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Conjecture de Dickson

 

 

En théorie des nombres, la conjecture de Dickson est une conjecture émise par Leonard Eugene Dickson, selon laquelle pour un ensemble fini de k suites arithmétiquesa₁ + nb₁, a₂ + nb₂, ..., a_k + nb_k avec b_i ≥ 1, il existe une infinité d'entiers positifs n pour lesquels les nombres correspondants sont tous premiers, excepté s'il existe une condition de congruence qui empêche cela (Ribenboim 1996, 6.I). Le cas k=1 est le théorème de Dirichlet.

Deux cas particuliers sont des conjectures célèbres et non résolues : l'existence d'une infinité de nombres premiers jumeaux (n et n+2 sont premiers), et d'une infinité denombres premiers de Sophie Germain (n et 2n+1 sont premiers).

La conjecture de Dickson a été par la suite généralisée par l'hypothèse H de Schinzel.

Références[modifier | modifier le code]

• (en) Cet article est partiellement ou en totalité issu de l’article de Wikipédia en anglais intitulé « Dickson's conjecture » (voir la liste des auteurs).
• (en) L. E. Dickson, A new extension of Dirichlet's theorem on prime numbers, vol. 33 : Messenger of mathematics, Macmillan and Co,‎ 1904, 155-161 p. (lire en ligne)
• (en) Paulo Ribenboim, The new book of prime number records, Berlin, New York, Springer-Verlag,‎ 1996 (ISBN 978-0-387-94457-9, lire en ligne)

Voir aussi[modifier | modifier le code]

Théorème de Green-Tao

• Portail de l’arithmétique et de la théorie des nombres

Catégories :

• Nombre premier
• Conjecture non résolue

| [+]

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
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44027,44029 #635
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46829,46831 #664
47057,47059 #665
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49031,49033 #689
49121,49123 #690
49169,49171 #691
49199,49201 #692
49277,49279 #693
49331,49333 #694
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51197,51199 #717
51239,51241 #718
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51347,51349 #720
51419,51421 #721
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51971,51973 #728
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52181,52183 #730
52289,52291 #731
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52541,52543 #733
52709,52711 #734
52859,52861 #735
52901,52903 #736
53087,53089 #737
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53171,53173 #739
53231,53233 #740
53267,53269 #741
53279,53281 #742
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53609,53611 #745
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53897,53899 #747
54011,54013 #748
54401,54403 #749
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56597,56599 #775
56711,56713 #776
56807,56809 #777
56891,56893 #778
56909,56911 #779
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57191,57193 #781
57221,57223 #782
57269,57271 #783
57329,57331 #784
57347,57349 #785
57527,57529 #786
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57899,57901 #789
58109,58111 #790
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58229,58231 #793
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58439,58441 #796
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58601,58603 #798
58787,58789 #799
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59219,59221 #805
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60089,60091 #812
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60257,60259 #815
60647,60649 #816
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60899,60901 #820
60917,60919 #821
61151,61153 #822
61331,61333 #823
61379,61381 #824
61469,61471 #825
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138449,138451 #1598
138569,138571 #1599
138797,138799 #1600
139121,139123 #1601
139199,139201 #1602
139301,139303 #1603
139367,139369 #1604
139457,139459 #1605
139589,139591 #1606
139661,139663 #1607
139967,139969 #1608
140069,140071 #1609
140417,140419 #1610
140549,140551 #1611
140627,140629 #1612
140681,140683 #1613
140729,140731 #1614
140759,140761 #1615
140837,140839 #1616
140867,140869 #1617
140891,140893 #1618
141179,141181 #1619
141221,141223 #1620
141497,141499 #1621
141509,141511 #1622
141677,141679 #1623
141707,141709 #1624
141767,141769 #1625
141851,141853 #1626
141959,141961 #1627
142097,142099 #1628
142157,142159 #1629
142589,142591 #1630
142607,142609 #1631
142697,142699 #1632
142757,142759 #1633
142787,142789 #1634
142871,142873 #1635
142979,142981 #1636
143111,143113 #1637
143261,143263 #1638
143501,143503 #1639
143567,143569 #1640
143651,143653 #1641
143831,143833 #1642
143879,143881 #1643
144071,144073 #1644
144161,144163 #1645
144167,144169 #1646
144407,144409 #1647
144479,144481 #1648
144539,144541 #1649
144887,144889 #1650
145007,145009 #1651
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
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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
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223007,223009 #2357
223049,223051 #2358
223061,223063 #2359
223217,223219 #2360
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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
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225161,225163 #2374
225221,225223 #2375
225287,225289 #2376
225341,225343 #2377
225347,225349 #2378
225371,225373 #2379
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225779,225781 #2385
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226199,226201 #2387
226379,226381 #2388
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226547,226549 #2390
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227111,227113 #2393
227189,227191 #2394
227231,227233 #2395
227471,227473 #2396
227531,227533 #2397
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227609,227611 #2399
227627,227629 #2400
227651,227653 #2401
228197,228199 #2402
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229751,229753 #2417
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230309,230311 #2423
230339,230341 #2424
230387,230389 #2425
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230771,230773 #2427
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231611,231613 #2438
231821,231823 #2439
231839,231841 #2440
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232187,232189 #2443
232409,232411 #2444
232457,232459 #2445
232709,232711 #2446
232751,232753 #2447
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233069,233071 #2449
233141,233143 #2450
233159,233161 #2451
233327,233329 #2452
233417,233419 #2453
233549,233551 #2454
233687,233689 #2455
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233939,233941 #2458
234191,234193 #2459
234317,234319 #2460
234341,234343 #2461
234461,234463 #2462
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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
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237161,237163 #2487
237689,237691 #2488
237857,237859 #2489
237971,237973 #2490
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238157,238159 #2493
238361,238363 #2494
238529,238531 #2495
238727,238729 #2496
238877,238879 #2497
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239231,239233 #2499
239387,239389 #2500
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239711,239713 #2502
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240047,240049 #2505
240257,240259 #2506
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240587,240589 #2508
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241067,241069 #2511
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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
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243431,243433 #2528
243587,243589 #2529
243671,243673 #2530
243701,243703 #2531
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244157,244159 #2533
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244379,244381 #2536
244637,244639 #2537
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244859,244861 #2540
245129,245131 #2541
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246611,246613 #2558
246641,246643 #2559
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247337,247339 #2564
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247649,247651 #2569
247691,247693 #2570
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249131,249133 #2582
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250049,250051 #2589
250499,250501 #2590
250739,250741 #2591
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251201,251203 #2597
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251231,251233 #2599
251261,251263 #2600
251429,251431 #2601
251609,251611 #2602
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251831,251833 #2605
251939,251941 #2606
251969,251971 #2607
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253607,253609 #2612
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308507,308509 #3069
308519,308521 #3070
308639,308641 #3071
308849,308851 #3072
308927,308929 #3073
309011,309013 #3074
309107,309109 #3075
309269,309271 #3076
309311,309313 #3077
309479,309481 #3078
309521,309523 #3079
309539,309541 #3080
309779,309781 #3081
309851,309853 #3082
309929,309931 #3083
310019,310021 #3084
310127,310129 #3085
310229,310231 #3086
310361,310363 #3087
310727,310729 #3088
310829,310831 #3089
311291,311293 #3090
311537,311539 #3091
311567,311569 #3092
311681,311683 #3093
311711,311713 #3094
311747,311749 #3095
311867,311869 #3096
312029,312031 #3097
312071,312073 #3098
312197,312199 #3099
312209,312211 #3100
312251,312253 #3101
312281,312283 #3102
312311,312313 #3103
312551,312553 #3104
312581,312583 #3105
312617,312619 #3106
312677,312679 #3107
312701,312703 #3108
312839,312841 #3109
312929,312931 #3110
312941,312943 #3111
313127,313129 #3112
313151,313153 #3113
313331,313333 #3114
313637,313639 #3115
313739,313741 #3116
313931,313933 #3117
313979,313981 #3118
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
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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
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330017,330019 #3274
330131,330133 #3275
330227,330229 #3276
330287,330289 #3277
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330329,330331 #3279
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330641,330643 #3281
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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
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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
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337901,337903 #3335
338159,338161 #3336
338267,338269 #3337
338321,338323 #3338
338339,338341 #3339
338411,338413 #3340
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339137,339139 #3342
339671,339673 #3343
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340061,340063 #3346
340127,340129 #3347
340337,340339 #3348
340451,340453 #3349
340577,340579 #3350
340787,340789 #3351
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340937,340939 #3353
341057,341059 #3354
341321,341323 #3355
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341771,341773 #3357
341951,341953 #3358
342047,342049 #3359
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342281,342283 #3363
342341,342343 #3364
342371,342373 #3365
342449,342451 #3366
342467,342469 #3367
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343307,343309 #3369
343379,343381 #3370
343391,343393 #3371
343529,343531 #3372
343559,343561 #3373
343589,343591 #3374
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343829,343831 #3377
344171,344173 #3378
344207,344209 #3379
344249,344251 #3380
344291,344293 #3381
344681,344683 #3382
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345017,345019 #3385
345227,345229 #3386
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345599,345601 #3388
345731,345733 #3389
345887,345889 #3390
346139,346141 #3391
346259,346261 #3392
346391,346393 #3393
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346559,346561 #3396
346649,346651 #3397
346667,346669 #3398
346961,346963 #3399
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347129,347131 #3402
347141,347143 #3403
347297,347299 #3404
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347729,347731 #3406
347771,347773 #3407
347957,347959 #3408
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348239,348241 #3411
348419,348421 #3412
348431,348433 #3413
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399149,399151 #3796
399239,399241 #3797
399281,399283 #3798
399389,399391 #3799
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1493729,1493731 #11558
1493741,1493743 #11559
1493927,1493929 #11560
1494047,1494049 #11561
1494347,1494349 #11562
1494371,1494373 #11563
1494401,1494403 #11564
1494461,1494463 #11565
1494509,1494511 #11566
1494641,1494643 #11567
1494677,1494679 #11568
1494707,1494709 #11569
1495157,1495159 #11570
1495379,1495381 #11571
1495631,1495633 #11572
1495829,1495831 #11573
1495859,1495861 #11574
1496069,1496071 #11575
1496477,1496479 #11576
1496489,1496491 #11577
1496567,1496569 #11578
1496639,1496641 #11579
1496939,1496941 #11580
1497149,1497151 #11581
1497227,1497229 #11582
1497281,1497283 #11583
1497719,1497721 #11584
1498139,1498141 #11585
1498349,1498351 #11586
1498529,1498531 #11587
1498619,1498621 #11588
1498799,1498801 #11589
1498811,1498813 #11590
1499219,1499221 #11591
1499357,1499359 #11592
1499549,1499551 #11593
1499567,1499569 #11594
1499609,1499611 #11595
1499681,1499683 #11596
1500041,1500043 #11597
1500071,1500073 #11598
1500347,1500349 #11599
1500407,1500409 #11600
1500467,1500469 #11601
1500647,1500649 #11602
1500701,1500703 #11603
1500767,1500769 #11604
1500797,1500799 #11605
1500857,1500859 #11606
1500929,1500931 #11607
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
1573139,1573141 #12093
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
1575731,1575733 #12112
1575989,1575991 #12113
1576037,1576039 #12114
1576241,1576243 #12115
1576499,1576501 #12116
1576649,1576651 #12117
1576721,1576723 #12118
1576889,1576891 #12119
1577099,1577101 #12120
1577117,1577119 #12121
1577201,1577203 #12122
1577291,1577293 #12123
1577297,1577299 #12124
1577507,1577509 #12125
1577531,1577533 #12126
1577657,1577659 #12127
1577699,1577701 #12128
1577999,1578001 #12129
1578821,1578823 #12130
1578851,1578853 #12131
1579139,1579141 #12132
1579217,1579219 #12133
1579397,1579399 #12134
1579619,1579621 #12135
1579637,1579639 #12136
1579931,1579933 #12137
1580417,1580419 #12138
1580429,1580431 #12139
1580459,1580461 #12140
1580651,1580653 #12141
1580771,1580773 #12142
1580849,1580851 #12143
1580921,1580923 #12144
1581077,1581079 #12145
1581191,1581193 #12146
1581707,1581709 #12147
1581719,1581721 #12148
1581857,1581859 #12149
1582079,1582081 #12150
1582169,1582171 #12151
1582391,1582393 #12152
1582577,1582579 #12153
1582811,1582813 #12154
1582961,1582963 #12155
1583291,1583293 #12156
1583357,1583359 #12157
1583651,1583653 #12158
1583807,1583809 #12159
1583861,1583863 #12160
1583927,1583929 #12161
1583999,1584001 #12162
1584101,1584103 #12163
1584137,1584139 #12164
1584257,1584259 #12165
1584431,1584433 #12166
1584437,1584439 #12167
1584641,1584643 #12168
1584701,1584703 #12169
1584827,1584829 #12170
1584899,1584901 #12171
1584929,1584931 #12172
1584941,1584943 #12173
1585007,1585009 #12174
1585289,1585291 #12175
1585481,1585483 #12176
1585697,1585699 #12177
1586111,1586113 #12178
1586309,1586311 #12179
1586537,1586539 #12180
1586621,1586623 #12181
1586771,1586773 #12182
1586789,1586791 #12183
1586867,1586869 #12184
1586951,1586953 #12185
1587737,1587739 #12186
1587869,1587871 #12187
1587959,1587961 #12188
1588049,1588051 #12189
1588187,1588189 #12190
1588511,1588513 #12191
1588661,1588663 #12192
1588751,1588753 #12193
1588757,1588759 #12194
1588859,1588861 #12195
1588877,1588879 #12196
1588901,1588903 #12197
1588931,1588933 #12198
1589207,1589209 #12199
1589249,1589251 #12200
1589297,1589299 #12201
1589387,1589389 #12202
1589501,1589503 #12203
1589561,1589563 #12204
1589669,1589671 #12205
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
1596377,1596379 #12243
1596629,1596631 #12244
1596737,1596739 #12245
1596869,1596871 #12246
1597067,1597069 #12247
1597109,1597111 #12248
1597619,1597621 #12249
1597721,1597723 #12250
1598237,1598239 #12251
1598447,1598449 #12252
1598501,1598503 #12253
1598789,1598791 #12254
1598897,1598899 #12255
1598951,1598953 #12256
1599461,1599463 #12257
1599509,1599511 #12258
1599707,1599709 #12259
1599839,1599841 #12260
1600097,1600099 #12261
1600217,1600219 #12262
1600241,1600243 #12263
1600631,1600633 #12264
1600787,1600789 #12265
1600811,1600813 #12266
1600889,1600891 #12267
1600967,1600969 #12268
1601207,1601209 #12269
1601267,1601269 #12270
1601441,1601443 #12271
1601627,1601629 #12272
1601729,1601731 #12273
1601777,1601779 #12274
1601867,1601869 #12275
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
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1802651,1802653 #13599
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1802837,1802839 #13601
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2020409,2020411 #15006
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2132309,2132311 #15704
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2133251,2133253 #15711
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2160029,2160031 #15883
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2162351,2162353 #15896
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2162579,2162581 #15898
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2163221,2163223 #15902
2163347,2163349 #15903
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2163671,2163673 #15906
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2177429,2177431 #15990
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2232509,2232511 #16310
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2240477,2240479 #16369
2240531,2240533 #16370
2240657,2240659 #16371
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2456747,2456749 #17685
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2456981,2456983 #17687
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2457317,2457319 #17689
2457347,2457349 #17690
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2458061,2458063 #17693
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2459921,2459923 #17711
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2462741,2462743 #17731
2463029,2463031 #17732
2463161,2463163 #17733
2463311,2463313 #17734
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2803781,2803783 #19781
2803817,2803819 #19782
2803937,2803939 #19783
2804027,2804029 #19784
2804057,2804059 #19785
2804141,2804143 #19786
2804237,2804239 #19787
2804309,2804311 #19788
2804519,2804521 #19789
2804567,2804569 #19790
2804729,2804731 #19791
2804831,2804833 #19792
2804939,2804941 #19793
2805041,2805043 #19794
2805161,2805163 #19795
2805167,2805169 #19796
2806121,2806123 #19797
2806247,2806249 #19798
2806367,2806369 #19799
2806379,2806381 #19800
2806457,2806459 #19801
2806691,2806693 #19802
2806787,2806789 #19803
2806847,2806849 #19804
2806961,2806963 #19805
2807087,2807089 #19806
2807177,2807179 #19807
2807477,2807479 #19808
2807549,2807551 #19809
2807591,2807593 #19810
2807657,2807659 #19811
2807879,2807881 #19812
2807927,2807929 #19813
2807969,2807971 #19814
2808059,2808061 #19815
2808359,2808361 #19816
2808497,2808499 #19817
2808719,2808721 #19818
2808761,2808763 #19819
2808809,2808811 #19820
2808917,2808919 #19821
2809271,2809273 #19822
2809307,2809309 #19823
2809349,2809351 #19824
2809451,2809453 #19825
2809487,2809489 #19826
2810009,2810011 #19827
2810369,2810371 #19828
2810411,2810413 #19829
2810501,2810503 #19830
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

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 p_d(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 R_d 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 R_d 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/

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File translated from T_EX by T_TH, version 3.01.
On 29 Jul 2002, 14:41.

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

LE MONDE SCIENCE ET TECHNO | 24.06.2013 à 15h40 |

 

"Peut-on collaborer massivement en mathématiques, faisant interagir des centaines de chercheurs vers un but unique ? C'est l'objectif de la plate-forme collaborative Polymath, lancée en 2009 par le mathématicien britannique Tim Gowers, qui a déjà plusieurs succès à son actif. L'amélioration en cours d'un récent résultat de théorie des nombres illustre l'efficacité de ce type de collaboration.

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

 

 

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 × 2^666 669 ± 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.

 

Sommaire

 [masquer] 

• 1Définition
• 2Liste des premiers nombres premiers jumeaux
• 3Quelques propriétés
• 4Records
• 5Conjecture des nombres premiers jumeaux
  • 5.1Résultats partiels
  • 5.2La conjecture de Hardy-Littlewood
• 6Notes et références
• 7Voir aussi
  • 7.1Articles connexes
  • 7.2Liens externes

 

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 × 2^666 669 ± 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 = 4 680^4,5. 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 p ≤ x tels que p + 2 soit aussi premier.

On note C₂ le nombre obtenu de la façon suivante :

¹¹

(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 :

(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,‎ 1949, 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,‎ 1919, 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, n^o 6,‎ juin 2015 (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

Voir aussi[modifier | modifier le code]

Articles connexes[modifier | modifier le code]

• Conjecture de Dickson
• Conjecture de Dubner
• Conjecture d'Elliott-Halberstam
• Nombres premiers cousins
• Nombres premiers sexy

Liens externes[modifier | modifier le code]

• (en) Twin Primes (Chris Caldwell)
• (en) Introduction to Twin Primes and Brun's Constant (Xavier Gourdon, Pascal Sebah)
• Liste des 20 000 premiers couples de nombres premiers jumeaux (p, p+2)
• (en) http://www.twinprimesearch.org
• (en) Projet Polymath 8 : le projet Polymath correspondant.

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Nombres premiers

• Portail de l’arithmétique et de la théorie des nombres

Catégories :

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Source : wikipedia

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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