalement trivialement valable pour l’union disjointe. Proposition 4 (Bel ordre sur l’union) Pour tout n > 1 et tous ensembles X1, . . . , Xn disjoints munis respectivement de beaux pr´eordres (resp. beaux ordres) 61, . . . , 6n, le pr´eordre (resp. l’ordre) union sur X = Sn k=1 Xk, d´efini par 6= Sn k=1 6k, est un beau pr´eordre (resp. bel ordre) sur X. Exemple 2 Bons ordres et beaux ordres. Une question brˆule certainement les l`evres du lecteur avis´e qui a certainement d´ej`a entendu parler de bons ordres, de relations bien fond´ees et se demande s’il existe un lien avec ces beaux ordres qu’il vient de d´ecouvrir. Une relation bien fond´ee sur un ensemble X est une relation binaire sur X2 telle que tout sous-ensemble non vide de X admette un ´el´ement “minimal” (au sens d’un ´el´ement sans ant´ec´edent par la relation ; en particulier, il n’y a pas n´ecessairement unicit´e de cet ´el´ement). Modulo l’axiome du choix d´ependant, cette d´efinition est ´equivalente `a la non existence de suite infinie d´ecroissante (on dit aussi que la relation est nœuth´erienne en th´eorie de la r´e´ecriture). Ainsi donc un pr´eordre est un beau pr´eordre si et seulement si l’ordre strict associ´e est bien fond´e et qu’il n’existe pas d’anti-chaˆıne infinie. Qu’en est-il des bons ordres ? Un bon ordre sur X est une ordre sur X tel que tout sousensemble non vide de X admette un plus petit ´el´ement (au sens d’un ´el´ement inf´erieur ou ´egal `a tous les autres). En particulier un tel ordre est total. L`a encore, modulo l’axiome du choix d´ependant, cette d´efinition est en fait ´equivalente `a dire que l’ordre est total et la relation d’ordre strict associ´ee est bien fond´ee. Un bon ordre est donc un bel ordre : il n’existe par d’anti-chaˆıne par totalit´e et la relation stricte est bien fond´ee. Ainsi donc tout ensemble bien ordonn´e est ´egalement muni d’un bel ordre. C’´etait par exemple le cas de N pour l’ordre usuel, mais c’est par exemple aussi le cas des N k pour l’ordre lexicographique. Quelques contre-exemples • L’ordre usuel sur Z, Q, R n’est pas un bel ordre (suites infinies strictement d´ecroissantes). • Les ordres produit et lexicographique sur Z k , Qk , R k (suites infinies strictement d´ecroissantes, ou anti-chaˆınes infinies dans le cas de l’ordre produit). • L’inclusion ⊆ sur un ensemble infini : les singletons forment une anti-chaˆıne infinie. 5/16 Beaux ordres et graphes Bastien Le Gloannec 3 Quelques r´esultats 3.1 Lemme de Higman Un r´esultat remarquable est que l’on peut ´etendre tout beau pr´eordre sur X `a l’ensemble X<ω des parties finies de X. On d´efinit en effet la relation 6 sur X<ω ainsi : pour tous A, B ∈ X<ω , A 6 B si et seulement s’il existe une injection f de A dans B telle que pour tout a ∈ A, a 6 f(a). On v´erifie ais´ement que cette relation est un pr´eordre sur X<ω : • R´eflexivit´e : il suffit de prendre f = idA. • Transitivit´e : si A 6 B par une injection f et B 6 C par une injection g, alors ∀a ∈ A, f(a) ∈ B donc g(f(a)) > f(a) > a et g ◦ f compos´ee d’injections reste injective. Dans le cas o`u l’on dispose initialement d’un bel ordre sur X, on obtient ´egalement un bel ordre sur X<ω. En effet, on h´erite de l’anti-sym´etrie : si A 6 B via f et B 6 A via g, alors ∀a ∈ A, g(f(a)) > a. g ◦ f est une bijection de A dans A et cette in´egalit´e exprime le fait que tout ´el´ement de a doive ˆetre envoy´e sur un ´el´ement depuis lequel il est accessible dans le DAG fini (car A fini) de la relation d’ordre (partielle) > sur A. Ainsi, les sources de ce DAG (´el´ements maximaux de A) sont n´ecessairement envoy´ees sur elles-mˆemes. Comme l’application est injective, on ne peut plus r´eutiliser ces sources pour continuer `a construire g ◦ f, on peut donc les retirer du graphe, faisant ainsi apparaˆıtre de nouvelles sources `a leur tour envoy´ees sur elles-mˆemes, et l’on it`ere. . . Finalement, ∀a ∈ A, g(f(a)) = a. Mais a = g(f(a)) > f(a) > a (par B 6 A puis A 6 B) donc f(a) = a et donc A = B. Lemme 1 (Higman – version ensembles) Soit X un ensemble muni d’un beau pr´eordre 6 alors le pr´eordre induit par 6 sur X<ω est un beau pr´eordre. Preuve Par l’absurde, supposons qu’il existe des mauvaises suites sur X<ω. On va construire une suite (Xn)n∈N d’´el´ements de X<ω par r´ecurrence. Supposons construite la suite jusqu’au rang i et supposons qu’elle v´erifie l’hypoth`ese suivante : X0, . . . , Xi est le d´ebut d’au moins une mauvaise suite sur X<ω. On alors choisit Xi+1 ∈ X<ω de cardinal minimal tel que X0, . . . , Xi , Xi+1 soit le d´ebut d’une mauvaise suite. La suite ainsi form´ee est bien sˆur une mauvaise suite (sinon il existe i < j tels que Xi 6 Xj et donc X0, . . . , Xi , . . . , Xj ne saurait ˆetre le d´ebut d’une mauvaise suite). A fortiori, on a donc ∀n ∈ N, Xn 6= ∅ (en remarquant que ∀A ∈ X<ω , ∅ 6 A). Pour tout n, on peut donc choisir xn ∈ Xn quelconque et poser Yn = Xn{an}. Par caract´erisation (iii) dans X muni d’un bel ordre, la suite (xn)n∈N admet une sous-suite (xϕ(n) )n∈N croissante. Par minimalit´e du cardinal dans le choix Aϕ(0), la suite X0, . . . , Xϕ(0)−1 , Yϕ(0), Yϕ(1), Yϕ(2), . . . est bonne et contient donc une bonne paire. Une telle paire ne peut ˆetre ni de la forme (Xi , Xj ) (puisque (Xn)n∈N est mauvaise) ni (Xi , Yj ) puisque Xj > Yj (et on aurait Xi 6 Yj < Xj ). Une bonne paire est donc de la forme (Yi , Yj ) et donc Yi 6 Yj via une 6/16 Beaux ordres et graphes Bastien Le Gloannec injection f de Yi vers Yj . On prolonge alors f en f 0 de Xi vers Xj en posant f 0 (xi) = xj (on a bien xj > xi car xi et xj sont issus de la suite (xϕ(n) )n∈N) on a construit une injection assurant que Xi 6 Xj , ce qui est absurde car la suite (Xn)n∈N est mauvaise. La r´eciproque du lemme de Higman est ´egalement vraie : il suffit de consid´erer les singletons. Il existe ´egalement une version mots du lemme de Higman que nous allons maintenant consid´erer. Si Σ un alphabet muni d’un beau pr´eordre 6, on peut ´etendre ce pr´eordre `a Σ∗ en posant, pour tous mots u = u1 . . . up et v = v1 . . . vq, u 6 v si et seulement si il existe une injection f de {1, . . . , p} dans {1, . . . , q} telle que pour tout 1 6 i 6 p, ui 6 vf(i) . Lemme 2 (Higman – version mots) Si Σ un alphabet muni d’un beau pr´eordre 6, alors le pr´eordre induit par 6 sur Σ ∗ est beau pr´eordre. Ce r´esultat pourrait se d´emontrer par une preuve totalement analogue `a la pr´ec´edente. Nous allons plutˆot proc´eder en r´eutilisant le r´esultat pr´ec´edent. Nous allons mˆeme montrer un peu plus : l’´equivalence des deux versions du lemme de Higman. Preuve (´equivalence des versions ensembles/mots) Il suffit de remarquer que pour toute permutation σ de {1, . . . , p} et toute permutation σ 0 de {1, . . . , q}, si l’on pose u 0 = uσ(1) . . . uσ(p) et v 0 = vσ0(1) . . . vσ0(q) alors si l’on a u 6 v via une injection f, on a ´egalement u 0 6 v 0 via l’injection σ 0−1 ◦ f ◦ σ. En d’autres termes, l’ordre des lettres n’a aucune importance, on peut les voir comme des ensembles finis de lettres, i.e. des ´el´ements de Σ∗<ω. L’´equivalence des ´enonc´es est alors imm´ediate. Il est `a noter que l’ordre des lettres n’´etant pas important, pour tout mot u et toute permutation u 0 de u, u 0 6= u, on aura tout de mˆeme u 6 u 0 et u 0 6 u sans avoir u = u 0 : l’anti-sym´etrie n’est pas v´erifi´ee, on ne peut avoir qu’un pr´eordre ici et pas d’ordre. Enfin, il existe en combinatoire sur les mots une autre version usuelle du lemme de Higman. On utilise pour cette derni`ere l’ordre suivant : u 6 v si et seulement si u est un sous-mot de v (au sens d’une suite extraite, aux lettres non n´ecessairement cons´ecutives dans v). On v´erifie ais´ement que cette relation sur les mots est cette fois un ordre et pas seulement un pr´eordre. Lemme 3 (Higman – version sous-mots) La relation de sous-mot 6 est un bel ordre sur Σ ∗ . Cela revient `a prendre l’´egalit´e comme relation et `a imposer de plus `a l’injection d’ˆetre croissante dans la version mots du lemme de Higman. On peut en fait montrer que le lemme de Higman est encore vrai si l’on impose `a l’injection d’ˆetre croissante. Il implique alors directement la version sous-mots (que l’on pourrait aussi d´emontrer directement en adaptant la preuve du th´eor`eme de Higman en une preuve plus simple par certains aspects, puisque l’on ne dispose plus d’un beau pr´eordre sur Σ, mais en assurant la croissance de l’injection). 7/16 Beaux ordres et graphes Bastien Le Gloannec 3.2 Th´eor`eme de Kruskal Dans cette section, nous allons ´etudier le th´eor`eme de Kruskal sur la classe des arbres finis. Avant toute chose, une remarque pr´eliminaire et importante pour toute la suite s’impose : les relations de mineur (not´ee 4) et mineur topologique sont des relations d’ordre (partielles) sur la classe des graphes finis. Ce fait est tr`es simple `a v´erifier. Th´eor`eme 3 (Kruskal, [2]) La relation de mineur topologique est un bel ordre sur les arbres finis. Il est `a noter que ce r´esultat ne restera cependant pas vrai sur la classe des graphes finis quelconques toute enti`ere comme nous allons le voir dans la section suivante. Afin de d´emontrer ce th´eor`eme, nous allons renforcer la notion de mineur topologique en d´efinissant la notion de mineur topologique enracin´e sur la classe des arbres. Rappelons si n´ecessaire les d´efinitions de la notions de mineur topologique. On dit qu’un graphe H est une subdivision d’un graphe G si H peut ˆetre obtenu `a partir de G en “subdivisant” des arˆetes, i.e. en rempla¸cant une arˆete de G par une chaˆıne de longueur arbitraire. On dit alors qu’un graphe G est un mineur topologique d’un graphe H s’il existe un sous-graphe H0 de H tel que H0 soit une subdivision de G. En d’autres termes, G est obtenu `a partir de H en supprimant des arˆetes, des sommets, et en contractant des chaˆınes. Etant donn´es ´ deux arbres enracin´es T et T 0 , de racines respectives r et r 0 (rappelons que l’enracinement induit un ordre naturel sur l’arbre), on dira que T 6 T 0 si et seulement s’il existe une isomorphisme ϕ d’une subdivision T0 de T (pour la mˆeme racine, ce qui induit un ordre sur T0) vers un sous-arbre T1 de T 0 qui pr´eserve l’ordre, i.e. telle que si x < y dans T alors ϕ(x) < ϕ(y) dans T1 ⊆ T 0 . Il est ais´e de v´erifier que l’on d´efinit bien un pr´eordre sur les arbres enracin´es ainsi. Essentiellement, la relation d´efinie est en tout points similaire `a la notion de mineur topologique, si ce n’est qu’elle pr´eserve l’orientation pour des arbres enracin´es. La Fig. 1 illustre cette notion. La m´ethode de preuve qui suit est totalement similaire `a celle mise en œuvre pour d´emontrer le lemme de Higman. D’ailleurs, nous n’h´esiterons pas `a renvoyer par endroit le lecteur `a cette derni`ere dans la preuve qui suit. Preuve (Kruskal, m´ethode de Nash-Williams, [6]) Nous allons d´emontrer que la relation de mineur topologique enracin´e est un beau pr´eordre sur les arbres finis enracin´es, ce qui implique naturellement ce que l’on veut d´emontrer du fait de l’´equivalence suivante : T 0 est un mineur topologique de T si et seulement si il existe un enracinement de T et un enracinement de T 0 tels que T 0 soit un mineur topologique enracin´e de T pour ces enracinements. Par l’absurde (i.e. on suppose l’existence de mauvaises suites), on proc`ede comme dans la preuve du lemme de Higman (s’y reporter si n´ecessaire) en construisant une suite (Tn)n∈N d’arbres enracin´es (de racines respectives les ´el´ements de la suite (rn)n∈N) en choisissant `a chaque ´etape i un plus petit arbre (en nombre de sommets) Ti de racine ri 8/16 Beaux ordres et graphes Bastien Le Gloannec Fig. 1 – Mineur topologique enracin´e, [1] tel que T0, . . . , Ti soit le d´ebut d’une mauvaise suite. L`a encore, (Tn)n∈N est une mauvaise suite. Pour tout i ∈ N, on pose Si l’ensemble des composantes connexes du graphe Ti dont on a retir´e la racine ri , chacune de ces composantes ´etant enracin´ee en le voisin de ri dans la composante, de sorte que l’ordre induit par l’enracinement reste exactement le mˆeme que dans Ti . On pose S = S n∈N Sn. Montrons alors que l’on a un beau pr´eordre sur S. Soit (tn)n∈N une suite quelconque de S en prenant, pour tout n, tn ∈ Sin . Soit m tel que im soit minimal parmi les in. D`es lors, la suite T0, . . . , Tim−1, tm, tm+1, tm+2, . . . est bonne (car tm ∈ Sim est une composante connexe issue de la suppression de rim dans Tim et a au moins un sommet de moins que Tim) et contient donc une bonne paire qui ne peut ˆetre que de la forme (ti , tj ) avec i < j. En effet, comme (Tn) est mauvaise, cela ne peut ˆetre (Ti , Tj ) et si c’´etait (Ti , tj ), alors Ti 6 tj < Tij avec i 6 im − 1 et par choix de m on a ij > im donc i < ij ce qui contredirait le fait que (Tn) soit mauvaise. On a donc trouv´e en (ti , tj ) une bonne paire dans la suite (tn) (a priori quelconque) de S qui est donc bien muni d’un beau pr´eordre. Puisque chaque Sn est une partie finie de S muni d’un beau pr´eordre, alors par lemme de Higman la suite (Sn) est bonne et admet donc une bonne paire (Si , Sj ), i < j, et donc Si 6 Sj via une injection f de Si dans Sj v´erifiant, pour tout t ∈ Si , t 6 f(t) via un certain isomorphisme ϕt . On pose ϕ le morphisme r´ealisant l’union de sous ces ϕt et on le prolonge `a Ti en posant ϕ(ri) = rj . L’ordre est ainsi pr´eserv´e (car on avait d´ej`a remarqu´e que l’ordre restait inchang´e dans les composantes connexes lorsque l’on effectuait la suppression de ri) et ϕ d´efinit naturellement un isomorphisme assurant Ti 6 Tj : on a trouv´e une bonne paire dans la mauvaise s´equence (Tn), d’o`u la 9/16 Beaux ordres et graphes Bastien Le Gloannec contradiction. 3.3 Contre-exemples On consid`ere dans cette section deux contre-exemples instructifs. Le premier montre que le th´eor`eme de Kruskal ne tient plus si l’on restreint la relation `a celle de sous-graphe connexe, et le second que le th´eor`eme des mineurs ne reste pas non plus vrai si l’on se limite `a la relation de mineur topologique. Contre-exemple 1 La relation de sous-graphe connexe n’est pas un bel ordre pour la classe des arbres finis. Notons que les propri´et´es de r´eflexivit´e, de transitivit´e et d’anti-sym´etrie sont trivialement v´erifi´ees pour cette relation. Remarquons ´egalement qu’un graphe n’a qu’un nombre fini de sous-graphes connexes (et ils sont tous de taille inf´erieure ou ´egale), par cons´equent il est inutile d’esp´erer obtenir une suite strictement d´ecroissante ici. Nous allons maintenant exhiber une anti-chaˆıne infinie de d’arbres. La Fig. 2 pr´esente une telle famille d’arbres deux `a deux incomparables pour la relation de sous graphe connexe. (a) T1 (b) T2 (c) T3 n arˆetes (d) Tn Fig. 2 – Anti-chaˆıne infinie pour la relation de sous-graphe connexe sur les arbres finis Contre exemple 2 La relation de mineur topologique n’est pas un bel ordre sur la classe des graphes finis. Rappelons que ce qui ´etait vrai sur la classe des arbres finis ne l’est donc plus lorsque l’on passe aux graphes quelconques. Mais cela sera par contre vrai sur la classe des graphes fini en ´elargissant la relation aux mineurs. L`a encore, comme dans l’exemple pr´ec´edent, il est inutile d’esp´erer obtenir une suite infinie strictement d´ecroissante, un graphe n’ayant qu’un nombre fini de mineurs topologiques. On cherche donc une anti-chaˆıne infinie, ce qui 10/16 Beaux ordres et graphes Bastien Le Gloannec est moins ´evident `a obtenir que dans le cas pr´ec´edent. L’id´ee est que pour montrer qu’un graphe H est un mineur topologique d’un graphe G, on peut exhiber un isomorphisme de graphe d’une subdivision de H vers une sous-graphe de G. Il n’est pas difficile de remarquer que ce morphisme envoie n´ecessairement tout sommet de H vers un sommet de degr´e sup´erieur ou ´egal dans G. Organiser judicieusement les degr´es est un moyen de forcer tel sommet `a ˆetre envoy´e sur tel autre sommet, en agen¸cant les sommets entre-eux de fa¸con `a ce qu’il ne soit pas possible qu’un graphe soit le mineur d’un autre (ici il y a 2 sommets de degr´es 6 par graphe qui sont forc´ement envoy´es les uns sur les autres, la chaine les s´eparant n’´etant pas bien contractable) on construit une anti-chaˆıne infinie d´ecrite sur la Fig. 3, et bas´ee sur une adaptation naturelle de l’exemple pr´ec´edent. (a) T1 (b) T2 (c) T3 n blocs (d) Tn Fig. 3 – Anti-chaˆıne infinie pour la relation de mineur topologique sur les graphes quelconques 4 Autour du th´eor`eme des mineurs Dans cette section, nous allons nous int´eresser au fameux th´eor`eme des mineurs et voir en quoi il est fondamentalement li´e `a la notion de bel ordre. Mais avant, pr´ecisons que nous dirons dans tout ce qui suit qu’une classe de graphes C est ferm´ee par mineurs si et seulement si tout mineur d’un graphe de C est encore dans C, i.e. la classe est stable par passage `a un mineur. 4.1 Approche et ´enonc´e Un r´esultat bien connu en th´eorie des graphes est que l’on peut caract´eriser la classe des graphes planaires comme l’ensemble des graphes n’admettant ni K5 ni K3,3 comme mineur. Ce r´esultat de 1930 est connu sous le nom de th´eor`eme de Kuratowski. 11/16 Beaux ordres et graphes Bastien Le Gloannec Th´eor`eme 4 (Kuratowski, [4]) Un graphe est planaire si et seulement s’il n’admet ni K5 ni K3,3 comme mineur. Ce r´esultat a particuli`erement marqu´e les th´eoriciens des graphes qui en ont longtemps cherch´e des g´en´eralisations. Il est en effet tr`es commode de disposer d’une telle caract´erisation par une famille finie de mineurs interdits : d’une part on peut montrer la non appartenance `a la classe d’un graphe en donnant pour certificat une s´equence de transformations conduisant au mineur interdit et d’autre part l’on peut tester l’appartenance `a la classe en temps polynomial. Malheureusement, encore auourd’hui bien peu de r´esultats explicites de ce genre sont connus et le th´eor`eme de Kuratowski reste de loin le plus embl´ematique. Toutefois, Wagner aurait conjectur´e d`es 1970 que toute classe de graphes ferm´ee par mineurs (i.e. telle que tout mineur d’un graphe de la classe est encore dedans) pouvait ˆetre caract´eris´ee par une famille finie de mineurs interdits, `a la mani`ere du th´eor`eme de Kuratowski. Ce r´esultat a ´et´e finalement ´et´e d´emontr´e par Robertson et Seymour `a travers une s´erie de vingts articles publi´es entre 1983 ([8]) et 2004 ([10]). Th´eor`eme 5 (Robertson & Seymour, [10]) Toute classe de graphes ferm´ee par mineurs peut ˆetre caract´eris´ee par une famille finie de mineurs interdits. Pour bien comprendre les enjeux de ce th´eor`eme, il est a noter que c’est bel et bien l’aspect fini de la famille de mineurs qui en est l’´el´ement important. Ce mˆeme r´esultat pour une famille infinie est une propri´et´e basique et bien connue ´enonc´e ci-dessous. Introduisons tout d’abord une notation utile. Pour K un ensemble de graphes, on d´efinit la classe Forb4(K) comme l’ensemble des graphes n’admettant aucun des ´el´ements de K comme mineur, i.e. la classe de graphes caract´eris´ee par un ensemble de mineurs interdits K. On rappelle que l’on note 4 la relation de mineur (et ≺ la relation stricte associ´ee). Inversement, pour tout classe C, on appelle ensemble de Kuratowski de C l’ensemble KC = {G graphe/G /∈ C et ∀H ≺ G, H ∈ C} i.e. l’ensemble des ´el´ements minimaux pour 4 dans le compl´ementaire C de C. Par construction mˆeme, les ´el´ements de KC forment une anti-chaˆıne pour 4. Proposition 5 Une classe de graphes est ferm´ee par mineurs si et seulement si on peut la caract´eriser par une famille (´eventuellement infinie) de mineurs interdits, auquel cas KC est une famille qui convient et c’est la famille minimale pour l’inclusion unique `a convenir. Preuve Si C est une classe de graphes ferm´ee par mineurs, il suffit de remarquer que C = Forb4(C) (pour C le compl´ementaire de la classe). Par ailleurs, KC est incluse dans toute autre famille K `a convenir car si un ´el´ement G de KC n’y ´etais pas, alors il serait interdit (car il doit l’ˆetre) en faisant intervenir un ´el´ement de K qui serait un mineur strict de G. Or par d´efinition mˆeme de KC, tout mineur strict de ses ´el´ements est dans C. D’o`u 12/16 Beaux ordres et graphes Bastien Le Gloannec la contradiction et donc KC ⊆ K. Par ailleurs KC convient ´egalement car s’il existait un ´el´ement G de C non interdit par KC, alors aucun de ses mineurs ne serait dans KC. Imaginons l’arbre des mineurs de G sur lequel G est la racine, suivent tous les mineurs stricts directs (issus d’une op´eration), puis les mineurs stricts directs des mineurs,etc (on autorise les r´ep´etitions de sommets correspondant `a un mˆeme graphe). Toutes les feuilles de l’arbre correspondent au graphe `a un sommet qui appartient ´evidemment `a tout classe de graphes ferm´ee par mineurs (suppos´ee implicitement non vide). Mais alors il existe dans le graphe des sommets appartenant `a C (ainsi qu’au moins la racine appartenant `a C). Par stabilit´e par mineurs de C, ces sommets ont tous leurs descendants dans C. Consid´erons un sommet S de profondeur maximale dans l’arbre parmi ceux correspondant `a un graphe de C. Tous ses descendants sont donc dans C. Ce sommet est donc dans KC et est un mineur de G. D’o`u la contradiction. R´eciproquement, tout Forb4(K) est trivialement ferm´e par mineurs. 4.2 Mineurs et beaux ordres Le th´eor`eme des mineurs a une autre formulation mettant plus en valeur ce qui nous pr´eoccupe, `a savoir les beaux ordres. Th´eor`eme 6 (des mineurs – deuxi`eme version) La relation de mineur est un bel ordre sur la classe des graphes finis. Il est int´eressant de montrer l’´equivalence des deux ´enonc´es de ce th´eor`eme. Preuve (´equivalence des ´enonc´es) Comme nous l’avons d´ej`a vu, si une classe C est ferm´ee par mineurs, alors C = Forb4(KC) o`u KC est une anti-chaˆıne pour 4. Mais alors, si KC n’est pas fini, alors on a trouv´e une anti-chaˆıne infinie et donc 4 ne saurait ˆetre un bel ordre sur la classe des graphes finis. On a montr´e par contrapos´ee que la deuxi`eme version implique la premi`ere (qui est clairement ´equivalente `a la finitude de KC en utilisant la proposition 5). R´eciproquement, supposons un instant par l’absurde qu’il existe une anti-chaˆıne infinie K. La classe C = Forb4(K) est close par mineurs et admet donc un ensemble de Kuratowski KC fini tel que C = Forb4(KC). Mais alors, KC est un ensemble de mineurs qui interdit l’ensemble des ´el´ements K. Et surtout, par proposition 5, KC ⊆ K. Il y a donc dans K infini une infinit´e d’´el´ements `a ne pas ˆetre dans KC et `a pourtant admettre pour mineur un ´el´ement de KC et donc de K, qui ne saurait donc ˆetre une anti-chaˆıne. D’o`u le r´esultat. 4.3 Aspects algorithmiques Le th´eor`eme suivant constitue une cons´equence algorithmique non n´egligeable du th´eor`eme des mineurs. 13/16 Beaux ordres et graphes Bastien Le Gloannec Th´eor`eme 7 L’appartenance d’un graphe `a une classe de ferm´ees par mineurs peut toujours ˆetre test´ee en temps polynomial. Ce r´esultat tr`es fort et g´en´eral repose sur une m´ethode algorithmique en O(n 3 ) (Seymour & Robertson, [9]). Toutefois la constante est ´enorme et d´epend fortement de la liste des mineurs exclus. Nous n’entrerons cependant pas dans les d´etails algorithmiques ici. 4.4 Une conjecture plus forte Dans les ann´ees 1980, Seymour a conjectur´e l’´enonc´e suivant. Conjecture 1 (Seymour) Tout graphe infini d´enombrable est un mineur strict de luimˆeme. Pr´ecisons le sens `a donner `a la notion de mineur strict ici : G infini est un mineur strict de lui mˆeme s’il existe une s´equence de transformations, parmi les suppressions de sommets, d’arˆetes et les contractions d’arˆetes, non vide et finie telle que le graphe obtenu soit isomorphe `a G. Cela peut sembler ´etonnant de prime abord, puisque l’on s’int´eresse ici `a des graphes infinis d´enombrables (un contre-exemple ind´enombrable a ´et´e d´ecouvert en 1990 par Oporowski, [7]), mais ce r´esultat impliquerait le th´eor`eme des mineurs. Preuve (le th´eor`eme des mineurs est un corollaire de la conjecture) Pour la classe des graphes connexes Par l’absurde, supposons la conjecture v´erifi´ee mais pas le th´eor`eme des mineurs. Comme nous l’avons d´ej`a vu, il n’y a jamais de suite infinie strictement d´ecroissante pour la relation 4. Par cons´equent, c’est qu’il existe une anti-chaˆıne infinie de graphes finis {G0, G1, . . .} (d´enombrable car l’ensemble des graphes finis est lui-mˆeme d´enombrable). Posons alors G = S i∈N Gi (sans cr´eer d’arˆetes entre-eux). G est un mineur strict de lui-mˆeme donc il existe une s´equence non vide de transformations qui produisent finalement un graphe G0 isomorphe `a G. Comme il y a au moins une transformation, au moins un graphe Gi est modifi´e. Mais Gi ne peut avoir ´et´e supprim´e compl`etement. En effet, s’il l’avait ´et´e, alors comme il est pr´esent dans G’, et qu’il est connexe, c’est qu’il a ´et´e obtenu `a partir d’un autre graphe (et d’un seul car on est dans le cas connexe) de l’anti-chaˆıne, graphe dont il serait donc mineur, ce qui est impossible par hypoth`ese. Mais alors il a ´et´e r´eduit sans totalement disparaˆıtre. Chaque composante connexe d’apr`es r´eduction (et il y en a au moins une) ´etant isomorphe `a un graphe de l’anti-chaˆıne, graphe qui est donc un mineur de G, ce qui est absurde. D’o`u le r´esultat dans le cas connexe. 14/16 Beaux ordres et graphes Bastien Le Gloannec Et dans le cas g´en´eral On peut voir un graphe fini quelconque comme l’ensemble de ses composantes connexes, i.e. comme une partie finie de la classe des graphes connexes. Il n’est pas difficile alors de remarquer que l’on a H 4 G avec H et G non n´ecessairement connexes si et seulement si G a au moins autant de composantes connexes que H et il existe une injection de H vers G envoyant chaque composante connexe c 0 de H vers une composante c de G telle que c 0 4 c, autrement dit c 0 4 f(c 0 ) (on r´eduit alors G en H en supprimant toutes les composantes qui n’appartiennent pas `a l’image de f, et en r´eduisant chaque f(c 0 ) en la composante c 0 de H). La relation de mineur sur les graphes finis est donc exactement la relation induite par la relation de mineur pour la classe graphes connexes (qui est un bel ordre) sur l’ensemble de ses parties finies. Par lemme de Higman, on en d´eduit que la relation 4 est un bel ordre sur la classe des graphes quelconques. 5 Conclusion L’´etude des beaux ordres a permis d’´etablir un certain nombre de th´eor`emes int´eressants, notamment, pour ce qui est de l’informatique fondamentale, en combinatoire sur les mots et en th´eorie des graphes comme nous avons pu l’illustrer tout au long de ce rapport. Le th´eor`eme des mineurs de Robertson et Seymour, qui est probablement le plus gros r´esultat de la th´eorie des graphes en l’´etat actuel de l’art, se ram`ene ainsi `a d´emontrer que la relation de mineur est un bel ordre sur la classe des graphes finis. Sur des structures combinatoires peu contraintes comme les graphes ou les mots, les beaux ordres peuvent ˆetre vus comme une alternative faible `a des notions plus fortes telles que les bons ordres, omnipr´esents en th´eorie des ensembles. C’est finalement un moyen fructueux de ramener des probl`emes combinatoires `a des objets math´ematiques bien connus et ´etudi´es. R´ef´erences [1] R. Diestel. Graph theory. Springer, 2005. [2] JB Kruskal. Well-quasi-ordering, the tree theorem, and Vazsonyi’s conjecture. Transactions of the American Mathematical Society, pages 210–225, 1960. [3] J.B. Kruskal. The theory of well-quasi-ordering : A frequently discovered concept. J. Combinatorial Theory Ser. A, 13(3) :297–305, 1972. [4] K. Kuratowski. Sur le probleme des courbes gauches en topologie. Fund. Math, 15(27) :1–283, 1930. [5] L. Lov´asz. Graph minor theory. Bulletin-American Mathematical Society, 43(1) :75, 2006. [6] C. Nash-Williams. On well-quasi-ordering finite trees. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 59, 1963. 15/16 Beaux ordres et graphes Bastien Le Gloannec [7] B. Oporowski. A counterexample to Seymour’s self-minor conjecture. Journal of Graph Theory, 14(5), 1990. [8] Neil Robertson and Paul D. Seymour. Graph minors. i. excluding a forest. J. Comb. Theory, Ser. B, 35(1) :39–61, 1983. [9] Neil Robertson and Paul D. Seymour. Graph minors .xiii. the disjoint paths problem. J. Comb. Theory, Ser. B, 63(1) :65–110, 1995. [10] Neil Robertson and Paul D. Seymour. Graph minors. xx. wagner’s conjecture. J. Comb. Theory, Ser. B, 92(2) :325–357, 2004. 16/16

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Graham Higman

Graham Higman

 
 
Graham Higman
Graham Higman.jpg

G. Higman en 1960

Naissance
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Formation
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Membre de
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membre de la Royal Society (d) +

Graham Higman (né le , mort le ) est un mathématicien britannique connu pour ses contributions à la théorie des groupes. Il est connu notamment pour le lemme de Higman qui donne une propriété sur la notion de sous-mot, analogue au théorème de Kruskal.

Il a fondé le Journal of Algebra (en) dont il a été le rédacteur de 1964 à 1984.

 

 

Distinctions[modifier | modifier le code]

Annexes[modifier | modifier le code]

Articles connexes[modifier | modifier le code]

Lien externe[modifier | modifier le code]

Source : wikipedia

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Lemme de Higman

Lemme de Higman

 
 

En mathématiques, le lemme de Higman est un résultat de la théorie des ordres qui affirme que, pour un ensemble X muni d'un bel ordre, l'ensemble X^* des mots finis sur Xmuni de l'ordre sous-mot est également un bel ordre. C'est un cas particulier du théorème de Kruskal sur les arbres, qui se généralise à son tour en le théorème de Robertson-Seymour sur les graphes.

Ce lemme est dû à Graham Higman qui l'a publié en 19521.

Notes et références[modifier | modifier le code]

  1. (Higman 1952)

Bibliographie[modifier | modifier le code]

Lien externe[modifier | modifier le code]

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Théorème de Kruskal

Théorème de Kruskal

 
 
Page d'aide sur l'homonymie Pour l’article homonyme, voir Théorème de Kruskal-Katona

En mathématiques, le théorème des arbres de Kruskal est un résultat de théorie des graphes conjecturé en 1937 par Andrew Vázsonyi (en) et démontré indépendamment en 1960 par Joseph Kruskal et S. Tarkowski1, affirmant que l'ensemble des arbres étiquetés par un ensemble muni d'un bel ordre est lui-même muni d'un bel ordre. Ce théorème est un cas particulier du théorème de Robertson-Seymour, dont il a constitué une des motivations.

En utilisant ce théorème, Harvey Friedman a pu définir des entiers « incompréhensiblement grands »2, qu'il a utilisé pour obtenir des résultats nouveaux d'indécidabilité.

 

 

Définitions préliminaires[modifier | modifier le code]

En théorie des graphes, un arbre est un graphe non orienté, acyclique et connexe ; on obtient un arbre enraciné en fixant l'un des sommets, qu'on appelle la racine de l'arbre. Enthéorie des ensembles, on définit une autre notion d'arbre, à partir d'une relation symétrique ; on démontre3 que, dans le cas fini, ces deux notions coïncident, et que tout arbre enraciné correspond à un ordre partiel unique défini sur l'ensemble des sommets, tel que tout sommet admette un unique prédécesseur4, sauf la racine, qui n'en a aucun (les arêtes du graphe étant exactement celles reliant chaque sommet à son prédécesseur) ; c'est cette représentation qui va servir à définir les applications qui font l'objet du théorème de Kruskal.

On dit qu'un arbre est étiqueté par un ensemble d'étiquettes X si on a défini une application x de l'ensemble des sommets de l'arbre vers X, autrement dit si on attache au sommet s l'étiquette x(s).

On dit qu'une application f entre ensembles partiellement ordonnés finis respecte les minorants si a= inf (b, c) entraîne f(a)= inf (f(b), f (c)), où inf(x, y) =z désigne la borne inférieure de x et y, c'est-à-dire le plus grand élément qui soit ≤ à x et à y ; on voit aisément que cela implique que f est strictement croissante, autrement dit que a<b entraînef(a)<f(b) et que f est une injection. Pour des arbres étiquetés par un ensemble X lui-même muni d'un ordre partiel noté ≤, on définit une notion de morphisme : une application fest un morphisme si elle respecte les minorants, et si elle respecte l'ordre des étiquettes, autrement dit si, pour tout sommet s du premier arbre, on a scriptstyle x(s)le x(f(s)). La relation « il existe un morphisme de A vers B » est une relation d'ordre partiel sur l'ensemble des arbres étiquetés, considérés à isomorphisme près5 (si l'on ne considère pas les arbres isomorphes comme identiques, la relation n'est plus antisymétrique, et on obtient seulement un préordre6) ; pour des arbres non étiquetés, on démontre que s'il existe un morphisme entre A et B, A est un mineur topologique de B.

Un ordre partiel (ou même un préordre) est appelé un bel ordre s'il ne contient aucune suite infinie strictement décroissante, ni aucune antichaîne infinie (définition qui généralise aux ordres partiels la notion de bon ordre définie pour les ensembles totalement ordonnés) ; c'est équivalent à dire que dans toute suite infinie d'éléments de l'ensemble x_1,x_2,dots,x_n,dots, il existe deux éléments x_i et x_j tels que i<j et x_ile x_j.

Énoncé[modifier | modifier le code]

Ces définitions permettent de formuler rigoureusement7 le

Théorème de Kruskal — Soit S un ensemble d'arbres étiquetés par un ensemble X d'étiquettes muni d'un bel ordre. La relation de préordre sur S : « scriptstyle Ale B si et seulement si il existe un morphisme de A vers B », est alors également un bel ordre.

L'existence d'une suite infinie strictement décroissante étant évidemment impossible (puisque, si A ≤ B et si A n'est pas isomorphe à B, A contient moins d'arêtes que B, ou des étiquettes plus petites), ce théorème revient donc à affirmer qu'il n'y a pas d'antichaînes infinies, c'est-à-dire d'ensemble infini d'arbres deux à deux incomparables par la relation ≤ (il convient cependant de remarquer qu'il existe des antichaînes finies aussi grandes que l'on veut).

Cas particuliers et généralisation[modifier | modifier le code]

Le lemme de Higman est un cas particulier de ce théorème, dont il existe de nombreuses généralisations, pour des arbres munis d'un plongement dans le plan, des arbres infinis, etc. Une généralisation bien plus puissante, concernant les graphes quelconques, est donnée par le théorème de Robertson-Seymour.

Les résultats d'indécidabilité de Friedman[modifier | modifier le code]

Harvey Friedman a remarqué8 que certains cas particuliers du théorème de Kruskal peuvent être énoncés dans l'arithmétique du premier ordre (la logique du premier ordrecorrespondant aux axiomes de Peano), mais que cette théorie est trop faible pour les démontrer, alors qu'ils se démontrent aisément en utilisant l'arithmétique du second ordre (en). Un exemple analogue est donné par le théorème de Goodstein, mais pour démontrer les énoncés de Friedman, une portion significativement plus grande de l'arithmétique du second ordre doit être utilisée9.

Soit P(n) l'affirmation

Il existe un m tel que si T1,...,Tm est une suite finie d'arbres (non étiquetés), avec Tk ayant (pout tout k) k+n sommets, alors il existe un couple (i,j) tel que i < j et Ti ≤ Tj.

Cette affirmation est un cas particulier du théorème de Kruskal, où la taille du premier arbre est fixée, et où la taille des arbres croît au plus petit rythme possible ; on dit souvent qu'il s'agit d'une forme finie du théorème de Kruskal.

Pour chaque n, les axiomes de Peano permettent de démontrer P(n), mais ces axiomes ne permettent pas de démontrer que « P(n) est vrai quel que soit n »10. De plus, la plus courte démonstration de P(n) a une longueur grandissant extrêmement vite en fonction de n, beaucoup plus vite que les fonctions récursives primitives ou que la fonction d'Ackermann par exemple.

Friedman a également utilisé la forme finie suivante du théorème de Kruskal pour les arbres étiquetés (avec des étiquettes non ordonnées), forme paramétrée, cette fois, par le nombre d'étiquettes :

Pour tout n, il existe un m tel que si T1,...,Tm est une suite finie d'arbres dont les sommets sont étiquetés parn symboles, chaque Ti ayant au plus i sommets, alors il existe un couple (i,j) tel que i < j et Ti ≤ Tj.

Dans ce cas, la relation ≤ signifie qu'il existe une application préservant les minorants, et envoyant chaque sommet sur un sommet ayant la même étiquette ; en théorie des graphes, ces applications sont souvent appelées des plongements.

Ce dernier théorème affirme l'existence d'une fonction à croissance rapide, que Friedman a nommée TREE, telle que TREE(n) est la longueur de la plus longue suite d'arbres àn étiquettes T1, ..., Tm dans laquelle chaque Ti a au plus i sommets, et telle qu'aucun arbre n'est plongeable dans un arbre ultérieur.

Les premières valeurs de TREE sont TREE(1) = 1, TREE(2) = 3, mais soudain TREE(3) explose à une valeur si gigantesque que la plupart des autres « grandes » constantes combinatoires, comme le nombre de Graham, sont ridiculement petites en comparaison. Ainsi, Friedman a défini (pour un autre problème plus simple) une famille de constantesn(k), et a montré que n(4) était beaucoup plus petit que TREE(3)2. Or n(4) est minoré par A(A(...A(1)...)), où le nombre de A est A(187196), A() étant une variante de la fonction d'Ackermann définie par : A(x) = 2↑↑...↑x avec un nombre de flèches de Knuth ↑ égal à x-1. À titre de comparaison, le nombre de Graham est de l'ordre de A64(4) ; pour mieux voir à quel point ce nombre est petit par rapport à AA(187196)(1), se reporter à l'article Hiérarchie de croissance rapide. Plus précisément, dans les notations de cette hiérarchie, on peut montrer que la vitesse de croissance de la fonction TREE est supérieure à celle de fΓ0, où Γ0 est l'ordinal de Feferman-Schütte, ce qui montre au passage à quel point cette fonction croît plus vite que la fonction de Goodstein, qui ne croît que comme fε0.

Tous ces résultats ont pour conséquence que les théorèmes précédents (tels que le fait que TREE soit une application, c'est-à-dire soit définie pour tout n) ne peuvent être démontrés que dans des théories assez fortes11 ; plus précisément, la force d'une théorie est mesurée par un ordinal (celui de l'arithmétique de Peano, par exemple, étant ε0), et des théorèmes ayant pour conséquence l'existence de fonctions croissant trop vite (plus rapidement que fε0 dans le cas des axiomes de Peano) ne peuvent être démontrés dans ces théories. Comme leur négation ne peut évidemment pas y être démontrée non plus (en supposant que la théorie où l'on a démontré ces théorèmes est cohérente), il en résulte que, par exemple dans l'arithmétique du premier ordre, ces théorèmes sont indécidables, ce qui a d'importantes conséquences métamathématiques, ces formes d'indécidabilité étant ressenties comme beaucoup plus naturelles que celles correspondant au théorème de Gödel11.

L'ordinal qui mesure la force du théorème de Kruskal est le petit ordinal de Veblen (en) (lequel est beaucoup plus grand que Γ0)12 ; il en résulte que l'on peut, par des constructions analogues à celles de Friedman, obtenir grâce à ce théorème des fonctions croissant plus vite que toute fα de la hiérarchie de croissance rapide, où α est un ordinal plus petit que l'ordinal de Veblen.

Notes[modifier | modifier le code]

  1. Kruskal 1960 ; une preuve courte en fut obtenue par Crispin Nash-Williams trois ans plus tard (Nash-Williams 1963)
  2. a et b Friedman décrit ces entiers comme « incompréhensiblement grands » ; n(p) reste cependant plus petit que TREE(3), même pour des valeurs énormes de p, telles que le nombre de Graham ; on trouvera une analyse plus serrée de ces encadrements dans ces notes de conférence [archive] (en), et des calculs plus précis des premières valeurs de n(k) dans cet autre article de Friedman [archive] (en) ; enfin, une estimation déjà moins imparfaite de TREE(3) figure sur cette page [archive] de MathOverflow.
  3. C. Berge, Graphes et hypergraphes, chapitre 3
  4. On dit que a est un prédécesseur de b (pour la relation d'ordre partiel <) si a<b et s'il n'existe aucun c tel que a<c<b.
  5. Elle est en effet réflexive (en utilisant le morphisme identité), et transitive (en composant les morphismes) ; de plus, s'il existe des morphismes de A vers B et de B vers A, A et B sont isomorphes.
  6. Voir par exemple N. Bourbaki, Éléments de mathématique : Théorie des ensembles [détail des éditions], ch. III, § 1, n° 2, p. 3, pour les définitions et les premières propriétés des ordres partiels. p. 3 pour les définitions et les premières propriétés des ordres partiels et des préordres
  7. On trouvera une présentation plus formalisée encore dans l'exposé de Jean Gallier [archive] (en anglais), dont cette section est largement inspirée ; toutefois, il réécrit la présentation initiale des théorèmes dans le langage des tree domains, ce qui peut demander un certain effort au lecteur non spécialiste...
  8. Friedman 2002
  9. En terme d'ordinaux, le théorème de Goodstein demande une récurrence jusqu'à l'ordinal varepsilon_0, alors que la fonctionTREE demande au moins l'ordinal de Feferman-Schütte, comme exposé plus loin.
  10. Voir l'article ω-cohérence (en) pour plus de détails sur d'autres situations de ce type.
  11. a et b Voir, par exemple, les analyses de Gallier 1991.
  12. On trouvera une description constructive de cet ordinal, sous forme d'un bon ordre explicite entre arbres finis, dans cet article de H. R. Jervell [archive] (en) (des dessins beaucoup plus nombreux d'arbres, avec les ordinaux correspondants, figurent dans ce document écrit par David Madore [archive] (en) [PDF]), et une démonstration du résultat lui-même dans cet article de Rathjen et Weiermann [archive]

Références[modifier | modifier le code]

  • (en) Harvey Friedman, Internal finite tree embeddings. Reflections on the foundations of mathematics (Stanford, CA, 1998), Urbana, IL, ASL, coll. « Lect. Notes Log. » (no 15),‎ , p. 60-91
  • (en) Jean H. Gallier, « What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory », Ann. Pure Appl. Logic, vol. 53, no 3,‎ ,p. 199-260 lien Math Reviews (texte intégral sous forme de trois documents PDF : partie 1 partie 2 partie 3).
  • (en) Stephen Simpson, « Nonprovability of certain combinatorial properties of finite trees », dans Harvey Friedman's Research on the Foundations of Mathematics, North-Holland, coll. « Studies in Logic and the Foundations of Mathematics »,‎ , p. 87-117

Voir aussi[modifier | modifier le code]

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The Cameron–Erd˝os Conjecture Ben Green1 Abstract

Voir le pdf : http://arxiv.org/pdf/math/0304058.pdf

 

arXiv:math/0304058v1 [math.NT] 4 Apr 2003 The Cameron–Erd˝os Conjecture Ben Green1 Abstract A subset A of the integers is said to be sum-free if there do not exist elements x,y,z ∈ A with x+y = z. It is shown that the number of sum-free subsets of {1,... ,N} is O(2N/2 ), confirming a well-known conjecture of Cameron and Erd˝os. 1. Introduction. If A is any subset of an abelian group then we say that A is sum-free if (A + A) ∩ A = ∅, that is if there do not exist x, y, z ∈ A for which x + y = z. The study of such sets goes back at least 30 years, and over 10 years ago Cameron and Erd˝os [4, 5] raised the question of enumerating the sum-free subsets of [N] = {1, . . . , N}. They noted that any set of odd integers is sum-free, as is any subset of {⌈N/2⌉, . . . , N}, but that it is hard to think of many sum-free sets which are not essentially of this form. Thus they advanced the following conjecture. Conjecture 1 (Cameron–Erd˝os) The number of sum-free subsets of [N] is O(2N/2 ). There has been some progress on this conjecture. Writing SF(N) for the collection of sum-free subsets of [N], Alon [1], Calkin [2] and Erd˝os and Granville (unpublished) showed independently that |SF(N)| = 2N/2+o(N) . (1) Results in a rather different direction were obtained by Freiman [7] and by Deshouillers, Freiman, S´os and Temkin [6]. In [7], for example, it was shown that the number of sum-free subsets of [N] with cardinality at least 5N/12 + 2 is at most O(2N/2 ). Let us also mention that Calkin and Taylor [3] showed that the number of subsets of [N] containing no solutions to x + y + z = w is O(22N/3 ), an estimate which is basically sharp. It is natural to ask for estimates for SF(Γ), the number of sum-free subsets of some finite abelian group Γ. When Γ = Z/pZ (p prime) this question is perhaps even more natural than the question of Cameron and Erd˝os. It was first considered explicitly by Lev and Schoen [12], who showed that |SF(Z/pZ)| ≤ 2 0.498p . Their result was improved by Ruzsa and the author [9], who obtained the estimate |SF(Z/pZ)| ≤ 2 p/3+o(p) . This is tight except for the o(p) term. For more general abelian groups Γ, work started by Lev, Luczak and Schoen [11] in the case |Γ| even was continued by Ruzsa and the author 1Supported by a Fellowship of Trinity College, Cambridge and a grant from the EPSRC, United Kingdom. Mathematics Subject Classification: 11B75. 1 [10], who obtained reasonably precise estimates for all abelian groups. The objective of the present paper is to prove the conjecture of Cameron and Erd˝os. Theorem 2 The number of sum-free subsets of [N] is asymptotically c(N)2N/2 , where c(N) takes two different constant values according as N is odd or even. It is extremely likely that our methods extend to give, for example, much tighter bounds on |SF(Z/pZ)| but we do not pursue such matters here. 2. A strategy for counting sum-free sets. The purpose of this section is to outline the broad strategy that we will use to count sum-free subsets of [N]. Our method falls conveniently into two parts, which are dealt with in detail in the two sections immediately following this one. We have tried to make these sections as independent as possible. Our strategy, then, is as follows. Part I. We find some family F of subsets of [N] with the following properties. Firstly, each A ∈ F is almost sum-free, meaning that the number of additive triples (triples with x+y = z) in A is o(N2 ). Secondly, F does not contain too many sets; in fact, |F| = 2o(N) . Finally, every sum-free subset of [N] is contained in some member of F. Part II. Given A ∈ SF(N), we consider some set A′ ∈ F with A ⊆ A′ . As F is so small, the number of A for which |A′ | ≤ 1 2 − 1 120  N is o(2N/2 ). If, however, |A′ | ≥ 1 2 − 1 120  N then it is possible to say something about the structure of A ′ , and hence about the structure of almost all A ∈ SF(N). What we will actually show is that almost all A ∈ SF(N) consist either entirely of odd numbers, or else are contained in the interval {⌈(N+1)/3⌉, . . . , N}. The author was delighted to discover that, in their original paper [4], Cameron and Erd˝os gave an elegant argument leading to an estimate for the number of sum-free subsets of {⌈(N + 1)/3⌉, . . . , N}. This argument, together with our work in the present paper, constitutes an affirmative solution to Conjecture 1. 3. Construction of F. Granularizations. In this section we complete Part I of the program outlined in §2 by constructing the family F. This was basically achieved in the paper of Ruzsa and the author [9]. Since it is not quite a trivial matter to isolate results from that paper in the form that we need them, we repeat some of the material from [9] here. We begin with a small amount of notation concerning Fourier transforms. We will be working on the group G = Z/pZ, where p is a prime. If f : G → C is a function and if r ∈ G then we define the Fourier transform ˆf by ˆf(r) = X x∈G f(x)e(rx/p) 2 where, as usual, e(θ) = e 2πiθ. If f, g are two functions then we define their convolution f ∗ g by (f ∗ g)(x) = X y∈G f(y)g(x − y). Observe that (f ∗ g)ˆ(r) = ˆf(r)ˆg(r). Finally, we remark that if A ⊆ G then we will identify A with its characteristic function: that is, we write A(x) = 1 if x ∈ A and A(x) = 0 otherwise. Observe that if A, B ⊆ G are two sets then (A ∗ B)(x) is the number of representations of x as a + b with a ∈ A, b ∈ B. Let p ∈ [2N, 4N] be a prime, let M be a positive integer, and let d ∈ (Z/pZ) ∗ . Let A ⊆ [N] be a set, and regard A as a subset of Z/pZ in the obvious manner. Suppose that |A| = αp. Consider a partition of Z/pZ into arithmetic progressions Ii , i ∈ Z/MZ, of common difference d defined by Ii = ( λd : ip M ≤ λ < (i + 1)p M ) , (2) where i denotes the least positive residue of i. Each of these progressions has length either L or L − 1, where L = ⌈p/M⌉. Let ǫ1 > 0 be a real number, and let T = {i ∈ Z/MZ | |A ∩ Ii | ≥ ǫ1|Ii |}. Finally, define the granularization A′ of A (with respect to the length d and the parameter ǫ1) by A ′ = [ i∈T Ii . It is easy to see that we have |A ′ A| ≤ ǫ1p. (3) One of the key results of [9] is that, provided d has a certain property (for which we will use the term “good length”) the set A′ retains some of the additive features of A. In fact, we will be able to show that A′ is almost sum-free. Let us now say what we mean by the statement “d is a good length”. Let ǫ2, ǫ3 > 0 be two further real numbers and set δ = 1 16 ǫ 2 1 ǫ2ǫ 1/2 3 α −1/2 . Let R, |R| = k, be the set of all r 6= 0 for which |Aˆ(r)| ≥ δp. We say that d is a good length for A (with respect to the parameters ǫ1, ǫ2, ǫ3) if kdr/pk ≤ 1 4L δp |Aˆ(r)| !1/2 (4) for all r ∈ R. The following proposition was (essentially) the main result of [9]. It clarifies the rˆole of ǫ2 and ǫ3, which have not so far featured. 3 Proposition 3 Suppose that d is a good length for A. Then the granularization A′ has the property that A + A contains all x for which A′ ∗ A′ (x) ≥ ǫ2p, with at most ǫ3p exceptions. Proof. We claim that if d is a good length then the function g(x) = 1 2L − 1 X L−1 j=−(L−1) e(jdx/p) (5) satisfies |Aˆ(x)||1 − g(x) 2 | ≤ δp (6) for all x. This automatically holds for x = 0, as g(x) = 1, and also whenever |Aˆ(x)| ≤ δp, since g(x) ∈ [−1, 1]. For any x ∈ Z/pZ we may estimate 1 − g(x) as follows. Writing ktk for the distance of t from the nearest integer we have the inequality 1 − cos 2πt ≤ 2π 2ktk 2 . It follows that 1 − g(x) = 2 2L − 1 X L−1 j=1  1 − cos 2πjdx p  ≤ 4π 2 2L − 1 X L−1 j=1 jdx p 2 ≤ 4π 2 2L − 1 dx p 2 X L−1 j=1 j 2 ≤ 2π 2L 2 3 dx p 2 . (7) Hence |Aˆ(x)||1 − g(x) 2 | ≤ 2|Aˆ(x)||1 − g(x)| ≤ 14L 2 kdx/pk 2 |Aˆ(x)| It is now easy to see that d being a good length is exactly the property required to make (6) hold. Now to establish the proposition we define a function a1 by a1(n) = 1 |P| (A ∗ P)(n) = 1 |P| |A ∩ (P + n)|, where P = {−(L − 1)d, . . . , 0, d, 2d, . . . ,(L − 1)d}. Observe that ˆa1(x) = Aˆ(x)g(x). Thus we 4 have, by two applications of Parseval’s identity, that X n |(A ∗ A)(n) − (a1 ∗ a1)(n)| 2 = p −1X x Aˆ(x) 2 − aˆ1(x) 2 2 = p −1X x |Aˆ(x)| 4 1 − g(x) 2 2 ≤ p −1  sup x |Aˆ(x)||1 − g(x) 2 | 2 X x |Aˆ(x)| 2 = αp  sup x |Aˆ(x)||1 − g(x) 2 | 2 . (8) (6) therefore implies that X n |(A ∗ A)(n) − (a1 ∗ a1)(n)| 2 ≤ αδ2 p 3 . (9) Now if n ∈ A′ then there is a progression of common difference d and length L containing n which contains at least ǫ1L/2 points of A. This progression is contained in [n − (L − 1)d, . . . , n + (L − 1)d]. Hence a1(n) is certainly at least ǫ1/4, and so a1(n) ≥ ǫ1A(n)/4 for all values of n. It follows immediately that (a1 ∗ a1)(n) ≥ ǫ 2 1 (A′ ∗ A′ )(n)/16 for all n, and hence that if A′ ∗ A′ (n) ≥ ǫ2p then a1 ∗ a1(n) ≥ ǫ 2 1 ǫ2p/16. We are to show that there are not many points n for which this is true whilst A ∗ A(n) = 0. Letting B denote the set of these “bad” points, observe that n ∈ B implies that |(A ∗ A)(n) − (a1 ∗ a1)(n)| 2 ≥ ǫ 4 1 ǫ 2 2p 2 256 . Substituting into (9) gives the bound |B| ≤ 256αδ2 ǫ 4 1 ǫ 2 2 p ≤ ǫ3p (this explains our choice of δ). We defer for a while the issue of whether there are any good lengths. Our next result says that the conclusion of Proposition 3 is enough to guarantee that if A is sum-free then A′ is almost sum-free. Proposition 4 Suppose that A is sum-free. Let ǫ > 0, set ǫ1 = ǫ, ǫ2 = ǫ 2 /144, ǫ3 = ǫ 2 /80 and let A′ be the granularization of A with respect to some good length d. Then A′ contains at most ǫp2 triples (x, y, z) with x + y = z. 5 Proof. The choice of p (that is, p ≥ 2N) guarantees that A is sum-free when considered as a subset of Z/pZ. Suppose without loss of generality that d = 1, and suppose for a contradiction that the proposition is false. Recall the notation we set up at the start of the section, particularly the definitions of the intervals Ii and the set T ⊆ Z/mZ. We begin by claiming that there are at least ǫM2/4 triples (i, j, k) ∈ T 3 for which i + j = k or k + 1. Indeed note that if x+y = z and if x ∈ Ii , y ∈ Ij and z ∈ Ik then i+j = k or k +1. However for a fixed triple (i, j, k) with this property there are at most 4p 2/M2 triples (x, y, z), so our claim follows from a simple double count. For definiteness suppose that there are at least ǫM2/8 triples (i, j, k) ∈ T 3 with i + j = k (the argument when there are many triples with i + j = k + 1 is very similar). Let K be the set of all k ∈ T for which T ∗ T(k) ≥ ǫM/16 so that, by an easy averaging argument, we have |K| ≥ ǫM/16. Suppose that i + j = k with i, j, k ∈ T, and suppose that z lies in the middle (1 − ǫ/2) of Ik. Then the number of representations of z as x + y with x ∈ Ii , y ∈ Ij is at least ǫp/8M. Therefore if k ∈ K we have A′ ∗A′ (z) ≥ ǫ 2p/144. Note, however, that since k ∈ T the middle (1 − ǫ/2) of Ik contains at least ǫp/4M points of A. We have now shown that there are at least ǫ 2 p/64 elements x ∈ A for which A ′ ∗ A ′ (x) ≥ ǫ 2p/144. By the property of A′ described in Proposition 3 we see that A + A contains an element of A, contrary to our assumption that A is sum-free. We now look at the issue of finding a good length. Proposition 5 Let A ⊆ Z/pZ have cardinality αp. A good length for A with parameters ǫ1, ǫ2, ǫ3 exists if p > (4L) 256α 2 ǫ −4 1 ǫ −2 2 ǫ −1 3 . (10) Proof. It follows by a standard application of the pigeonhole principle that a d satisfying (4) exists if p > (4L) k Y r∈R |Aˆ(r)| δp !1/2 . (11) We claim that this inequality is a consequence of the hypothesis on p, L, ǫ1, ǫ2 and ǫ3 in the statement of the proposition. Indeed, observe that Parseval’s identity implies that X r∈R |Aˆ(r)| 2 ≤ αp2 , (12) from which the arithmetic-geometric mean inequality gives Y r∈R |Aˆ(r)| ≤  αp2 k k/2 . It follows that the right side of (11) is at most (4Lα1/4 δ −1/2 k −1/4 ) k , (13) 6 which is an increasing function of k in the range k <  256L4 e  α δ 2 . However another consequence of (12) is the inequality k < α/δ2 , and hence (13) is itself bounded above by (4L) α/δ2 . Recalling our choice of δ confirms the claim, and hence there is a d for which (4) holds. To get the conclusion of Proposition 4 we required ǫ1 = ǫ, ǫ2 = ǫ 2 /144 and ǫ3 = ǫ 2 /80. It is an easy but slightly tedious task to check that if we put ǫ = (log N) −1/11 and M =  N exp(−(log N) 1/12)  then, at least for N sufficiently large, A has at least one good length. For the remainder of the section we assume that the parameters ǫ and M take these values. We are now in a position to define our family of sets F. Take F to consist of all sets which can be formed in the following manner. For all d ∈ (Z/pZ) ∗ consider the decomposition (2) of Z/pZ into progressions Ii (i ∈ Z/mZ) with common difference d. Let G be the collection of sets which are unions of progressions Ii , for some d. Now throw away from G all those sets which have more than ǫp2 additive triples, giving a new collection H. Finally, let F consist of all subsets of [N] which can be obtained by adding at most ǫp elements to some H ∩ [N], H ∈ H. This may seem complicated. It turns out, however, that we can rather easily establish the following rather clean proposition concerning F which contains all the information we need for subsequent sections. Proposition 6 The family F has the following properties: (i) Every member of F has at most o(N2 ) additive triples; (ii) If A is sum-free then A is contained in some member of F; (iii) |F| ≤ 2 o(N) . Proof. (i) By definition every set in H has at most ǫp2 additive triples, and thus the same is true of sets of the form H ∩ [N], H ∈ H. By adding ǫp elements to such an H, we cannot create more than 3ǫp2 new additive triples. The result follows from the fact that p ≤ 4N. (ii) Set ǫ1 = ǫ, ǫ2 = ǫ 2/144 and ǫ3 = ǫ 2/80. Choose a good length d for A with respect to ǫ1, ǫ2, ǫ3, and consider the granularization A′ with respect to d and ǫ1. By Proposition 4 this lies in H, and the result follows from (3). (iii) There are p −1 choices for d, and then 2M ways to pick elements of G. Thus |H| ≤ p2 M, and so |F| is at most p2 M times the number of subsets of [N] of size at most ǫN. This is clearly 2o(N) . 4. The structure of almost sum-free sets. In this section we study large almost sumfree sets. The results may be regarded as “almost” versions of the results of Freiman [7]. Freiman’s methods do not seem to generalise easily to almost sum-free sets, so we have been forced to devise our own arguments. We will need one further piece of notation. If K is a positive real number and if A ⊆ G is a subset of an abelian group, we will write D(A, K) for the set of all x ∈ G which have at least K representations as a − a ′ with a, a′ ∈ A. We call 7 this the set of K-popular differences of A. In this section the objects Ii , M and ǫ are not the same as in the previous section. Proposition 7 Let ǫ = o(N) and suppose that A ⊆ [N] has at most ǫN2 additive triples, and that |A| = ( 1 2 − η)N where η ≤ 1/50 (η is allowed to be negative). Then one of the following alternatives occurs: (i) With the possible exception of at most 32ǫ 1/8N elements, A is contained in some interval of length ( 1 2 + 3η + 60ǫ 1/8 )N; (ii) At most 54ǫ 1/8N elements of A are even. Throughout what follows we shall assume that |A| = ( 1 2 − η)N, η ≤ 1/50 and that A has at most ǫN2 additive triples. Lemma 8 We have 1 2 |D(A, ǫ1/2N)| + |A| ≤ N(1 + 2ǫ 1/2 ). (14) Proof. We have D(A, ǫ1/2N) ∩ Z>0  ∩ A ≤ ǫ 1/2N, or else A would contain more than ǫN2 additive triples. The result follows quickly from this and the observation that d is K-popular if and only if −d is. Lemma 9 For all but at most 8ǫ 1/4N values of a, at least |A| − 16ǫ 1/4N of the differences a − a ′ with a ′ ∈ A lie in D(A, 32ǫ 1/2N). Proof. Consider the graph on vertex set A in which a is joined to a ′ if a − a ′ is not in D(A, 32ǫ 1/2N). It has at most 64ǫ 1/2N2 edges. The number of vertices with degree more than 16ǫ 1/4N is thus at most 8ǫ 1/4N. The next lemma, which is an application of basic graph theory, is [11], §4, Proposition 1. We specialise the result to the case we need. Lemma 10 (Lev, Luczak, Schoen) Let S be a subset of an abelian group Γ, |Γ| ≤ N. Suppose that |D(S, 8ǫ 1/2N)| ≤ 2|S| − 16ǫ 1/4N. Then there is a set X ⊆ S, |S X| ≤ 4ǫ 1/4N, with X − X ⊆ D(S, 8ǫ 1/2N). Now partition [N] into intervals Ii such that the smallest element of Ii is ⌊2iǫ1/8N⌋. Let j be minimal so that Ij contains at least 9ǫ 1/4N points of A. Then, by Lemma 9, there is some m ∈ Ii such that at least |A| − 16ǫ 1/4N of the differences a − m, a ∈ A, lie in D(A, 32ǫ 1/2N). Let k be maximal so that Ik contains at least 9ǫ 1/4N points of A. Again, there is M ∈ Ik so that at least |A| − 16ǫ 1/4N of the differences a − m, a ∈ A, lie in D(A, 32ǫ 1/2N). Clearly for at least |A| − 32ǫ 1/4N values of a both a − m and a − M are popular. Furthermore |A ∩ [1, m]| ≤ 9ǫ 1/4N ǫ 1/8 ≤ 9ǫ 1/8N, and a similar inequality holds for |A ∩[M, N]|. Thus there is a set B ⊆ A, with the following properties: 8 (i) B is contained in {m + 1, . . . , M}; (ii) |B| ≥ |A| − 50ǫ 1/8N; (iii) For all b ∈ B the differences b − m and b − M are both in D(A, 32ǫ 1/2N). Observe that the first and second of these points imply that M − m > N/4. Now let t = M − m. Following Lev and Smeliansky [13], consider the projection map π : Z → Z/tZ. We note some simple facts about this map in a lemma. Lemma 11 (i) |π(A)| ≥ |A| − 50ǫ 1/8N; (ii) Let δ > 0. If d ∈ D(A, 4δN) then π(d) ∈ D(π(A), δN); (iii) If d ∈ D(π(A), 8δN) then some element of π −1 (d) lies in D(A, δN). Proof. (i) Clearly |π(A)| ≥ |π(B)| = |B|. (ii) If d = a−a ′ then π(d) = π(a)−π(a ′ ). For different representations of d as a−a ′ , certain of these representations of π(d) may be the same. However, since t > N/4, no element of Z/tZ has more than 4 preimages under π which lie in A. The result follows. (iii) If π(d) = π(ai) − π(a ′ i ) then ai − a ′ i = d + λit for some λi ∈ Z. As t > N/4 and −N < ai − a ′ i < N there are at most 8 possible values for λi . Thus for at least one value of i there are δN solutions to ai − a ′ i = d + λit. It is immediate from part (iii) of this lemma that |D(A, δN)| ≥ |D(π(A), 8δN)|. However we can do better than this, since for several d at least two of the elements π −1 (d) are popular differences. Indeed, for any b ∈ B we have b−m, b−M ∈ D(A, 32ǫ 1/2N), but b−m ≡ b−M (mod t). Certainly π(b − m) ∈ D(π(A), 8ǫ 1/2N) by Lemma 11(ii). Thus, by Lemma 11(iii), we have |D(A, ǫ1/2N)| ≥ |D(π(A), 8ǫ 1/2N)| + |B| ≥ |D(π(A), 8ǫ 1/2N)| + |A| − 50ǫ 1/8N. (15) Combining this with (14) gives |D(π(A), 8ǫ 1/2N)| ≤ 2N(1 + 30ǫ 1/8 ) − 3|A|. (16) Now we must have |D(π(A), 8ǫ 1/2N)| ≤ 2|π(A)| − 16ǫ 1/4N, since otherwise (16) and Lemma 11(i) would give |A| ≤ ( 2 5 + 100ǫ 1/8 )N, which is contrary to our assumption about |A|. Thus Lemma 10 applies, and we may pass to a subset X ⊆ π(A) with |X| ≥ |A| − 54ǫ 1/8N (17) and X − X ⊆ D(π(A), 8ǫ 1/2N). (18) We distinguish three further cases. 9 Case 1. |X| ≥ t/2. Then X − X is all of Z/tZ, and so (16) and (18) yield t ≤ 1 2 + 3η + 60ǫ 1/8  N. (19) But we know that, with at most 18ǫ 1/8N exceptions, the elements of A lie in the interval {m + 1, . . . , M} which has length t. This is alternative (i) of Proposition 7. Case 2. |X − X| ≥ 2|X| − t/3. Then (16),(17) and (18) give |A| ≤ ( 7 15 + 40ǫ 1/8 )N, contrary to assumption. Case 3. |X − X| < 2|X| − t/3. Then, by Kneser’s theorem on the addition of sets in abelian groups (see [14], Theorem 4.2), X−X is a union of cosets of some subgroup H ≤ Z/tZ of index 2. Thus t is even and π −1 (X) consists of integers of just one parity. That is, either at least |A|−54ǫ 1/8N elements of A are odd, or else at least that many are even. The latter possibility is, however, easily excluded; any subset of {2, 4, 6, . . . , 2⌊N/2⌋} of cardinality at least 12N/25 contains at least N2/100 additive triples. This concludes the proof of Proposition 7. An immediate corollary of Propositions 6 and 7 is the following result of Alon [1], Calkin [2] and Erd˝os and Granville (unpublished). Proposition 12 (Alon,Calkin,Erd˝os–Granville) |SF(N)| = 2N/2+o(N) . Proof. It follows from Proposition 7 that if F ⊆ [N] has o(N2 ) triples then |F| ≤ ( 1 2+o(1))N. The result now follows from Proposition 6. A much more important corollary for us will be the following description of almost all sum-free subsets of [N]. Corollary 13 With o(2N/2 ) exceptions, all sum-free subsets of [N] consist entirely of odd numbers, or else are contained in {⌈(N + 1)/3⌉, . . . , N}. Proof. Let A ∈ SF(N), and let F ∈ F contain A. The number of A for which |F| ≤ 1 2 − 1 120  N is certainly o(2N/2 ), so suppose that |F| ≥ 1 2 − 1 120  N. Proposition 7 then applies. Suppose first of all that alternative (ii) of that proposition holds, so that A contains o(N) even numbers. Suppose that A contains at least one even number, t say. If t < N/2 then we may select ⌊N/8⌋ disjoint pairs (x, x + t) of odd numbers, and A cannot contain both of the elements of any of them since it is sum-free. The number of choices for A is thus no more than 2N/4+o(N) 3 N/8 = o(2N/2 ). If t ≥ N/2 then a very similar argument applies with pairs (x, t − x). Thus all but o(2N/2 ) of the sum-free sets with o(N) even numbers consist entirely of odd numbers. Now suppose that alternative (i) of Proposition 7 holds. If A ∩ 1 − 1 120  N, N ≤ 32ǫ 1/8N (20) 10 then, using the fact that A ∩ [1, 1 − 1 120  N] is sum-free together with Theorem 12, we see that there are just o(2N/2 ) possibilities for A. Suppose, then, that (20) fails to hold. Since Proposition 7, (i), holds we infer that A is contained in the interval [ 1 2 − 1 30 − 256ǫ 1/8  N, N] with the exception of at most 32ǫ 1/8N elements. Suppose that A contains some element t ∈ {1, . . . , ⌊(N + 1)/3⌋}. Then we may select ⌊N/12⌋ − 4 totally disjoint pairs (x, x + t) with x ≥ N/2, and A can contain at most one element from each of them. This means that the number of choices for A is no more than 3N/122 ( 1 3 + 1 30 +o(1))N which, it can be checked, is o(2N/2 ). As we remarked in the introduction, Cameron and Erd˝os [4] addressed the issue of counting sum-free subsets of {⌈(N + 1)/3⌉, . . . , N}. They discovered that the number of such sets is asymptotically c(N)2N/2 , where c(N) takes two different constant values depending on whether N is odd or even. Combining their result with the work of this paper, then, leads to Theorem 2. Concluding remarks. The paper [4] of Cameron and Erd˝os can be hard to locate and so we have written up their argument and posted it on the web [8]. The author would like to thank Imre Ruzsa for the many conversations which led to the papers [9, 10] and which, naturally, have had a significant bearing on the present work. References [1] Alon, N., Independent sets in regular graphs and sum-free subsets of abelian groups, Israel Jour. Math. 73 (1991) 247 – 256. [2] Calkin, N.J.,On the number of sum-free sets, Bull. London Math. Soc. 22 (1990), no. 2, 141–144. . [3] Calkin, N. J., Taylor, A. C. Counting sets of integers, no k of which sum to another, J. Number Theory 57 (1996), no. 2, 323–327. [4] Cameron, P.J; Erd˝os, P. On the number of sets of integers with various properties, Number theory (Banff, AB, 1988), 61–79, de Gruyter, Berlin, 1990. [5] Cameron, P.J and Erd˝os, P. Notes on sum-free and related sets, Recent trends in combinatorics (M´atrah´aza, 1995), 95–107, CUP, Cambridge, 2001. [6] Deshouillers, J-M; Freiman, G. A; S´os, V; Temkin, M; On the structure of sum-free sets. II, Structure theory of set addition. Ast´erisque 258, (1999), xii, 149–161. [7] Freiman, G. A. On the structure and the number of sum-free sets, Journ´ees Arithm´etiques, 1991 (Geneva). Ast´erisque 209, (1992), 13, 195–201. 11 [8] Green, B.J. Notes on an argument of Cameron and Erd˝os, available at: http://www.dpmms.cam.ac.uk/˜bjg23/papers/ce.pdf. [9] Green, B.J. and Ruzsa, I.Z. Counting sumsets and sum-free sets in Z/pZ, preprint. [10] Green, B.J. and Ruzsa, I.Z. Counting sum-free sets in abelian groups, preprint. [11] Lev, V.F., Luczak, T. and Schoen, T. Sum-free sets in abelian groups, Israel Jour. Math. 125(2001) 347 – 367. [12] Lev, V.F. and Schoen, T. Cameron-Erd˝os modulo a prime, Finite Fields Appl. 8 (2002), no. 1, 108–119. [13] Lev, V. F. and Smeliansky, P. Y. On addition of two distinct sets of integers, Acta Arith. 70 (1995), no. 1, 85–91. [14] Nathanson, M.B. Additive number theory: inverse problems and the geometry of sumsets, Graduate Texts in Mathematics 165, Springer-Verlag, New York 1996. Ben Green Trinity College, Cambridge, England. email: bjg23@hermes.cam.ac.uk 12

 

 

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The Book of Numbers Par John H. Conway,Richard Guy

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A Ramsey Problem on Hypercubes

A Ramsey Problem on Hypercubes

Consider the vertex set of the unit n-dimensional cube embedded in n-dimensional Euclidean space. Join each pair of vertices by a straight line segment, creating a complete graph. Color the edges of this complete graph in two colors. Graham and Rothschild proved that if n is sufficently large, there is a monochromatic complete graph of order 4 in some plane. They posed the problem of determining how large n must be. Their lower bound was 6 and their upper bound came to be known as Graham's number. Below we give links to colorings that improve the lower bound to 11. In the coloring matrices, we use colors 0 and 1.

 

The following link points to a coloring matrix for n=10 wherein 1/3 of the edges are bicolored. When an edge is bicolored, it can belong to cliques of either color. The matrix is here. Here, we use colors 1 and 2. An entry of 3 indicates a bicolored edge. This topic is discussed in a paper that appeared in the Journal of Discrete and Computational Geometry.

Geoff Exoo

 

Source : 

http://isu.indstate.edu/ge/GEOMETRY/cubes.html

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Peter Cameron (mathématicien)

Peter Cameron (mathématicien)

 
 
Page d'aide sur l'homonymie Pour les articles homonymes, voir Cameron.
Peter Cameron
Description de l'image PeterCameron.JPG.
Naissance (68 ans)
Toowoomba, Queensland(Australie)
Domicile Oxford
Nationalité Drapeau d'Australie Australien
Champs Mathématiques
Institutions Université d'Oxford
Queen Mary, University of London (en)
Diplôme Université du Queensland
Université d'Oxford
Distinctions Prix Whitehead (1979)
Médaille Euler (2003)

Peter Jephson Cameron (né le ) est un mathématicien australien qui a travaillé en théorie des groupes, encombinatoire, sur la théorie des codes, et la théorie des modèles. Il est actuellement professeur de mathématiques àQueen Mary, University of London (en).

Cameron a reçu un B.Sc. à l'université du Queensland et un D.Phil. en 1971 à l'université d'Oxford, sous la direction dePeter Neumann (de).

 

 

Œuvre[modifier | modifier le code]

Cameron s'est spécialisé en algèbre et en combinatoire ; il a écrit des livres sur la combinatoire, l'algèbre, les groupes de permutations, et la logique, et a produit plus de 250 articles1. Il a un nombre d'Erdős de 12. Son nom est attaché à laconjecture de Cameron–Erdős (en).

Bibliographie[modifier | modifier le code]

  • Permutation Groups, CUP, 1999
  • Combinatorics: topics, techniques, algorithms, CUP, 1994
  • Sets, logics and combinatorics, Springer 1999
  • Oligomorphic permutation groups, CUP, 1990 (ISBN 0521388368)
  • Introduction to Algebra, Oxford University Press 1989, 2. éd. 2008
  • avec Jacobus van Lint (de) : Graph theory, coding theory and block designs, CUP, 1975
  • avec Jacobus van Lint : Graphs, codes and designs, CUP, 1980
  • Parallelisms of complete designs, CUP, 1976

Distinctions[modifier | modifier le code]

Notes et références[modifier | modifier le code]

(en) Cet article est partiellement ou en totalité issu de l’article de Wikipédia en anglais intitulé « Peter Cameron » (voir la liste des auteurs).

  1. Recent publications of Peter J. Cameron [archive]
  2. The Erdös Number Project [archive]

Liens externes[modifier | modifier le code]

Source : wikipedia

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