21/11/2015
Quand des objets #mathématiques abstraits donnent des images étonnantes..
Diaporama #CNRSleJournal : Quand des objets #mathématiques abstraits donnent des images étonnantes...
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Conjecture de Dickson
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, ..., ak + nbk avec bi ≥ 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, , 155-161 p. (lire en ligne)
- (en) Paulo Ribenboim, The new book of prime number records, Berlin, New York, Springer-Verlag, (ISBN 978-0-387-94457-9, lire en ligne)
Voir aussi[modifier | modifier le code]
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Liste des 20 000 premiers couples de nombres premiers jumeaux (p, p+2)
Source : http://arnflo.se/~site_files/Other/twinprimes
# 20000 first twin primes # Calculated: 12/09-10 By Oscar Arnflo # Processing time: 626.478574038 seconds 3,5 #1 5,7 #2 11,13 #3 17,19 #4 29,31 #5 41,43 #6 59,61 #7 71,73 #8 101,103 #9 107,109 #10 137,139 #11 149,151 #12 179,181 #13 191,193 #14 197,199 #15 227,229 #16 239,241 #17 269,271 #18 281,283 #19 311,313 #20 347,349 #21 419,421 #22 431,433 #23 461,463 #24 521,523 #25 569,571 #26 599,601 #27 617,619 #28 641,643 #29 659,661 #30 809,811 #31 821,823 #32 827,829 #33 857,859 #34 881,883 #35 1019,1021 #36 1031,1033 #37 1049,1051 #38 1061,1063 #39 1091,1093 #40 1151,1153 #41 1229,1231 #42 1277,1279 #43 1289,1291 #44 1301,1303 #45 1319,1321 #46 1427,1429 #47 1451,1453 #48 1481,1483 #49 1487,1489 #50 1607,1609 #51 1619,1621 #52 1667,1669 #53 1697,1699 #54 1721,1723 #55 1787,1789 #56 1871,1873 #57 1877,1879 #58 1931,1933 #59 1949,1951 #60 1997,1999 #61 2027,2029 #62 2081,2083 #63 2087,2089 #64 2111,2113 #65 2129,2131 #66 2141,2143 #67 2237,2239 #68 2267,2269 #69 2309,2311 #70 2339,2341 #71 2381,2383 #72 2549,2551 #73 2591,2593 #74 2657,2659 #75 2687,2689 #76 2711,2713 #77 2729,2731 #78 2789,2791 #79 2801,2803 #80 2969,2971 #81 2999,3001 #82 3119,3121 #83 3167,3169 #84 3251,3253 #85 3257,3259 #86 3299,3301 #87 3329,3331 #88 3359,3361 #89 3371,3373 #90 3389,3391 #91 3461,3463 #92 3467,3469 #93 3527,3529 #94 3539,3541 #95 3557,3559 #96 3581,3583 #97 3671,3673 #98 3767,3769 #99 3821,3823 #100 3851,3853 #101 3917,3919 #102 3929,3931 #103 4001,4003 #104 4019,4021 #105 4049,4051 #106 4091,4093 #107 4127,4129 #108 4157,4159 #109 4217,4219 #110 4229,4231 #111 4241,4243 #112 4259,4261 #113 4271,4273 #114 4337,4339 #115 4421,4423 #116 4481,4483 #117 4517,4519 #118 4547,4549 #119 4637,4639 #120 4649,4651 #121 4721,4723 #122 4787,4789 #123 4799,4801 #124 4931,4933 #125 4967,4969 #126 5009,5011 #127 5021,5023 #128 5099,5101 #129 5231,5233 #130 5279,5281 #131 5417,5419 #132 5441,5443 #133 5477,5479 #134 5501,5503 #135 5519,5521 #136 5639,5641 #137 5651,5653 #138 5657,5659 #139 5741,5743 #140 5849,5851 #141 5867,5869 #142 5879,5881 #143 6089,6091 #144 6131,6133 #145 6197,6199 #146 6269,6271 #147 6299,6301 #148 6359,6361 #149 6449,6451 #150 6551,6553 #151 6569,6571 #152 6659,6661 #153 6689,6691 #154 6701,6703 #155 6761,6763 #156 6779,6781 #157 6791,6793 #158 6827,6829 #159 6869,6871 #160 6947,6949 #161 6959,6961 #162 7127,7129 #163 7211,7213 #164 7307,7309 #165 7331,7333 #166 7349,7351 #167 7457,7459 #168 7487,7489 #169 7547,7549 #170 7559,7561 #171 7589,7591 #172 7757,7759 #173 7877,7879 #174 7949,7951 #175 8009,8011 #176 8087,8089 #177 8219,8221 #178 8231,8233 #179 8291,8293 #180 8387,8389 #181 8429,8431 #182 8537,8539 #183 8597,8599 #184 8627,8629 #185 8819,8821 #186 8837,8839 #187 8861,8863 #188 8969,8971 #189 8999,9001 #190 9011,9013 #191 9041,9043 #192 9239,9241 #193 9281,9283 #194 9341,9343 #195 9419,9421 #196 9431,9433 #197 9437,9439 #198 9461,9463 #199 9629,9631 #200 9677,9679 #201 9719,9721 #202 9767,9769 #203 9857,9859 #204 9929,9931 #205 10007,10009 #206 10037,10039 #207 10067,10069 #208 10091,10093 #209 10139,10141 #210 10271,10273 #211 10301,10303 #212 10331,10333 #213 10427,10429 #214 10457,10459 #215 10499,10501 #216 10529,10531 #217 10709,10711 #218 10859,10861 #219 10889,10891 #220 10937,10939 #221 11057,11059 #222 11069,11071 #223 11117,11119 #224 11159,11161 #225 11171,11173 #226 11351,11353 #227 11489,11491 #228 11549,11551 #229 11699,11701 #230 11717,11719 #231 11777,11779 #232 11831,11833 #233 11939,11941 #234 11969,11971 #235 12041,12043 #236 12071,12073 #237 12107,12109 #238 12161,12163 #239 12239,12241 #240 12251,12253 #241 12377,12379 #242 12539,12541 #243 12611,12613 #244 12821,12823 #245 12917,12919 #246 13001,13003 #247 13007,13009 #248 13217,13219 #249 13337,13339 #250 13397,13399 #251 13679,13681 #252 13691,13693 #253 13709,13711 #254 13721,13723 #255 13757,13759 #256 13829,13831 #257 13877,13879 #258 13901,13903 #259 13931,13933 #260 13997,13999 #261 14009,14011 #262 14081,14083 #263 14249,14251 #264 14321,14323 #265 14387,14389 #266 14447,14449 #267 14549,14551 #268 14561,14563 #269 14591,14593 #270 14627,14629 #271 14867,14869 #272 15137,15139 #273 15269,15271 #274 15287,15289 #275 15329,15331 #276 15359,15361 #277 15581,15583 #278 15641,15643 #279 15647,15649 #280 15731,15733 #281 15737,15739 #282 15887,15889 #283 15971,15973 #284 16061,16063 #285 16067,16069 #286 16139,16141 #287 16187,16189 #288 16229,16231 #289 16361,16363 #290 16451,16453 #291 16631,16633 #292 16649,16651 #293 16691,16693 #294 16829,16831 #295 16901,16903 #296 16979,16981 #297 17027,17029 #298 17189,17191 #299 17207,17209 #300 17291,17293 #301 17387,17389 #302 17417,17419 #303 17489,17491 #304 17579,17581 #305 17597,17599 #306 17657,17659 #307 17681,17683 #308 17747,17749 #309 17789,17791 #310 17837,17839 #311 17909,17911 #312 17921,17923 #313 17957,17959 #314 17987,17989 #315 18041,18043 #316 18047,18049 #317 18059,18061 #318 18119,18121 #319 18131,18133 #320 18251,18253 #321 18287,18289 #322 18311,18313 #323 18521,18523 #324 18539,18541 #325 18911,18913 #326 18917,18919 #327 19079,19081 #328 19139,19141 #329 19181,19183 #330 19211,19213 #331 19379,19381 #332 19421,19423 #333 19427,19429 #334 19469,19471 #335 19541,19543 #336 19697,19699 #337 19751,19753 #338 19841,19843 #339 19889,19891 #340 19961,19963 #341 19991,19993 #342 20021,20023 #343 20147,20149 #344 20231,20233 #345 20357,20359 #346 20441,20443 #347 20477,20479 #348 20507,20509 #349 20549,20551 #350 20639,20641 #351 20717,20719 #352 20747,20749 #353 20771,20773 #354 20807,20809 #355 20897,20899 #356 20981,20983 #357 21011,21013 #358 21017,21019 #359 21059,21061 #360 21191,21193 #361 21317,21319 #362 21377,21379 #363 21491,21493 #364 21521,21523 #365 21557,21559 #366 21587,21589 #367 21599,21601 #368 21611,21613 #369 21647,21649 #370 21737,21739 #371 21839,21841 #372 22037,22039 #373 22091,22093 #374 22109,22111 #375 22157,22159 #376 22271,22273 #377 22277,22279 #378 22367,22369 #379 22481,22483 #380 22541,22543 #381 22571,22573 #382 22619,22621 #383 22637,22639 #384 22697,22699 #385 22739,22741 #386 22859,22861 #387 22961,22963 #388 23027,23029 #389 23039,23041 #390 23057,23059 #391 23201,23203 #392 23291,23293 #393 23369,23371 #394 23537,23539 #395 23561,23563 #396 23627,23629 #397 23669,23671 #398 23687,23689 #399 23741,23743 #400 23831,23833 #401 23909,23911 #402 24107,24109 #403 24179,24181 #404 24371,24373 #405 24419,24421 #406 24917,24919 #407 24977,24979 #408 25031,25033 #409 25169,25171 #410 25301,25303 #411 25307,25309 #412 25409,25411 #413 25469,25471 #414 25577,25579 #415 25601,25603 #416 25799,25801 #417 25847,25849 #418 25931,25933 #419 25997,25999 #420 26111,26113 #421 26249,26251 #422 26261,26263 #423 26681,26683 #424 26699,26701 #425 26711,26713 #426 26729,26731 #427 26861,26863 #428 26879,26881 #429 26891,26893 #430 26951,26953 #431 27059,27061 #432 27107,27109 #433 27239,27241 #434 27281,27283 #435 27407,27409 #436 27479,27481 #437 27527,27529 #438 27539,27541 #439 27581,27583 #440 27689,27691 #441 27737,27739 #442 27749,27751 #443 27791,27793 #444 27917,27919 #445 27941,27943 #446 28097,28099 #447 28109,28111 #448 28181,28183 #449 28277,28279 #450 28307,28309 #451 28349,28351 #452 28409,28411 #453 28547,28549 #454 28571,28573 #455 28619,28621 #456 28661,28663 #457 28751,28753 #458 29021,29023 #459 29129,29131 #460 29207,29209 #461 29387,29389 #462 29399,29401 #463 29567,29569 #464 29669,29671 #465 29759,29761 #466 29879,29881 #467 30011,30013 #468 30089,30091 #469 30137,30139 #470 30269,30271 #471 30389,30391 #472 30467,30469 #473 30491,30493 #474 30557,30559 #475 30839,30841 #476 30851,30853 #477 30869,30871 #478 31079,31081 #479 31121,31123 #480 31151,31153 #481 31181,31183 #482 31247,31249 #483 31319,31321 #484 31391,31393 #485 31511,31513 #486 31541,31543 #487 31721,31723 #488 31727,31729 #489 31769,31771 #490 31847,31849 #491 32027,32029 #492 32057,32059 #493 32117,32119 #494 32141,32143 #495 32189,32191 #496 32297,32299 #497 32321,32323 #498 32369,32371 #499 32411,32413 #500 32441,32443 #501 32531,32533 #502 32561,32563 #503 32609,32611 #504 32717,32719 #505 32801,32803 #506 32831,32833 #507 32909,32911 #508 32939,32941 #509 32969,32971 #510 33071,33073 #511 33149,33151 #512 33179,33181 #513 33287,33289 #514 33329,33331 #515 33347,33349 #516 33587,33589 #517 33599,33601 #518 33617,33619 #519 33749,33751 #520 33767,33769 #521 33809,33811 #522 33827,33829 #523 34031,34033 #524 34127,34129 #525 34157,34159 #526 34211,34213 #527 34259,34261 #528 34301,34303 #529 34367,34369 #530 34469,34471 #531 34499,34501 #532 34511,34513 #533 34589,34591 #534 34649,34651 #535 34757,34759 #536 34841,34843 #537 34847,34849 #538 34961,34963 #539 35051,35053 #540 35081,35083 #541 35279,35281 #542 35447,35449 #543 35507,35509 #544 35531,35533 #545 35591,35593 #546 35729,35731 #547 35801,35803 #548 35837,35839 #549 35897,35899 #550 36011,36013 #551 36107,36109 #552 36341,36343 #553 36467,36469 #554 36527,36529 #555 36779,36781 #556 36791,36793 #557 36899,36901 #558 36929,36931 #559 37019,37021 #560 37199,37201 #561 37307,37309 #562 37337,37339 #563 37361,37363 #564 37547,37549 #565 37571,37573 #566 37589,37591 #567 37691,37693 #568 37781,37783 #569 37811,37813 #570 37991,37993 #571 38237,38239 #572 38327,38329 #573 38447,38449 #574 38459,38461 #575 38567,38569 #576 38609,38611 #577 38651,38653 #578 38669,38671 #579 38711,38713 #580 38747,38749 #581 38921,38923 #582 39041,39043 #583 39161,39163 #584 39227,39229 #585 39239,39241 #586 39341,39343 #587 39371,39373 #588 39509,39511 #589 39827,39829 #590 39839,39841 #591 40037,40039 #592 40127,40129 #593 40151,40153 #594 40427,40429 #595 40529,40531 #596 40637,40639 #597 40697,40699 #598 40847,40849 #599 41141,41143 #600 41177,41179 #601 41201,41203 #602 41231,41233 #603 41387,41389 #604 41411,41413 #605 41519,41521 #606 41609,41611 #607 41759,41761 #608 41849,41851 #609 41957,41959 #610 41981,41983 #611 42017,42019 #612 42071,42073 #613 42179,42181 #614 42221,42223 #615 42281,42283 #616 42407,42409 #617 42461,42463 #618 42569,42571 #619 42641,42643 #620 42701,42703 #621 42839,42841 #622 42899,42901 #623 43049,43051 #624 43319,43321 #625 43397,43399 #626 43541,43543 #627 43577,43579 #628 43607,43609 #629 43649,43651 #630 43781,43783 #631 43787,43789 #632 43889,43891 #633 43961,43963 #634 44027,44029 #635 44087,44089 #636 44129,44131 #637 44201,44203 #638 44267,44269 #639 44279,44281 #640 44381,44383 #641 44531,44533 #642 44621,44623 #643 44699,44701 #644 44771,44773 #645 45119,45121 #646 45137,45139 #647 45179,45181 #648 45317,45319 #649 45341,45343 #650 45587,45589 #651 45821,45823 #652 46049,46051 #653 46091,46093 #654 46181,46183 #655 46271,46273 #656 46307,46309 #657 46349,46351 #658 46439,46441 #659 46589,46591 #660 46679,46681 #661 46769,46771 #662 46817,46819 #663 46829,46831 #664 47057,47059 #665 47147,47149 #666 47351,47353 #667 47387,47389 #668 47417,47419 #669 47657,47659 #670 47699,47701 #671 47711,47713 #672 47741,47743 #673 47777,47779 #674 47807,47809 #675 48119,48121 #676 48311,48313 #677 48407,48409 #678 48479,48481 #679 48539,48541 #680 48647,48649 #681 48677,48679 #682 48731,48733 #683 48779,48781 #684 48821,48823 #685 48857,48859 #686 48869,48871 #687 48989,48991 #688 49031,49033 #689 49121,49123 #690 49169,49171 #691 49199,49201 #692 49277,49279 #693 49331,49333 #694 49367,49369 #695 49391,49393 #696 49409,49411 #697 49529,49531 #698 49547,49549 #699 49667,49669 #700 49739,49741 #701 49787,49789 #702 49919,49921 #703 49937,49939 #704 49991,49993 #705 50021,50023 #706 50051,50053 #707 50129,50131 #708 50261,50263 #709 50459,50461 #710 50549,50551 #711 50591,50593 #712 50891,50893 #713 50969,50971 #714 51059,51061 #715 51131,51133 #716 51197,51199 #717 51239,51241 #718 51341,51343 #719 51347,51349 #720 51419,51421 #721 51437,51439 #722 51479,51481 #723 51719,51721 #724 51767,51769 #725 51827,51829 #726 51869,51871 #727 51971,51973 #728 52067,52069 #729 52181,52183 #730 52289,52291 #731 52361,52363 #732 52541,52543 #733 52709,52711 #734 52859,52861 #735 52901,52903 #736 53087,53089 #737 53147,53149 #738 53171,53173 #739 53231,53233 #740 53267,53269 #741 53279,53281 #742 53549,53551 #743 53591,53593 #744 53609,53611 #745 53717,53719 #746 53897,53899 #747 54011,54013 #748 54401,54403 #749 54419,54421 #750 54497,54499 #751 54539,54541 #752 54581,54583 #753 54629,54631 #754 54917,54919 #755 55049,55051 #756 55217,55219 #757 55331,55333 #758 55337,55339 #759 55439,55441 #760 55619,55621 #761 55631,55633 #762 55661,55663 #763 55817,55819 #764 55901,55903 #765 55931,55933 #766 56039,56041 #767 56099,56101 #768 56207,56209 #769 56237,56239 #770 56267,56269 #771 56477,56479 #772 56501,56503 #773 56531,56533 #774 56597,56599 #775 56711,56713 #776 56807,56809 #777 56891,56893 #778 56909,56911 #779 56921,56923 #780 57191,57193 #781 57221,57223 #782 57269,57271 #783 57329,57331 #784 57347,57349 #785 57527,57529 #786 57557,57559 #787 57791,57793 #788 57899,57901 #789 58109,58111 #790 58151,58153 #791 58169,58171 #792 58229,58231 #793 58367,58369 #794 58391,58393 #795 58439,58441 #796 58451,58453 #797 58601,58603 #798 58787,58789 #799 58907,58909 #800 59009,59011 #801 59021,59023 #802 59051,59053 #803 59207,59209 #804 59219,59221 #805 59357,59359 #806 59417,59419 #807 59441,59443 #808 59471,59473 #809 59627,59629 #810 59669,59671 #811 60089,60091 #812 60101,60103 #813 60167,60169 #814 60257,60259 #815 60647,60649 #816 60659,60661 #817 60761,60763 #818 60887,60889 #819 60899,60901 #820 60917,60919 #821 61151,61153 #822 61331,61333 #823 61379,61381 #824 61469,61471 #825 61559,61561 #826 61979,61981 #827 62129,62131 #828 62141,62143 #829 62189,62191 #830 62297,62299 #831 62927,62929 #832 62969,62971 #833 62981,62983 #834 62987,62989 #835 63029,63031 #836 63197,63199 #837 63311,63313 #838 63389,63391 #839 63419,63421 #840 63587,63589 #841 63599,63601 #842 63647,63649 #843 63689,63691 #844 63839,63841 #845 64151,64153 #846 64187,64189 #847 64301,64303 #848 64451,64453 #849 64577,64579 #850 64661,64663 #851 64781,64783 #852 64877,64879 #853 64919,64921 #854 65027,65029 #855 65099,65101 #856 65171,65173 #857 65267,65269 #858 65447,65449 #859 65519,65521 #860 65537,65539 #861 65579,65581 #862 65699,65701 #863 65717,65719 #864 65729,65731 #865 65837,65839 #866 65927,65929 #867 65981,65983 #868 66107,66109 #869 66359,66361 #870 66569,66571 #871 66749,66751 #872 66851,66853 #873 66947,66949 #874 67139,67141 #875 67187,67189 #876 67211,67213 #877 67217,67219 #878 67271,67273 #879 67409,67411 #880 67427,67429 #881 67577,67579 #882 67757,67759 #883 67931,67933 #884 68111,68113 #885 68207,68209 #886 68279,68281 #887 68447,68449 #888 68489,68491 #889 68711,68713 #890 68819,68821 #891 68879,68881 #892 68897,68899 #893 69029,69031 #894 69149,69151 #895 69191,69193 #896 69257,69259 #897 69401,69403 #898 69491,69493 #899 69497,69499 #900 69737,69739 #901 69761,69763 #902 69827,69829 #903 69857,69859 #904 69929,69931 #905 70001,70003 #906 70121,70123 #907 70139,70141 #908 70181,70183 #909 70199,70201 #910 70379,70381 #911 70457,70459 #912 70487,70489 #913 70571,70573 #914 70619,70621 #915 70841,70843 #916 70877,70879 #917 70919,70921 #918 70949,70951 #919 70979,70981 #920 70997,70999 #921 71261,71263 #922 71327,71329 #923 71339,71341 #924 71387,71389 #925 71411,71413 #926 71471,71473 #927 71549,71551 #928 71711,71713 #929 71807,71809 #930 71879,71881 #931 72089,72091 #932 72101,72103 #933 72167,72169 #934 72221,72223 #935 72227,72229 #936 72251,72253 #937 72269,72271 #938 72467,72469 #939 72647,72649 #940 72671,72673 #941 72869,72871 #942 73037,73039 #943 73061,73063 #944 73361,73363 #945 73607,73609 #946 73679,73681 #947 73847,73849 #948 74099,74101 #949 74159,74161 #950 74201,74203 #951 74381,74383 #952 74411,74413 #953 74507,74509 #954 74609,74611 #955 74717,74719 #956 74729,74731 #957 74759,74761 #958 75011,75013 #959 75167,75169 #960 75209,75211 #961 75389,75391 #962 75401,75403 #963 75539,75541 #964 75617,75619 #965 75707,75709 #966 75989,75991 #967 76001,76003 #968 76079,76081 #969 76157,76159 #970 76259,76261 #971 76367,76369 #972 76421,76423 #973 76541,76543 #974 76649,76651 #975 76829,76831 #976 76871,76873 #977 76961,76963 #978 77237,77239 #979 77261,77263 #980 77267,77269 #981 77417,77419 #982 77477,77479 #983 77489,77491 #984 77549,77551 #985 77687,77689 #986 77711,77713 #987 78137,78139 #988 78191,78193 #989 78437,78439 #990 78509,78511 #991 78539,78541 #992 78569,78571 #993 78779,78781 #994 78887,78889 #995 78977,78979 #996 79151,79153 #997 79229,79231 #998 79397,79399 #999 79559,79561 #1000 79631,79633 #1001 79691,79693 #1002 79697,79699 #1003 79811,79813 #1004 79841,79843 #1005 79901,79903 #1006 79997,79999 #1007 80147,80149 #1008 80207,80209 #1009 80231,80233 #1010 80447,80449 #1011 80471,80473 #1012 80489,80491 #1013 80627,80629 #1014 80669,80671 #1015 80681,80683 #1016 80747,80749 #1017 80777,80779 #1018 80831,80833 #1019 80909,80911 #1020 81017,81019 #1021 81041,81043 #1022 81047,81049 #1023 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159191,159193 #1792 159347,159349 #1793 159539,159541 #1794 159569,159571 #1795 159629,159631 #1796 159671,159673 #1797 159737,159739 #1798 159791,159793 #1799 159869,159871 #1800 159977,159979 #1801 160031,160033 #1802 160079,160081 #1803 160091,160093 #1804 160481,160483 #1805 160619,160621 #1806 160637,160639 #1807 160649,160651 #1808 160709,160711 #1809 160751,160753 #1810 160877,160879 #1811 160967,160969 #1812 161339,161341 #1813 161459,161461 #1814 161561,161563 #1815 161639,161641 #1816 161729,161731 #1817 161741,161743 #1818 161771,161773 #1819 161879,161881 #1820 161921,161923 #1821 161969,161971 #1822 162287,162289 #1823 162389,162391 #1824 162527,162529 #1825 162749,162751 #1826 162821,162823 #1827 162971,162973 #1828 163019,163021 #1829 163061,163063 #1830 163127,163129 #1831 163169,163171 #1832 163307,163309 #1833 163409,163411 #1834 163481,163483 #1835 163859,163861 #1836 163979,163981 #1837 163991,163993 #1838 164147,164149 #1839 164231,164233 #1840 164249,164251 #1841 164429,164431 #1842 164447,164449 #1843 164621,164623 #1844 164837,164839 #1845 164999,165001 #1846 165047,165049 #1847 165311,165313 #1848 165551,165553 #1849 165587,165589 #1850 165701,165703 #1851 165707,165709 #1852 165719,165721 #1853 166301,166303 #1854 166349,166351 #1855 166601,166603 #1856 166667,166669 #1857 166739,166741 #1858 166781,166783 #1859 166841,166843 #1860 166847,166849 #1861 167021,167023 #1862 167117,167119 #1863 167267,167269 #1864 167309,167311 #1865 167339,167341 #1866 167441,167443 #1867 167621,167623 #1868 167777,167779 #1869 167861,167863 #1870 168449,168451 #1871 168599,168601 #1872 168629,168631 #1873 168899,168901 #1874 169007,169009 #1875 169067,169069 #1876 169217,169219 #1877 169241,169243 #1878 169319,169321 #1879 169691,169693 #1880 169751,169753 #1881 169889,169891 #1882 170099,170101 #1883 170351,170353 #1884 170369,170371 #1885 170537,170539 #1886 170759,170761 #1887 171047,171049 #1888 171077,171079 #1889 171161,171163 #1890 171167,171169 #1891 171251,171253 #1892 171401,171403 #1893 171467,171469 #1894 171539,171541 #1895 171671,171673 #1896 171761,171763 #1897 172169,172171 #1898 172217,172219 #1899 172421,172423 #1900 172439,172441 #1901 172517,172519 #1902 173021,173023 #1903 173189,173191 #1904 173207,173209 #1905 173291,173293 #1906 173357,173359 #1907 173429,173431 #1908 173669,173671 #1909 173741,173743 #1910 173777,173779 #1911 174017,174019 #1912 174047,174049 #1913 174077,174079 #1914 174257,174259 #1915 174329,174331 #1916 174467,174469 #1917 174569,174571 #1918 174761,174763 #1919 174929,174931 #1920 174989,174991 #1921 175067,175069 #1922 175079,175081 #1923 175391,175393 #1924 175631,175633 #1925 175757,175759 #1926 175781,175783 #1927 175937,175939 #1928 175961,175963 #1929 175991,175993 #1930 176021,176023 #1931 176051,176053 #1932 176087,176089 #1933 176159,176161 #1934 176327,176329 #1935 176417,176419 #1936 176459,176461 #1937 176507,176509 #1938 176549,176551 #1939 176597,176599 #1940 176609,176611 #1941 176711,176713 #1942 176777,176779 #1943 176789,176791 #1944 176807,176809 #1945 176921,176923 #1946 177011,177013 #1947 177209,177211 #1948 177431,177433 #1949 177677,177679 #1950 177761,177763 #1951 177839,177841 #1952 177887,177889 #1953 178037,178039 #1954 178067,178069 #1955 178091,178093 #1956 178247,178249 #1957 178259,178261 #1958 178349,178351 #1959 178439,178441 #1960 178487,178489 #1961 178559,178561 #1962 178601,178603 #1963 178691,178693 #1964 178817,178819 #1965 178907,178909 #1966 178931,178933 #1967 179381,179383 #1968 179579,179581 #1969 179591,179593 #1970 179657,179659 #1971 179687,179689 #1972 179717,179719 #1973 179819,179821 #1974 179897,179899 #1975 179951,179953 #1976 179999,180001 #1977 180071,180073 #1978 180179,180181 #1979 180239,180241 #1980 180287,180289 #1981 180539,180541 #1982 180749,180751 #1983 180797,180799 #1984 181001,181003 #1985 181061,181063 #1986 181199,181201 #1987 181211,181213 #1988 181301,181303 #1989 181397,181399 #1990 181457,181459 #1991 181499,181501 #1992 181607,181609 #1993 181667,181669 #1994 181757,181759 #1995 181787,181789 #1996 181871,181873 #1997 181889,181891 #1998 182009,182011 #1999 182027,182029 #2000 182057,182059 #2001 182099,182101 #2002 182129,182131 #2003 182177,182179 #2004 182339,182341 #2005 182387,182389 #2006 182471,182473 #2007 182639,182641 #2008 182657,182659 #2009 182711,182713 #2010 182927,182929 #2011 183089,183091 #2012 183299,183301 #2013 183317,183319 #2014 183437,183439 #2015 183497,183499 #2016 183509,183511 #2017 183569,183571 #2018 183707,183709 #2019 183761,183763 #2020 183917,183919 #2021 183971,183973 #2022 184187,184189 #2023 184271,184273 #2024 184487,184489 #2025 184607,184609 #2026 184631,184633 #2027 184649,184651 #2028 184829,184831 #2029 184901,184903 #2030 184967,184969 #2031 184997,184999 #2032 185069,185071 #2033 185369,185371 #2034 185531,185533 #2035 185567,185569 #2036 185681,185683 #2037 185747,185749 #2038 185819,185821 #2039 185831,185833 #2040 185957,185959 #2041 186161,186163 #2042 186227,186229 #2043 186299,186301 #2044 186377,186379 #2045 186479,186481 #2046 186581,186583 #2047 186647,186649 #2048 186707,186709 #2049 186761,186763 #2050 186869,186871 #2051 187067,187069 #2052 187127,187129 #2053 187139,187141 #2054 187217,187219 #2055 187337,187339 #2056 187469,187471 #2057 187631,187633 #2058 187637,187639 #2059 187907,187909 #2060 188831,188833 #2061 188861,188863 #2062 188939,188941 #2063 189017,189019 #2064 189041,189043 #2065 189149,189151 #2066 189251,189253 #2067 189347,189349 #2068 189389,189391 #2069 189437,189439 #2070 189491,189493 #2071 189617,189619 #2072 189797,189799 #2073 189851,189853 #2074 189947,189949 #2075 190367,190369 #2076 190577,190579 #2077 190667,190669 #2078 190709,190711 #2079 190889,190891 #2080 191141,191143 #2081 191249,191251 #2082 191297,191299 #2083 191339,191341 #2084 191447,191449 #2085 191459,191461 #2086 191507,191509 #2087 191531,191533 #2088 191561,191563 #2089 191669,191671 #2090 191747,191749 #2091 191801,191803 #2092 191831,191833 #2093 192191,192193 #2094 192317,192319 #2095 192341,192343 #2096 192461,192463 #2097 192497,192499 #2098 192581,192583 #2099 192611,192613 #2100 192629,192631 #2101 192887,192889 #2102 192977,192979 #2103 193181,193183 #2104 193379,193381 #2105 193601,193603 #2106 193811,193813 #2107 193859,193861 #2108 193871,193873 #2109 193937,193939 #2110 194069,194071 #2111 194267,194269 #2112 194681,194683 #2113 194861,194863 #2114 194867,194869 #2115 195047,195049 #2116 195161,195163 #2117 195341,195343 #2118 195539,195541 #2119 195731,195733 #2120 195737,195739 #2121 195929,195931 #2122 195971,195973 #2123 196169,196171 #2124 196277,196279 #2125 196499,196501 #2126 196541,196543 #2127 196661,196663 #2128 196769,196771 #2129 196871,196873 #2130 196991,196993 #2131 197159,197161 #2132 197297,197299 #2133 197339,197341 #2134 197369,197371 #2135 197381,197383 #2136 197567,197569 #2137 197597,197599 #2138 197711,197713 #2139 197891,197893 #2140 197957,197959 #2141 197969,197971 #2142 198221,198223 #2143 198257,198259 #2144 198347,198349 #2145 198437,198439 #2146 198461,198463 #2147 198827,198829 #2148 198839,198841 #2149 198899,198901 #2150 198941,198943 #2151 199037,199039 #2152 199151,199153 #2153 199487,199489 #2154 199499,199501 #2155 199601,199603 #2156 199739,199741 #2157 199751,199753 #2158 199811,199813 #2159 199931,199933 #2160 200381,200383 #2161 200867,200869 #2162 200927,200929 #2163 200987,200989 #2164 201119,201121 #2165 201209,201211 #2166 201401,201403 #2167 201449,201451 #2168 201491,201493 #2169 201497,201499 #2170 201767,201769 #2171 201821,201823 #2172 201827,201829 #2173 202061,202063 #2174 202127,202129 #2175 202289,202291 #2176 202637,202639 #2177 202751,202753 #2178 202877,202879 #2179 202931,202933 #2180 203207,203209 #2181 203309,203311 #2182 203321,203323 #2183 203339,203341 #2184 203351,203353 #2185 203381,203383 #2186 203417,203419 #2187 203429,203431 #2188 203459,203461 #2189 203657,203659 #2190 203771,203773 #2191 203807,203809 #2192 203909,203911 #2193 203969,203971 #2194 204161,204163 #2195 204299,204301 #2196 204329,204331 #2197 204359,204361 #2198 204437,204439 #2199 204509,204511 #2200 204599,204601 #2201 204749,204751 #2202 204791,204793 #2203 204857,204859 #2204 205031,205033 #2205 205211,205213 #2206 205397,205399 #2207 205421,205423 #2208 205661,205663 #2209 205949,205951 #2210 205991,205993 #2211 206081,206083 #2212 206177,206179 #2213 206249,206251 #2214 206279,206281 #2215 206411,206413 #2216 206639,206641 #2217 206819,206821 #2218 206909,206911 #2219 206951,206953 #2220 207197,207199 #2221 207239,207241 #2222 207329,207331 #2223 207341,207343 #2224 207479,207481 #2225 207509,207511 #2226 207521,207523 #2227 207671,207673 #2228 207719,207721 #2229 207797,207799 #2230 207971,207973 #2231 208001,208003 #2232 208139,208141 #2233 208277,208279 #2234 208391,208393 #2235 208457,208459 #2236 208499,208501 #2237 208511,208513 #2238 208589,208591 #2239 208697,208699 #2240 208889,208891 #2241 208931,208933 #2242 208961,208963 #2243 208991,208993 #2244 209201,209203 #2245 209267,209269 #2246 209357,209359 #2247 209567,209569 #2248 209579,209581 #2249 209621,209623 #2250 209717,209719 #2251 209819,209821 #2252 209927,209929 #2253 210191,210193 #2254 210317,210319 #2255 210359,210361 #2256 210401,210403 #2257 210599,210601 #2258 210809,210811 #2259 210911,210913 #2260 211049,211051 #2261 211061,211063 #2262 211151,211153 #2263 211217,211219 #2264 211229,211231 #2265 211499,211501 #2266 211571,211573 #2267 211661,211663 #2268 211691,211693 #2269 211877,211879 #2270 211889,211891 #2271 211931,211933 #2272 212207,212209 #2273 212669,212671 #2274 212867,212869 #2275 213131,213133 #2276 213287,213289 #2277 213359,213361 #2278 213611,213613 #2279 213947,213949 #2280 214007,214009 #2281 214031,214033 #2282 214211,214213 #2283 214481,214483 #2284 214517,214519 #2285 214559,214561 #2286 214787,214789 #2287 215141,215143 #2288 215351,215353 #2289 215459,215461 #2290 215687,215689 #2291 215981,215983 #2292 216317,216319 #2293 216371,216373 #2294 216551,216553 #2295 216569,216571 #2296 216647,216649 #2297 216779,216781 #2298 216899,216901 #2299 216917,216919 #2300 217001,217003 #2301 217199,217201 #2302 217307,217309 #2303 217337,217339 #2304 217361,217363 #2305 217367,217369 #2306 217409,217411 #2307 217517,217519 #2308 217559,217561 #2309 217577,217579 #2310 217907,217909 #2311 217979,217981 #2312 218081,218083 #2313 218417,218419 #2314 218459,218461 #2315 218549,218551 #2316 218627,218629 #2317 218717,218719 #2318 218969,218971 #2319 218987,218989 #2320 219017,219019 #2321 219311,219313 #2322 219407,219409 #2323 219647,219649 #2324 219677,219679 #2325 219761,219763 #2326 219797,219799 #2327 219941,219943 #2328 219977,219979 #2329 220019,220021 #2330 220469,220471 #2331 220511,220513 #2332 220859,220861 #2333 220877,220879 #2334 220901,220903 #2335 220931,220933 #2336 221069,221071 #2337 221171,221173 #2338 221201,221203 #2339 221399,221401 #2340 221411,221413 #2341 221537,221539 #2342 221621,221623 #2343 221657,221659 #2344 221717,221719 #2345 221951,221953 #2346 221987,221989 #2347 222041,222043 #2348 222107,222109 #2349 222149,222151 #2350 222161,222163 #2351 222197,222199 #2352 222347,222349 #2353 222791,222793 #2354 222839,222841 #2355 222977,222979 #2356 223007,223009 #2357 223049,223051 #2358 223061,223063 #2359 223217,223219 #2360 223241,223243 #2361 223337,223339 #2362 223439,223441 #2363 223547,223549 #2364 223679,223681 #2365 223757,223759 #2366 223829,223831 #2367 223841,223843 #2368 223919,223921 #2369 224069,224071 #2370 224129,224131 #2371 224909,224911 #2372 225077,225079 #2373 225161,225163 #2374 225221,225223 #2375 225287,225289 #2376 225341,225343 #2377 225347,225349 #2378 225371,225373 #2379 225527,225529 #2380 225581,225583 #2381 225611,225613 #2382 225749,225751 #2383 225767,225769 #2384 225779,225781 #2385 225941,225943 #2386 226199,226201 #2387 226379,226381 #2388 226451,226453 #2389 226547,226549 #2390 226817,226819 #2391 226901,226903 #2392 227111,227113 #2393 227189,227191 #2394 227231,227233 #2395 227471,227473 #2396 227531,227533 #2397 227567,227569 #2398 227609,227611 #2399 227627,227629 #2400 227651,227653 #2401 228197,228199 #2402 228299,228301 #2403 228419,228421 #2404 228509,228511 #2405 228521,228523 #2406 228617,228619 #2407 228731,228733 #2408 228797,228799 #2409 228881,228883 #2410 228911,228913 #2411 228959,228961 #2412 229247,229249 #2413 229547,229549 #2414 229589,229591 #2415 229637,229639 #2416 229751,229753 #2417 229769,229771 #2418 229847,229849 #2419 229937,229939 #2420 229961,229963 #2421 229979,229981 #2422 230309,230311 #2423 230339,230341 #2424 230387,230389 #2425 230561,230563 #2426 230771,230773 #2427 230861,230863 #2428 230939,230941 #2429 230999,231001 #2430 231017,231019 #2431 231107,231109 #2432 231269,231271 #2433 231347,231349 #2434 231431,231433 #2435 231461,231463 #2436 231479,231481 #2437 231611,231613 #2438 231821,231823 #2439 231839,231841 #2440 232049,232051 #2441 232079,232081 #2442 232187,232189 #2443 232409,232411 #2444 232457,232459 #2445 232709,232711 #2446 232751,232753 #2447 232961,232963 #2448 233069,233071 #2449 233141,233143 #2450 233159,233161 #2451 233327,233329 #2452 233417,233419 #2453 233549,233551 #2454 233687,233689 #2455 233879,233881 #2456 233921,233923 #2457 233939,233941 #2458 234191,234193 #2459 234317,234319 #2460 234341,234343 #2461 234461,234463 #2462 234527,234529 #2463 234539,234541 #2464 234587,234589 #2465 234809,234811 #2466 234959,234961 #2467 234977,234979 #2468 235007,235009 #2469 235241,235243 #2470 235307,235309 #2471 235439,235441 #2472 235661,235663 #2473 235787,235789 #2474 235811,235813 #2475 235889,235891 #2476 236207,236209 #2477 236477,236479 #2478 236699,236701 #2479 236771,236773 #2480 236867,236869 #2481 236879,236881 #2482 236891,236893 #2483 236981,236983 #2484 237071,237073 #2485 237089,237091 #2486 237161,237163 #2487 237689,237691 #2488 237857,237859 #2489 237971,237973 #2490 238037,238039 #2491 238079,238081 #2492 238157,238159 #2493 238361,238363 #2494 238529,238531 #2495 238727,238729 #2496 238877,238879 #2497 238919,238921 #2498 239231,239233 #2499 239387,239389 #2500 239429,239431 #2501 239711,239713 #2502 239849,239851 #2503 240041,240043 #2504 240047,240049 #2505 240257,240259 #2506 240347,240349 #2507 240587,240589 #2508 240881,240883 #2509 241049,241051 #2510 241067,241069 #2511 241259,241261 #2512 241361,241363 #2513 241391,241393 #2514 241511,241513 #2515 241559,241561 #2516 241601,241603 #2517 241781,241783 #2518 241919,241921 #2519 241979,241981 #2520 242057,242059 #2521 242171,242173 #2522 242447,242449 #2523 242519,242521 #2524 242729,242731 #2525 243119,243121 #2526 243401,243403 #2527 243431,243433 #2528 243587,243589 #2529 243671,243673 #2530 243701,243703 #2531 243707,243709 #2532 244157,244159 #2533 244217,244219 #2534 244301,244303 #2535 244379,244381 #2536 244637,244639 #2537 244667,244669 #2538 244841,244843 #2539 244859,244861 #2540 245129,245131 #2541 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274709,274711 #2792 274829,274831 #2793 275129,275131 #2794 275159,275161 #2795 275321,275323 #2796 275447,275449 #2797 275459,275461 #2798 275489,275491 #2799 275579,275581 #2800 275591,275593 #2801 275921,275923 #2802 275939,275941 #2803 276041,276043 #2804 276047,276049 #2805 276371,276373 #2806 276587,276589 #2807 276671,276673 #2808 276779,276781 #2809 276821,276823 #2810 276917,276919 #2811 277097,277099 #2812 277259,277261 #2813 277427,277429 #2814 277547,277549 #2815 277577,277579 #2816 277601,277603 #2817 277637,277639 #2818 277787,277789 #2819 277889,277891 #2820 278147,278149 #2821 278207,278209 #2822 278489,278491 #2823 278501,278503 #2824 278561,278563 #2825 278609,278611 #2826 278687,278689 #2827 278741,278743 #2828 278807,278809 #2829 278879,278881 #2830 278909,278911 #2831 279119,279121 #2832 279479,279481 #2833 279551,279553 #2834 279707,279709 #2835 280097,280099 #2836 280337,280339 #2837 280409,280411 #2838 280547,280549 #2839 280589,280591 #2840 280697,280699 #2841 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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 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1039037,1039039 #8460 1039067,1039069 #8461 1039109,1039111 #8462 1039349,1039351 #8463 1039427,1039429 #8464 1040057,1040059 #8465 1040069,1040071 #8466 1040159,1040161 #8467 1040189,1040191 #8468 1040447,1040449 #8469 1040579,1040581 #8470 1040657,1040659 #8471 1040747,1040749 #8472 1040777,1040779 #8473 1041119,1041121 #8474 1041149,1041151 #8475 1041167,1041169 #8476 1041221,1041223 #8477 1041239,1041241 #8478 1041281,1041283 #8479 1041449,1041451 #8480 1041617,1041619 #8481 1041671,1041673 #8482 1042121,1042123 #8483 1042241,1042243 #8484 1042271,1042273 #8485 1042331,1042333 #8486 1042607,1042609 #8487 1042631,1042633 #8488 1042901,1042903 #8489 1043111,1043113 #8490 1043291,1043293 #8491 1043591,1043593 #8492 1043597,1043599 #8493 1043759,1043761 #8494 1043837,1043839 #8495 1043897,1043899 #8496 1043921,1043923 #8497 1044179,1044181 #8498 1044287,1044289 #8499 1044737,1044739 #8500 1044749,1044751 #8501 1044779,1044781 #8502 1044809,1044811 #8503 1045061,1045063 #8504 1045151,1045153 #8505 1045307,1045309 #8506 1045391,1045393 #8507 1045409,1045411 #8508 1045547,1045549 #8509 1045571,1045573 #8510 1045727,1045729 #8511 1045799,1045801 #8512 1046051,1046053 #8513 1046189,1046191 #8514 1046237,1046239 #8515 1046369,1046371 #8516 1046447,1046449 #8517 1046597,1046599 #8518 1046657,1046659 #8519 1047041,1047043 #8520 1047131,1047133 #8521 1047197,1047199 #8522 1047281,1047283 #8523 1047311,1047313 #8524 1047467,1047469 #8525 1047587,1047589 #8526 1047647,1047649 #8527 1047689,1047691 #8528 1047701,1047703 #8529 1047881,1047883 #8530 1048007,1048009 #8531 1048049,1048051 #8532 1048127,1048129 #8533 1048217,1048219 #8534 1048571,1048573 #8535 1048889,1048891 #8536 1049129,1049131 #8537 1049141,1049143 #8538 1049171,1049173 #8539 1049471,1049473 #8540 1049681,1049683 #8541 1049861,1049863 #8542 1049897,1049899 #8543 1050011,1050013 #8544 1050167,1050169 #8545 1050239,1050241 #8546 1050449,1050451 #8547 1050737,1050739 #8548 1050851,1050853 #8549 1050899,1050901 #8550 1051007,1051009 #8551 1051079,1051081 #8552 1051151,1051153 #8553 1051469,1051471 #8554 1051619,1051621 #8555 1051847,1051849 #8556 1052039,1052041 #8557 1052279,1052281 #8558 1052327,1052329 #8559 1052531,1052533 #8560 1052561,1052563 #8561 1052801,1052803 #8562 1052897,1052899 #8563 1053179,1053181 #8564 1053257,1053259 #8565 1053509,1053511 #8566 1053581,1053583 #8567 1053737,1053739 #8568 1053989,1053991 #8569 1054169,1054171 #8570 1054199,1054201 #8571 1054301,1054303 #8572 1054439,1054441 #8573 1054607,1054609 #8574 1054721,1054723 #8575 1055141,1055143 #8576 1055189,1055191 #8577 1055231,1055233 #8578 1055267,1055269 #8579 1055501,1055503 #8580 1055609,1055611 #8581 1056047,1056049 #8582 1056071,1056073 #8583 1056269,1056271 #8584 1056371,1056373 #8585 1056479,1056481 #8586 1056719,1056721 #8587 1057181,1057183 #8588 1057391,1057393 #8589 1057487,1057489 #8590 1057577,1057579 #8591 1057631,1057633 #8592 1057739,1057741 #8593 1058009,1058011 #8594 1058147,1058149 #8595 1058339,1058341 #8596 1058381,1058383 #8597 1058591,1058593 #8598 1058747,1058749 #8599 1058807,1058809 #8600 1058999,1059001 #8601 1059059,1059061 #8602 1059257,1059259 #8603 1059437,1059439 #8604 1059701,1059703 #8605 1060019,1060021 #8606 1060349,1060351 #8607 1060391,1060393 #8608 1060571,1060573 #8609 1060721,1060723 #8610 1060991,1060993 #8611 1061141,1061143 #8612 1061771,1061773 #8613 1061867,1061869 #8614 1061909,1061911 #8615 1062251,1062253 #8616 1062407,1062409 #8617 1062599,1062601 #8618 1062671,1062673 #8619 1062779,1062781 #8620 1062869,1062871 #8621 1062911,1062913 #8622 1062947,1062949 #8623 1062977,1062979 #8624 1063157,1063159 #8625 1063241,1063243 #8626 1063397,1063399 #8627 1063847,1063849 #8628 1063871,1063873 #8629 1063919,1063921 #8630 1063961,1063963 #8631 1063967,1063969 #8632 1064177,1064179 #8633 1064339,1064341 #8634 1064471,1064473 #8635 1064519,1064521 #8636 1064669,1064671 #8637 1064939,1064941 #8638 1064951,1064953 #8639 1065011,1065013 #8640 1065017,1065019 #8641 1065089,1065091 #8642 1065131,1065133 #8643 1065527,1065529 #8644 1065899,1065901 #8645 1066139,1066141 #8646 1066157,1066159 #8647 1066409,1066411 #8648 1066619,1066621 #8649 1066979,1066981 #8650 1067327,1067329 #8651 1067489,1067491 #8652 1067567,1067569 #8653 1067747,1067749 #8654 1067849,1067851 #8655 1068101,1068103 #8656 1068251,1068253 #8657 1068257,1068259 #8658 1068407,1068409 #8659 1068437,1068439 #8660 1068497,1068499 #8661 1068629,1068631 #8662 1068701,1068703 #8663 1068707,1068709 #8664 1068719,1068721 #8665 1068887,1068889 #8666 1069127,1069129 #8667 1069217,1069219 #8668 1069427,1069429 #8669 1069499,1069501 #8670 1069571,1069573 #8671 1069919,1069921 #8672 1069931,1069933 #8673 1069949,1069951 #8674 1070009,1070011 #8675 1070231,1070233 #8676 1070339,1070341 #8677 1070429,1070431 #8678 1070567,1070569 #8679 1070681,1070683 #8680 1071149,1071151 #8681 1071227,1071229 #8682 1071311,1071313 #8683 1071377,1071379 #8684 1071569,1071571 #8685 1071641,1071643 #8686 1071659,1071661 #8687 1071977,1071979 #8688 1072229,1072231 #8689 1072457,1072459 #8690 1072829,1072831 #8691 1072931,1072933 #8692 1072997,1072999 #8693 1073141,1073143 #8694 1073351,1073353 #8695 1073381,1073383 #8696 1073507,1073509 #8697 1073711,1073713 #8698 1073789,1073791 #8699 1073879,1073881 #8700 1073909,1073911 #8701 1073951,1073953 #8702 1074107,1074109 #8703 1074251,1074253 #8704 1074287,1074289 #8705 1074377,1074379 #8706 1074509,1074511 #8707 1074641,1074643 #8708 1074707,1074709 #8709 1074761,1074763 #8710 1074917,1074919 #8711 1074971,1074973 #8712 1074989,1074991 #8713 1075091,1075093 #8714 1075169,1075171 #8715 1075337,1075339 #8716 1075619,1075621 #8717 1075649,1075651 #8718 1075691,1075693 #8719 1075727,1075729 #8720 1075757,1075759 #8721 1075769,1075771 #8722 1076111,1076113 #8723 1076279,1076281 #8724 1076399,1076401 #8725 1076501,1076503 #8726 1076771,1076773 #8727 1077299,1077301 #8728 1077539,1077541 #8729 1077719,1077721 #8730 1077761,1077763 #8731 1077821,1077823 #8732 1077911,1077913 #8733 1078109,1078111 #8734 1078151,1078153 #8735 1078331,1078333 #8736 1078367,1078369 #8737 1078409,1078411 #8738 1078787,1078789 #8739 1079009,1079011 #8740 1079357,1079359 #8741 1079471,1079473 #8742 1079669,1079671 #8743 1079777,1079779 #8744 1079927,1079929 #8745 1080089,1080091 #8746 1080269,1080271 #8747 1080449,1080451 #8748 1080479,1080481 #8749 1080557,1080559 #8750 1080647,1080649 #8751 1080899,1080901 #8752 1080941,1080943 #8753 1080971,1080973 #8754 1081097,1081099 #8755 1081121,1081123 #8756 1081229,1081231 #8757 1081277,1081279 #8758 1081679,1081681 #8759 1081709,1081711 #8760 1081721,1081723 #8761 1081937,1081939 #8762 1081979,1081981 #8763 1082141,1082143 #8764 1082231,1082233 #8765 1082381,1082383 #8766 1082531,1082533 #8767 1082579,1082581 #8768 1082969,1082971 #8769 1083077,1083079 #8770 1083191,1083193 #8771 1083287,1083289 #8772 1083317,1083319 #8773 1083449,1083451 #8774 1083611,1083613 #8775 1083911,1083913 #8776 1083947,1083949 #8777 1084217,1084219 #8778 1084469,1084471 #8779 1085111,1085113 #8780 1085141,1085143 #8781 1085351,1085353 #8782 1085429,1085431 #8783 1086089,1086091 #8784 1086101,1086103 #8785 1086191,1086193 #8786 1086257,1086259 #8787 1086299,1086301 #8788 1086389,1086391 #8789 1086509,1086511 #8790 1086557,1086559 #8791 1086989,1086991 #8792 1087379,1087381 #8793 1087451,1087453 #8794 1087517,1087519 #8795 1087589,1087591 #8796 1087787,1087789 #8797 1087841,1087843 #8798 1088159,1088161 #8799 1088237,1088239 #8800 1088387,1088389 #8801 1088447,1088449 #8802 1088621,1088623 #8803 1088639,1088641 #8804 1088669,1088671 #8805 1088957,1088959 #8806 1089461,1089463 #8807 1089677,1089679 #8808 1089917,1089919 #8809 1089941,1089943 #8810 1090097,1090099 #8811 1090127,1090129 #8812 1090151,1090153 #8813 1090211,1090213 #8814 1090421,1090423 #8815 1090457,1090459 #8816 1090469,1090471 #8817 1090709,1090711 #8818 1090757,1090759 #8819 1090877,1090879 #8820 1090889,1090891 #8821 1090937,1090939 #8822 1091021,1091023 #8823 1091147,1091149 #8824 1091159,1091161 #8825 1091219,1091221 #8826 1091261,1091263 #8827 1091267,1091269 #8828 1091369,1091371 #8829 1091399,1091401 #8830 1091411,1091413 #8831 1091549,1091551 #8832 1091729,1091731 #8833 1091807,1091809 #8834 1092041,1092043 #8835 1092059,1092061 #8836 1092389,1092391 #8837 1092461,1092463 #8838 1092731,1092733 #8839 1092827,1092829 #8840 1092851,1092853 #8841 1092989,1092991 #8842 1093061,1093063 #8843 1093067,1093069 #8844 1093109,1093111 #8845 1093199,1093201 #8846 1093529,1093531 #8847 1093637,1093639 #8848 1093679,1093681 #8849 1093751,1093753 #8850 1093991,1093993 #8851 1093997,1093999 #8852 1094057,1094059 #8853 1094099,1094101 #8854 1094129,1094131 #8855 1094549,1094551 #8856 1094669,1094671 #8857 1094801,1094803 #8858 1094831,1094833 #8859 1094921,1094923 #8860 1095047,1095049 #8861 1095221,1095223 #8862 1095401,1095403 #8863 1095581,1095583 #8864 1095779,1095781 #8865 1095791,1095793 #8866 1095839,1095841 #8867 1095959,1095961 #8868 1096097,1096099 #8869 1096349,1096351 #8870 1096559,1096561 #8871 1096829,1096831 #8872 1096859,1096861 #8873 1096967,1096969 #8874 1097111,1097113 #8875 1097141,1097143 #8876 1097321,1097323 #8877 1097441,1097443 #8878 1097651,1097653 #8879 1097849,1097851 #8880 1097891,1097893 #8881 1098191,1098193 #8882 1098311,1098313 #8883 1098479,1098481 #8884 1098509,1098511 #8885 1098707,1098709 #8886 1099079,1099081 #8887 1099247,1099249 #8888 1099391,1099393 #8889 1099409,1099411 #8890 1099487,1099489 #8891 1099619,1099621 #8892 1099727,1099729 #8893 1099841,1099843 #8894 1100039,1100041 #8895 1100147,1100149 #8896 1100441,1100443 #8897 1100681,1100683 #8898 1100831,1100833 #8899 1100837,1100839 #8900 1101407,1101409 #8901 1101431,1101433 #8902 1101509,1101511 #8903 1101671,1101673 #8904 1101689,1101691 #8905 1101929,1101931 #8906 1102427,1102429 #8907 1102481,1102483 #8908 1102679,1102681 #8909 1102691,1102693 #8910 1102727,1102729 #8911 1102901,1102903 #8912 1103279,1103281 #8913 1103339,1103341 #8914 1103579,1103581 #8915 1103987,1103989 #8916 1104137,1104139 #8917 1104377,1104379 #8918 1104557,1104559 #8919 1104659,1104661 #8920 1104737,1104739 #8921 1104749,1104751 #8922 1104767,1104769 #8923 1104821,1104823 #8924 1105061,1105063 #8925 1105337,1105339 #8926 1105547,1105549 #8927 1105607,1105609 #8928 1105649,1105651 #8929 1105691,1105693 #8930 1105757,1105759 #8931 1105961,1105963 #8932 1105997,1105999 #8933 1106099,1106101 #8934 1106177,1106179 #8935 1106447,1106449 #8936 1106489,1106491 #8937 1106627,1106629 #8938 1106687,1106689 #8939 1106837,1106839 #8940 1107047,1107049 #8941 1107107,1107109 #8942 1107317,1107319 #8943 1107569,1107571 #8944 1107581,1107583 #8945 1107677,1107679 #8946 1107791,1107793 #8947 1107851,1107853 #8948 1108169,1108171 #8949 1108361,1108363 #8950 1108487,1108489 #8951 1108559,1108561 #8952 1108571,1108573 #8953 1108691,1108693 #8954 1108727,1108729 #8955 1108817,1108819 #8956 1108907,1108909 #8957 1108997,1108999 #8958 1109159,1109161 #8959 1109399,1109401 #8960 1109489,1109491 #8961 1109531,1109533 #8962 1109609,1109611 #8963 1109789,1109791 #8964 1110269,1110271 #8965 1110311,1110313 #8966 1110521,1110523 #8967 1110539,1110541 #8968 1110587,1110589 #8969 1110917,1110919 #8970 1110929,1110931 #8971 1110971,1110973 #8972 1111181,1111183 #8973 1111211,1111213 #8974 1111637,1111639 #8975 1112129,1112131 #8976 1112141,1112143 #8977 1112339,1112341 #8978 1112381,1112383 #8979 1112567,1112569 #8980 1112651,1112653 #8981 1112729,1112731 #8982 1112777,1112779 #8983 1112831,1112833 #8984 1112897,1112899 #8985 1113197,1113199 #8986 1113317,1113319 #8987 1113401,1113403 #8988 1113701,1113703 #8989 1114037,1114039 #8990 1114271,1114273 #8991 1114301,1114303 #8992 1114721,1114723 #8993 1114907,1114909 #8994 1115027,1115029 #8995 1115237,1115239 #8996 1115267,1115269 #8997 1115297,1115299 #8998 1115327,1115329 #8999 1115417,1115419 #9000 1115447,1115449 #9001 1115531,1115533 #9002 1115579,1115581 #9003 1115711,1115713 #9004 1115771,1115773 #9005 1116317,1116319 #9006 1116569,1116571 #9007 1116749,1116751 #9008 1116851,1116853 #9009 1116887,1116889 #9010 1117031,1117033 #9011 1117307,1117309 #9012 1117481,1117483 #9013 1117601,1117603 #9014 1117607,1117609 #9015 1117679,1117681 #9016 1117757,1117759 #9017 1117811,1117813 #9018 1117817,1117819 #9019 1117931,1117933 #9020 1118009,1118011 #9021 1118021,1118023 #9022 1118147,1118149 #9023 1118567,1118569 #9024 1118807,1118809 #9025 1118861,1118863 #9026 1118867,1118869 #9027 1119047,1119049 #9028 1119527,1119529 #9029 1119821,1119823 #9030 1119947,1119949 #9031 1120157,1120159 #9032 1120289,1120291 #9033 1120319,1120321 #9034 1120499,1120501 #9035 1120517,1120519 #9036 1120541,1120543 #9037 1120547,1120549 #9038 1120661,1120663 #9039 1120739,1120741 #9040 1120781,1120783 #9041 1121189,1121191 #9042 1121387,1121389 #9043 1121831,1121833 #9044 1121837,1121839 #9045 1122089,1122091 #9046 1122131,1122133 #9047 1122137,1122139 #9048 1122179,1122181 #9049 1122281,1122283 #9050 1123079,1123081 #9051 1123217,1123219 #9052 1123349,1123351 #9053 1123427,1123429 #9054 1123667,1123669 #9055 1123691,1123693 #9056 1123739,1123741 #9057 1124267,1124269 #9058 1124351,1124353 #9059 1124441,1124443 #9060 1124831,1124833 #9061 1124867,1124869 #9062 1125167,1125169 #9063 1125359,1125361 #9064 1125431,1125433 #9065 1125557,1125559 #9066 1125569,1125571 #9067 1125911,1125913 #9068 1126031,1126033 #9069 1126397,1126399 #9070 1126439,1126441 #9071 1126457,1126459 #9072 1126577,1126579 #9073 1126661,1126663 #9074 1126667,1126669 #9075 1126859,1126861 #9076 1127309,1127311 #9077 1127381,1127383 #9078 1127801,1127803 #9079 1127981,1127983 #9080 1128089,1128091 #9081 1128107,1128109 #9082 1128287,1128289 #9083 1128299,1128301 #9084 1128371,1128373 #9085 1128497,1128499 #9086 1128599,1128601 #9087 1128641,1128643 #9088 1128761,1128763 #9089 1128779,1128781 #9090 1128821,1128823 #9091 1128899,1128901 #9092 1128947,1128949 #9093 1128977,1128979 #9094 1129109,1129111 #9095 1129211,1129213 #9096 1129439,1129441 #9097 1129487,1129489 #9098 1129559,1129561 #9099 1129787,1129789 #9100 1129859,1129861 #9101 1130429,1130431 #9102 1130579,1130581 #9103 1130627,1130629 #9104 1130639,1130641 #9105 1130807,1130809 #9106 1130951,1130953 #9107 1131047,1131049 #9108 1131077,1131079 #9109 1131131,1131133 #9110 1131269,1131271 #9111 1131329,1131331 #9112 1131341,1131343 #9113 1131419,1131421 #9114 1131749,1131751 #9115 1131827,1131829 #9116 1131881,1131883 #9117 1131917,1131919 #9118 1131959,1131961 #9119 1132139,1132141 #9120 1132601,1132603 #9121 1132991,1132993 #9122 1133147,1133149 #9123 1133189,1133191 #9124 1133261,1133263 #9125 1133357,1133359 #9126 1133477,1133479 #9127 1133621,1133623 #9128 1133651,1133653 #9129 1133681,1133683 #9130 1134149,1134151 #9131 1134239,1134241 #9132 1134311,1134313 #9133 1134389,1134391 #9134 1134479,1134481 #9135 1134557,1134559 #9136 1135007,1135009 #9137 1135019,1135021 #9138 1135061,1135063 #9139 1135091,1135093 #9140 1135427,1135429 #9141 1135859,1135861 #9142 1135919,1135921 #9143 1135997,1135999 #9144 1136087,1136089 #9145 1136327,1136329 #9146 1136459,1136461 #9147 1136717,1136719 #9148 1136831,1136833 #9149 1136981,1136983 #9150 1136999,1137001 #9151 1137137,1137139 #9152 1137161,1137163 #9153 1137527,1137529 #9154 1137551,1137553 #9155 1137611,1137613 #9156 1137809,1137811 #9157 1137881,1137883 #9158 1137887,1137889 #9159 1138367,1138369 #9160 1138391,1138393 #9161 1138409,1138411 #9162 1138427,1138429 #9163 1138589,1138591 #9164 1138637,1138639 #9165 1138679,1138681 #9166 1138829,1138831 #9167 1138997,1138999 #9168 1139141,1139143 #9169 1139291,1139293 #9170 1139471,1139473 #9171 1139519,1139521 #9172 1139681,1139683 #9173 1139771,1139773 #9174 1139849,1139851 #9175 1139861,1139863 #9176 1139909,1139911 #9177 1140101,1140103 #9178 1140569,1140571 #9179 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1185929,1185931 #9495 1186049,1186051 #9496 1186349,1186351 #9497 1186439,1186441 #9498 1186517,1186519 #9499 1186697,1186699 #9500 1186739,1186741 #9501 1186811,1186813 #9502 1187309,1187311 #9503 1187339,1187341 #9504 1187411,1187413 #9505 1187507,1187509 #9506 1187687,1187689 #9507 1187699,1187701 #9508 1187801,1187803 #9509 1187819,1187821 #9510 1187939,1187941 #9511 1187999,1188001 #9512 1188071,1188073 #9513 1188149,1188151 #9514 1188167,1188169 #9515 1188287,1188289 #9516 1188359,1188361 #9517 1188527,1188529 #9518 1188557,1188559 #9519 1188839,1188841 #9520 1189061,1189063 #9521 1189469,1189471 #9522 1189481,1189483 #9523 1189577,1189579 #9524 1189631,1189633 #9525 1189649,1189651 #9526 1189757,1189759 #9527 1190069,1190071 #9528 1190261,1190263 #9529 1190489,1190491 #9530 1190507,1190509 #9531 1190699,1190701 #9532 1190807,1190809 #9533 1190897,1190899 #9534 1190951,1190953 #9535 1191011,1191013 #9536 1191077,1191079 #9537 1191107,1191109 #9538 1191611,1191613 #9539 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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 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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 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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 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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 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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 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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 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1736387,1736389 #13173 1736417,1736419 #13174 1736459,1736461 #13175 1736687,1736689 #13176 1736849,1736851 #13177 1736879,1736881 #13178 1737101,1737103 #13179 1737401,1737403 #13180 1737431,1737433 #13181 1737521,1737523 #13182 1737611,1737613 #13183 1737677,1737679 #13184 1738019,1738021 #13185 1738127,1738129 #13186 1738169,1738171 #13187 1738379,1738381 #13188 1738421,1738423 #13189 1738589,1738591 #13190 1738901,1738903 #13191 1738967,1738969 #13192 1738991,1738993 #13193 1739039,1739041 #13194 1739207,1739209 #13195 1739357,1739359 #13196 1739399,1739401 #13197 1739471,1739473 #13198 1739579,1739581 #13199 1739867,1739869 #13200 1740047,1740049 #13201 1740119,1740121 #13202 1740197,1740199 #13203 1740257,1740259 #13204 1740437,1740439 #13205 1740521,1740523 #13206 1740689,1740691 #13207 1740701,1740703 #13208 1741151,1741153 #13209 1741319,1741321 #13210 1741379,1741381 #13211 1741697,1741699 #13212 1741877,1741879 #13213 1742171,1742173 #13214 1742537,1742539 #13215 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1748489,1748491 #13259 1749029,1749031 #13260 1749047,1749049 #13261 1749071,1749073 #13262 1749089,1749091 #13263 1749149,1749151 #13264 1749257,1749259 #13265 1749269,1749271 #13266 1749467,1749469 #13267 1749491,1749493 #13268 1749497,1749499 #13269 1749701,1749703 #13270 1749959,1749961 #13271 1750127,1750129 #13272 1750181,1750183 #13273 1750379,1750381 #13274 1750499,1750501 #13275 1750871,1750873 #13276 1750979,1750981 #13277 1751039,1751041 #13278 1751291,1751293 #13279 1751411,1751413 #13280 1751567,1751569 #13281 1752011,1752013 #13282 1752029,1752031 #13283 1752227,1752229 #13284 1752269,1752271 #13285 1752599,1752601 #13286 1752629,1752631 #13287 1752701,1752703 #13288 1752719,1752721 #13289 1752941,1752943 #13290 1752977,1752979 #13291 1753049,1753051 #13292 1753289,1753291 #13293 1753439,1753441 #13294 1753517,1753519 #13295 1753559,1753561 #13296 1753649,1753651 #13297 1753901,1753903 #13298 1753979,1753981 #13299 1754171,1754173 #13300 1754579,1754581 #13301 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2009879,2009881 #14936 2009921,2009923 #14937 2009981,2009983 #14938 2010137,2010139 #14939 2010581,2010583 #14940 2010971,2010973 #14941 2011019,2011021 #14942 2011127,2011129 #14943 2011199,2011201 #14944 2011391,2011393 #14945 2011439,2011441 #14946 2011697,2011699 #14947 2012009,2012011 #14948 2012159,2012161 #14949 2012447,2012449 #14950 2012531,2012533 #14951 2012639,2012641 #14952 2012711,2012713 #14953 2012741,2012743 #14954 2012819,2012821 #14955 2013227,2013229 #14956 2013287,2013289 #14957 2013299,2013301 #14958 2013617,2013619 #14959 2013707,2013709 #14960 2013749,2013751 #14961 2014097,2014099 #14962 2014139,2014141 #14963 2014217,2014219 #14964 2014301,2014303 #14965 2014457,2014459 #14966 2014799,2014801 #14967 2014811,2014813 #14968 2014919,2014921 #14969 2015087,2015089 #14970 2015177,2015179 #14971 2015201,2015203 #14972 2015267,2015269 #14973 2015441,2015443 #14974 2015777,2015779 #14975 2015831,2015833 #14976 2015861,2015863 #14977 2016029,2016031 #14978 2016137,2016139 #14979 2016197,2016199 #14980 2016359,2016361 #14981 2016401,2016403 #14982 2016407,2016409 #14983 2016671,2016673 #14984 2016821,2016823 #14985 2016851,2016853 #14986 2017187,2017189 #14987 2017469,2017471 #14988 2017709,2017711 #14989 2017751,2017753 #14990 2018111,2018113 #14991 2018171,2018173 #14992 2018249,2018251 #14993 2018381,2018383 #14994 2018591,2018593 #14995 2018747,2018749 #14996 2018897,2018899 #14997 2019011,2019013 #14998 2019131,2019133 #14999 2019401,2019403 #15000 2019461,2019463 #15001 2019707,2019709 #15002 2019767,2019769 #15003 2020001,2020003 #15004 2020391,2020393 #15005 2020409,2020411 #15006 2020661,2020663 #15007 2020721,2020723 #15008 2020727,2020729 #15009 2020817,2020819 #15010 2021081,2021083 #15011 2021597,2021599 #15012 2021627,2021629 #15013 2021651,2021653 #15014 2021777,2021779 #15015 2021837,2021839 #15016 2022017,2022019 #15017 2022047,2022049 #15018 2022101,2022103 #15019 2022281,2022283 #15020 2022329,2022331 #15021 2022401,2022403 #15022 2022617,2022619 #15023 2022659,2022661 #15024 2022749,2022751 #15025 2022989,2022991 #15026 2023157,2023159 #15027 2023529,2023531 #15028 2023577,2023579 #15029 2023829,2023831 #15030 2023841,2023843 #15031 2024177,2024179 #15032 2024219,2024221 #15033 2024261,2024263 #15034 2024327,2024329 #15035 2024369,2024371 #15036 2024417,2024419 #15037 2024597,2024599 #15038 2024831,2024833 #15039 2024861,2024863 #15040 2025251,2025253 #15041 2025347,2025349 #15042 2025629,2025631 #15043 2025641,2025643 #15044 2025719,2025721 #15045 2025899,2025901 #15046 2026151,2026153 #15047 2026181,2026183 #15048 2026391,2026393 #15049 2026469,2026471 #15050 2026727,2026729 #15051 2027021,2027023 #15052 2027099,2027101 #15053 2027159,2027161 #15054 2027237,2027239 #15055 2027447,2027449 #15056 2027567,2027569 #15057 2027897,2027899 #15058 2027951,2027953 #15059 2028107,2028109 #15060 2028119,2028121 #15061 2028137,2028139 #15062 2028197,2028199 #15063 2028239,2028241 #15064 2028371,2028373 #15065 2028701,2028703 #15066 2028779,2028781 #15067 2029019,2029021 #15068 2029121,2029123 #15069 2029241,2029243 #15070 2029439,2029441 #15071 2029499,2029501 #15072 2029667,2029669 #15073 2029721,2029723 #15074 2029799,2029801 #15075 2029829,2029831 #15076 2029871,2029873 #15077 2029889,2029891 #15078 2030051,2030053 #15079 2030099,2030101 #15080 2030309,2030311 #15081 2030381,2030383 #15082 2030459,2030461 #15083 2030657,2030659 #15084 2030879,2030881 #15085 2030909,2030911 #15086 2031569,2031571 #15087 2031977,2031979 #15088 2032109,2032111 #15089 2032157,2032159 #15090 2032271,2032273 #15091 2032361,2032363 #15092 2032559,2032561 #15093 2032619,2032621 #15094 2032649,2032651 #15095 2032661,2032663 #15096 2032859,2032861 #15097 2032937,2032939 #15098 2032967,2032969 #15099 2033201,2033203 #15100 2033279,2033281 #15101 2033357,2033359 #15102 2033441,2033443 #15103 2033459,2033461 #15104 2033531,2033533 #15105 2033609,2033611 #15106 2033951,2033953 #15107 2034209,2034211 #15108 2034491,2034493 #15109 2034839,2034841 #15110 2035067,2035069 #15111 2035211,2035213 #15112 2035301,2035303 #15113 2035511,2035513 #15114 2035667,2035669 #15115 2035841,2035843 #15116 2036129,2036131 #15117 2036339,2036341 #15118 2036807,2036809 #15119 2036831,2036833 #15120 2036861,2036863 #15121 2036891,2036893 #15122 2036939,2036941 #15123 2037017,2037019 #15124 2037071,2037073 #15125 2037149,2037151 #15126 2037251,2037253 #15127 2037281,2037283 #15128 2037377,2037379 #15129 2037491,2037493 #15130 2037851,2037853 #15131 2038019,2038021 #15132 2038427,2038429 #15133 2038577,2038579 #15134 2038637,2038639 #15135 2038919,2038921 #15136 2039171,2039173 #15137 2039351,2039353 #15138 2039621,2039623 #15139 2039909,2039911 #15140 2039927,2039929 #15141 2040107,2040109 #15142 2040149,2040151 #15143 2040191,2040193 #15144 2040251,2040253 #15145 2040431,2040433 #15146 2040539,2040541 #15147 2040557,2040559 #15148 2040917,2040919 #15149 2041199,2041201 #15150 2042399,2042401 #15151 2042849,2042851 #15152 2042981,2042983 #15153 2043191,2043193 #15154 2043257,2043259 #15155 2043539,2043541 #15156 2043719,2043721 #15157 2043749,2043751 #15158 2043761,2043763 #15159 2044067,2044069 #15160 2044127,2044129 #15161 2044277,2044279 #15162 2044487,2044489 #15163 2044787,2044789 #15164 2044841,2044843 #15165 2044919,2044921 #15166 2045009,2045011 #15167 2045189,2045191 #15168 2045357,2045359 #15169 2045567,2045569 #15170 2045609,2045611 #15171 2045651,2045653 #15172 2045759,2045761 #15173 2045837,2045839 #15174 2046047,2046049 #15175 2046311,2046313 #15176 2046389,2046391 #15177 2046719,2046721 #15178 2046827,2046829 #15179 2046971,2046973 #15180 2047037,2047039 #15181 2047061,2047063 #15182 2047091,2047093 #15183 2047181,2047183 #15184 2047217,2047219 #15185 2047349,2047351 #15186 2047811,2047813 #15187 2048327,2048329 #15188 2049041,2049043 #15189 2049119,2049121 #15190 2049347,2049349 #15191 2049407,2049409 #15192 2049449,2049451 #15193 2049491,2049493 #15194 2049611,2049613 #15195 2049821,2049823 #15196 2050031,2050033 #15197 2050229,2050231 #15198 2050331,2050333 #15199 2050337,2050339 #15200 2050511,2050513 #15201 2050817,2050819 #15202 2051111,2051113 #15203 2051171,2051173 #15204 2051249,2051251 #15205 2051279,2051281 #15206 2051321,2051323 #15207 2051417,2051419 #15208 2051459,2051461 #15209 2051477,2051479 #15210 2051627,2051629 #15211 2051891,2051893 #15212 2052047,2052049 #15213 2052179,2052181 #15214 2052329,2052331 #15215 2052749,2052751 #15216 2052857,2052859 #15217 2053067,2053069 #15218 2053109,2053111 #15219 2053211,2053213 #15220 2053421,2053423 #15221 2053619,2053621 #15222 2053769,2053771 #15223 2053871,2053873 #15224 2054009,2054011 #15225 2054021,2054023 #15226 2054231,2054233 #15227 2054249,2054251 #15228 2054579,2054581 #15229 2054627,2054629 #15230 2054849,2054851 #15231 2055101,2055103 #15232 2055197,2055199 #15233 2055479,2055481 #15234 2055509,2055511 #15235 2055707,2055709 #15236 2056079,2056081 #15237 2056139,2056141 #15238 2056277,2056279 #15239 2056751,2056753 #15240 2056841,2056843 #15241 2056907,2056909 #15242 2056961,2056963 #15243 2057021,2057023 #15244 2057177,2057179 #15245 2057381,2057383 #15246 2057399,2057401 #15247 2057477,2057479 #15248 2057597,2057599 #15249 2057609,2057611 #15250 2057777,2057779 #15251 2058011,2058013 #15252 2058191,2058193 #15253 2058557,2058559 #15254 2058701,2058703 #15255 2058839,2058841 #15256 2058869,2058871 #15257 2059271,2059273 #15258 2059709,2059711 #15259 2059721,2059723 #15260 2059817,2059819 #15261 2059859,2059861 #15262 2059931,2059933 #15263 2060099,2060101 #15264 2060159,2060161 #15265 2060249,2060251 #15266 2060447,2060449 #15267 2060561,2060563 #15268 2060579,2060581 #15269 2060627,2060629 #15270 2060801,2060803 #15271 2060879,2060881 #15272 2061077,2061079 #15273 2061179,2061181 #15274 2061287,2061289 #15275 2061599,2061601 #15276 2062001,2062003 #15277 2062007,2062009 #15278 2062061,2062063 #15279 2062199,2062201 #15280 2062517,2062519 #15281 2062757,2062759 #15282 2062871,2062873 #15283 2062889,2062891 #15284 2063057,2063059 #15285 2063249,2063251 #15286 2063291,2063293 #15287 2063459,2063461 #15288 2063561,2063563 #15289 2063729,2063731 #15290 2063771,2063773 #15291 2063777,2063779 #15292 2063861,2063863 #15293 2064149,2064151 #15294 2064371,2064373 #15295 2064527,2064529 #15296 2064581,2064583 #15297 2064761,2064763 #15298 2064947,2064949 #15299 2065571,2065573 #15300 2065577,2065579 #15301 2065667,2065669 #15302 2065727,2065729 #15303 2065799,2065801 #15304 2066081,2066083 #15305 2066177,2066179 #15306 2066201,2066203 #15307 2066507,2066509 #15308 2066549,2066551 #15309 2066681,2066683 #15310 2066759,2066761 #15311 2066969,2066971 #15312 2067071,2067073 #15313 2067209,2067211 #15314 2067719,2067721 #15315 2067797,2067799 #15316 2067851,2067853 #15317 2068037,2068039 #15318 2068061,2068063 #15319 2068487,2068489 #15320 2068499,2068501 #15321 2068637,2068639 #15322 2068751,2068753 #15323 2068811,2068813 #15324 2069381,2069383 #15325 2069531,2069533 #15326 2069909,2069911 #15327 2069957,2069959 #15328 2069987,2069989 #15329 2070041,2070043 #15330 2070179,2070181 #15331 2070239,2070241 #15332 2070317,2070319 #15333 2070461,2070463 #15334 2070611,2070613 #15335 2070641,2070643 #15336 2070797,2070799 #15337 2071259,2071261 #15338 2071427,2071429 #15339 2071721,2071723 #15340 2071799,2071801 #15341 2071997,2071999 #15342 2072129,2072131 #15343 2072207,2072209 #15344 2072429,2072431 #15345 2072489,2072491 #15346 2072699,2072701 #15347 2073101,2073103 #15348 2073119,2073121 #15349 2073347,2073349 #15350 2073359,2073361 #15351 2073389,2073391 #15352 2073647,2073649 #15353 2073809,2073811 #15354 2074139,2074141 #15355 2074199,2074201 #15356 2074349,2074351 #15357 2074481,2074483 #15358 2074517,2074519 #15359 2074871,2074873 #15360 2074949,2074951 #15361 2075261,2075263 #15362 2075537,2075539 #15363 2075657,2075659 #15364 2075741,2075743 #15365 2075831,2075833 #15366 2075837,2075839 #15367 2075867,2075869 #15368 2075999,2076001 #15369 2076407,2076409 #15370 2076419,2076421 #15371 2076617,2076619 #15372 2077319,2077321 #15373 2077637,2077639 #15374 2077709,2077711 #15375 2077769,2077771 #15376 2077811,2077813 #15377 2078159,2078161 #15378 2078309,2078311 #15379 2078339,2078341 #15380 2078507,2078509 #15381 2078927,2078929 #15382 2079017,2079019 #15383 2079071,2079073 #15384 2079167,2079169 #15385 2079191,2079193 #15386 2079197,2079199 #15387 2079239,2079241 #15388 2079401,2079403 #15389 2079461,2079463 #15390 2079599,2079601 #15391 2079629,2079631 #15392 2079941,2079943 #15393 2080451,2080453 #15394 2080541,2080543 #15395 2080847,2080849 #15396 2080961,2080963 #15397 2081159,2081161 #15398 2081249,2081251 #15399 2081351,2081353 #15400 2081921,2081923 #15401 2082131,2082133 #15402 2082497,2082499 #15403 2082737,2082739 #15404 2082851,2082853 #15405 2082887,2082889 #15406 2083019,2083021 #15407 2083421,2083423 #15408 2083451,2083453 #15409 2083511,2083513 #15410 2083517,2083519 #15411 2083769,2083771 #15412 2083847,2083849 #15413 2084231,2084233 #15414 2084441,2084443 #15415 2084447,2084449 #15416 2084501,2084503 #15417 2084567,2084569 #15418 2084609,2084611 #15419 2084981,2084983 #15420 2085131,2085133 #15421 2085227,2085229 #15422 2085287,2085289 #15423 2085701,2085703 #15424 2085737,2085739 #15425 2085929,2085931 #15426 2086079,2086081 #15427 2086109,2086111 #15428 2086211,2086213 #15429 2086349,2086351 #15430 2086361,2086363 #15431 2086421,2086423 #15432 2086457,2086459 #15433 2086547,2086549 #15434 2086571,2086573 #15435 2086757,2086759 #15436 2086829,2086831 #15437 2087219,2087221 #15438 2087231,2087233 #15439 2087381,2087383 #15440 2087387,2087389 #15441 2087669,2087671 #15442 2087711,2087713 #15443 2087807,2087809 #15444 2088011,2088013 #15445 2088131,2088133 #15446 2088341,2088343 #15447 2088407,2088409 #15448 2088419,2088421 #15449 2088587,2088589 #15450 2088599,2088601 #15451 2088641,2088643 #15452 2088719,2088721 #15453 2088731,2088733 #15454 2088869,2088871 #15455 2088971,2088973 #15456 2089037,2089039 #15457 2089049,2089051 #15458 2089091,2089093 #15459 2089271,2089273 #15460 2089391,2089393 #15461 2089541,2089543 #15462 2090069,2090071 #15463 2090279,2090281 #15464 2090327,2090329 #15465 2090351,2090353 #15466 2090717,2090719 #15467 2091149,2091151 #15468 2091239,2091241 #15469 2091281,2091283 #15470 2091317,2091319 #15471 2091707,2091709 #15472 2092019,2092021 #15473 2092427,2092429 #15474 2092589,2092591 #15475 2092661,2092663 #15476 2092721,2092723 #15477 2092799,2092801 #15478 2092859,2092861 #15479 2092997,2092999 #15480 2093321,2093323 #15481 2093489,2093491 #15482 2093699,2093701 #15483 2094107,2094109 #15484 2094341,2094343 #15485 2094359,2094361 #15486 2094749,2094751 #15487 2094809,2094811 #15488 2095361,2095363 #15489 2095397,2095399 #15490 2095697,2095699 #15491 2096009,2096011 #15492 2096231,2096233 #15493 2096399,2096401 #15494 2096429,2096431 #15495 2096597,2096599 #15496 2096789,2096791 #15497 2096909,2096911 #15498 2096957,2096959 #15499 2097131,2097133 #15500 2097257,2097259 #15501 2097287,2097289 #15502 2097449,2097451 #15503 2097479,2097481 #15504 2097671,2097673 #15505 2097857,2097859 #15506 2098079,2098081 #15507 2098169,2098171 #15508 2098277,2098279 #15509 2098697,2098699 #15510 2098739,2098741 #15511 2098781,2098783 #15512 2099219,2099221 #15513 2099477,2099479 #15514 2099939,2099941 #15515 2100191,2100193 #15516 2100227,2100229 #15517 2100407,2100409 #15518 2100587,2100589 #15519 2101091,2101093 #15520 2101247,2101249 #15521 2101259,2101261 #15522 2101481,2101483 #15523 2101499,2101501 #15524 2101667,2101669 #15525 2101871,2101873 #15526 2101907,2101909 #15527 2102171,2102173 #15528 2102249,2102251 #15529 2102459,2102461 #15530 2102531,2102533 #15531 2103149,2103151 #15532 2103611,2103613 #15533 2103791,2103793 #15534 2104019,2104021 #15535 2104757,2104759 #15536 2105069,2105071 #15537 2105267,2105269 #15538 2105357,2105359 #15539 2105417,2105419 #15540 2105729,2105731 #15541 2106197,2106199 #15542 2106227,2106229 #15543 2106341,2106343 #15544 2106617,2106619 #15545 2106677,2106679 #15546 2106779,2106781 #15547 2106917,2106919 #15548 2106989,2106991 #15549 2107319,2107321 #15550 2107529,2107531 #15551 2107601,2107603 #15552 2107661,2107663 #15553 2107667,2107669 #15554 2108087,2108089 #15555 2108549,2108551 #15556 2108597,2108599 #15557 2108759,2108761 #15558 2108807,2108809 #15559 2108879,2108881 #15560 2108927,2108929 #15561 2109011,2109013 #15562 2109101,2109103 #15563 2109617,2109619 #15564 2109797,2109799 #15565 2109869,2109871 #15566 2110019,2110021 #15567 2110151,2110153 #15568 2110187,2110189 #15569 2110289,2110291 #15570 2110529,2110531 #15571 2110751,2110753 #15572 2110859,2110861 #15573 2110877,2110879 #15574 2110949,2110951 #15575 2111309,2111311 #15576 2111357,2111359 #15577 2111507,2111509 #15578 2111531,2111533 #15579 2111729,2111731 #15580 2111801,2111803 #15581 2111969,2111971 #15582 2112191,2112193 #15583 2112569,2112571 #15584 2112827,2112829 #15585 2113037,2113039 #15586 2113289,2113291 #15587 2113469,2113471 #15588 2113511,2113513 #15589 2113667,2113669 #15590 2113679,2113681 #15591 2113757,2113759 #15592 2114039,2114041 #15593 2114087,2114089 #15594 2114249,2114251 #15595 2114507,2114509 #15596 2114531,2114533 #15597 2114711,2114713 #15598 2114741,2114743 #15599 2114969,2114971 #15600 2115077,2115079 #15601 2115131,2115133 #15602 2115227,2115229 #15603 2115317,2115319 #15604 2115431,2115433 #15605 2116019,2116021 #15606 2116097,2116099 #15607 2116559,2116561 #15608 2116571,2116573 #15609 2116577,2116579 #15610 2116691,2116693 #15611 2116799,2116801 #15612 2116811,2116813 #15613 2116901,2116903 #15614 2116949,2116951 #15615 2116967,2116969 #15616 2117039,2117041 #15617 2117051,2117053 #15618 2117237,2117239 #15619 2117429,2117431 #15620 2117651,2117653 #15621 2117699,2117701 #15622 2118029,2118031 #15623 2118089,2118091 #15624 2118119,2118121 #15625 2118299,2118301 #15626 2119259,2119261 #15627 2119307,2119309 #15628 2119589,2119591 #15629 2119877,2119879 #15630 2119919,2119921 #15631 2119937,2119939 #15632 2119967,2119969 #15633 2120099,2120101 #15634 2120351,2120353 #15635 2120549,2120551 #15636 2120849,2120851 #15637 2121191,2121193 #15638 2121239,2121241 #15639 2121737,2121739 #15640 2121941,2121943 #15641 2122511,2122513 #15642 2122709,2122711 #15643 2122721,2122723 #15644 2122961,2122963 #15645 2123081,2123083 #15646 2123237,2123239 #15647 2123279,2123281 #15648 2123741,2123743 #15649 2123879,2123881 #15650 2123969,2123971 #15651 2124011,2124013 #15652 2124041,2124043 #15653 2124359,2124361 #15654 2124401,2124403 #15655 2124467,2124469 #15656 2124839,2124841 #15657 2125469,2125471 #15658 2125601,2125603 #15659 2125679,2125681 #15660 2125691,2125693 #15661 2126027,2126029 #15662 2126039,2126041 #15663 2126147,2126149 #15664 2126429,2126431 #15665 2126849,2126851 #15666 2126897,2126899 #15667 2127269,2127271 #15668 2127287,2127289 #15669 2127341,2127343 #15670 2127647,2127649 #15671 2127689,2127691 #15672 2127947,2127949 #15673 2127971,2127973 #15674 2128547,2128549 #15675 2128559,2128561 #15676 2128601,2128603 #15677 2128667,2128669 #15678 2128781,2128783 #15679 2128871,2128873 #15680 2128991,2128993 #15681 2129261,2129263 #15682 2129291,2129293 #15683 2129399,2129401 #15684 2129507,2129509 #15685 2129549,2129551 #15686 2129597,2129599 #15687 2129819,2129821 #15688 2129849,2129851 #15689 2130239,2130241 #15690 2130341,2130343 #15691 2130437,2130439 #15692 2130617,2130619 #15693 2130671,2130673 #15694 2130701,2130703 #15695 2130767,2130769 #15696 2131319,2131321 #15697 2131427,2131429 #15698 2131601,2131603 #15699 2131691,2131693 #15700 2131979,2131981 #15701 2132231,2132233 #15702 2132279,2132281 #15703 2132309,2132311 #15704 2132321,2132323 #15705 2132591,2132593 #15706 2132657,2132659 #15707 2132759,2132761 #15708 2132981,2132983 #15709 2133029,2133031 #15710 2133251,2133253 #15711 2133431,2133433 #15712 2133539,2133541 #15713 2133587,2133589 #15714 2133611,2133613 #15715 2133797,2133799 #15716 2134019,2134021 #15717 2134241,2134243 #15718 2134259,2134261 #15719 2134961,2134963 #15720 2135099,2135101 #15721 2135519,2135521 #15722 2135687,2135689 #15723 2135699,2135701 #15724 2135717,2135719 #15725 2136107,2136109 #15726 2136131,2136133 #15727 2136137,2136139 #15728 2136191,2136193 #15729 2136287,2136289 #15730 2136311,2136313 #15731 2136359,2136361 #15732 2136389,2136391 #15733 2136437,2136439 #15734 2136557,2136559 #15735 2136599,2136601 #15736 2136731,2136733 #15737 2136989,2136991 #15738 2137151,2137153 #15739 2137409,2137411 #15740 2137547,2137549 #15741 2137979,2137981 #15742 2138249,2138251 #15743 2138399,2138401 #15744 2138501,2138503 #15745 2138831,2138833 #15746 2138987,2138989 #15747 2139407,2139409 #15748 2139461,2139463 #15749 2139497,2139499 #15750 2139539,2139541 #15751 2139659,2139661 #15752 2139857,2139859 #15753 2140001,2140003 #15754 2140601,2140603 #15755 2140847,2140849 #15756 2140967,2140969 #15757 2141057,2141059 #15758 2141297,2141299 #15759 2141591,2141593 #15760 2141801,2141803 #15761 2141807,2141809 #15762 2141897,2141899 #15763 2142167,2142169 #15764 2142227,2142229 #15765 2142251,2142253 #15766 2142521,2142523 #15767 2142641,2142643 #15768 2143199,2143201 #15769 2143259,2143261 #15770 2143481,2143483 #15771 2143487,2143489 #15772 2143541,2143543 #15773 2143571,2143573 #15774 2143829,2143831 #15775 2143859,2143861 #15776 2144249,2144251 #15777 2144369,2144371 #15778 2144477,2144479 #15779 2144489,2144491 #15780 2144501,2144503 #15781 2144507,2144509 #15782 2144687,2144689 #15783 2144717,2144719 #15784 2144729,2144731 #15785 2144897,2144899 #15786 2144951,2144953 #15787 2145191,2145193 #15788 2145287,2145289 #15789 2145329,2145331 #15790 2145359,2145361 #15791 2145629,2145631 #15792 2145641,2145643 #15793 2145707,2145709 #15794 2145821,2145823 #15795 2146091,2146093 #15796 2146139,2146141 #15797 2146691,2146693 #15798 2146787,2146789 #15799 2147021,2147023 #15800 2147051,2147053 #15801 2147279,2147281 #15802 2147501,2147503 #15803 2147861,2147863 #15804 2147909,2147911 #15805 2147987,2147989 #15806 2148071,2148073 #15807 2148401,2148403 #15808 2148449,2148451 #15809 2148527,2148529 #15810 2148659,2148661 #15811 2148737,2148739 #15812 2149139,2149141 #15813 2149247,2149249 #15814 2149349,2149351 #15815 2149619,2149621 #15816 2149991,2149993 #15817 2150009,2150011 #15818 2150207,2150209 #15819 2150417,2150419 #15820 2150639,2150641 #15821 2150717,2150719 #15822 2150879,2150881 #15823 2151011,2151013 #15824 2151137,2151139 #15825 2151269,2151271 #15826 2151509,2151511 #15827 2151701,2151703 #15828 2152229,2152231 #15829 2152307,2152309 #15830 2152427,2152429 #15831 2152481,2152483 #15832 2152817,2152819 #15833 2152847,2152849 #15834 2153057,2153059 #15835 2153069,2153071 #15836 2153111,2153113 #15837 2153297,2153299 #15838 2153561,2153563 #15839 2154041,2154043 #15840 2154329,2154331 #15841 2154539,2154541 #15842 2154641,2154643 #15843 2154791,2154793 #15844 2154851,2154853 #15845 2155007,2155009 #15846 2155271,2155273 #15847 2155511,2155513 #15848 2155961,2155963 #15849 2156039,2156041 #15850 2156309,2156311 #15851 2156459,2156461 #15852 2156597,2156599 #15853 2156681,2156683 #15854 2156849,2156851 #15855 2157119,2157121 #15856 2157149,2157151 #15857 2157341,2157343 #15858 2157557,2157559 #15859 2157677,2157679 #15860 2157731,2157733 #15861 2157737,2157739 #15862 2157767,2157769 #15863 2157821,2157823 #15864 2157899,2157901 #15865 2158181,2158183 #15866 2158367,2158369 #15867 2158547,2158549 #15868 2158577,2158579 #15869 2158589,2158591 #15870 2158601,2158603 #15871 2158697,2158699 #15872 2158769,2158771 #15873 2158841,2158843 #15874 2159081,2159083 #15875 2159231,2159233 #15876 2159237,2159239 #15877 2159249,2159251 #15878 2159327,2159329 #15879 2159669,2159671 #15880 2159819,2159821 #15881 2159957,2159959 #15882 2160029,2160031 #15883 2160131,2160133 #15884 2160209,2160211 #15885 2160461,2160463 #15886 2160617,2160619 #15887 2160881,2160883 #15888 2161127,2161129 #15889 2161301,2161303 #15890 2161637,2161639 #15891 2161697,2161699 #15892 2162057,2162059 #15893 2162087,2162089 #15894 2162189,2162191 #15895 2162351,2162353 #15896 2162507,2162509 #15897 2162579,2162581 #15898 2162957,2162959 #15899 2163011,2163013 #15900 2163041,2163043 #15901 2163221,2163223 #15902 2163347,2163349 #15903 2163479,2163481 #15904 2163569,2163571 #15905 2163671,2163673 #15906 2163827,2163829 #15907 2163881,2163883 #15908 2164037,2164039 #15909 2164607,2164609 #15910 2164619,2164621 #15911 2165027,2165029 #15912 2165081,2165083 #15913 2165321,2165323 #15914 2165531,2165533 #15915 2165591,2165593 #15916 2165771,2165773 #15917 2165957,2165959 #15918 2166119,2166121 #15919 2166509,2166511 #15920 2166917,2166919 #15921 2166947,2166949 #15922 2167019,2167021 #15923 2167091,2167093 #15924 2167259,2167261 #15925 2167367,2167369 #15926 2167439,2167441 #15927 2167469,2167471 #15928 2167769,2167771 #15929 2167937,2167939 #15930 2168057,2168059 #15931 2168291,2168293 #15932 2168519,2168521 #15933 2168651,2168653 #15934 2168657,2168659 #15935 2168669,2168671 #15936 2168687,2168689 #15937 2168711,2168713 #15938 2168861,2168863 #15939 2168951,2168953 #15940 2168987,2168989 #15941 2169029,2169031 #15942 2169071,2169073 #15943 2169311,2169313 #15944 2169359,2169361 #15945 2169467,2169469 #15946 2169509,2169511 #15947 2169617,2169619 #15948 2170109,2170111 #15949 2170241,2170243 #15950 2170409,2170411 #15951 2170937,2170939 #15952 2171159,2171161 #15953 2171621,2171623 #15954 2171759,2171761 #15955 2172089,2172091 #15956 2172227,2172229 #15957 2172239,2172241 #15958 2172827,2172829 #15959 2172851,2172853 #15960 2172869,2172871 #15961 2172977,2172979 #15962 2173079,2173081 #15963 2173151,2173153 #15964 2173361,2173363 #15965 2173529,2173531 #15966 2173571,2173573 #15967 2173649,2173651 #15968 2173727,2173729 #15969 2173877,2173879 #15970 2174399,2174401 #15971 2174591,2174593 #15972 2174609,2174611 #15973 2174699,2174701 #15974 2174771,2174773 #15975 2175449,2175451 #15976 2175599,2175601 #15977 2175659,2175661 #15978 2175791,2175793 #15979 2175851,2175853 #15980 2176409,2176411 #15981 2176547,2176549 #15982 2176631,2176633 #15983 2176637,2176639 #15984 2176829,2176831 #15985 2176871,2176873 #15986 2177009,2177011 #15987 2177237,2177239 #15988 2177321,2177323 #15989 2177429,2177431 #15990 2177447,2177449 #15991 2177501,2177503 #15992 2177507,2177509 #15993 2177519,2177521 #15994 2177597,2177599 #15995 2177687,2177689 #15996 2178131,2178133 #15997 2178149,2178151 #15998 2178257,2178259 #15999 2178641,2178643 #16000 2178677,2178679 #16001 2178731,2178733 #16002 2179139,2179141 #16003 2179607,2179609 #16004 2179649,2179651 #16005 2180177,2180179 #16006 2180219,2180221 #16007 2180681,2180683 #16008 2180921,2180923 #16009 2181071,2181073 #16010 2181227,2181229 #16011 2181329,2181331 #16012 2181461,2181463 #16013 2181539,2181541 #16014 2181869,2181871 #16015 2182007,2182009 #16016 2182097,2182099 #16017 2182559,2182561 #16018 2182601,2182603 #16019 2182811,2182813 #16020 2182991,2182993 #16021 2183339,2183341 #16022 2183507,2183509 #16023 2183579,2183581 #16024 2183681,2183683 #16025 2183771,2183773 #16026 2183789,2183791 #16027 2183807,2183809 #16028 2183957,2183959 #16029 2184197,2184199 #16030 2184317,2184319 #16031 2184359,2184361 #16032 2184407,2184409 #16033 2184647,2184649 #16034 2184989,2184991 #16035 2185187,2185189 #16036 2185199,2185201 #16037 2185427,2185429 #16038 2185697,2185699 #16039 2185871,2185873 #16040 2185919,2185921 #16041 2186099,2186101 #16042 2186837,2186839 #16043 2187959,2187961 #16044 2187971,2187973 #16045 2188031,2188033 #16046 2188157,2188159 #16047 2188169,2188171 #16048 2188409,2188411 #16049 2188607,2188609 #16050 2188787,2188789 #16051 2188871,2188873 #16052 2189027,2189029 #16053 2189219,2189221 #16054 2189321,2189323 #16055 2189417,2189419 #16056 2189459,2189461 #16057 2189741,2189743 #16058 2189879,2189881 #16059 2189987,2189989 #16060 2190077,2190079 #16061 2190191,2190193 #16062 2190269,2190271 #16063 2190479,2190481 #16064 2190521,2190523 #16065 2190581,2190583 #16066 2190821,2190823 #16067 2191067,2191069 #16068 2191169,2191171 #16069 2191337,2191339 #16070 2191457,2191459 #16071 2191949,2191951 #16072 2192051,2192053 #16073 2192129,2192131 #16074 2192249,2192251 #16075 2192339,2192341 #16076 2192621,2192623 #16077 2192651,2192653 #16078 2192789,2192791 #16079 2192849,2192851 #16080 2193311,2193313 #16081 2193419,2193421 #16082 2193479,2193481 #16083 2193599,2193601 #16084 2193641,2193643 #16085 2193701,2193703 #16086 2193881,2193883 #16087 2193887,2193889 #16088 2193941,2193943 #16089 2194019,2194021 #16090 2194319,2194321 #16091 2194529,2194531 #16092 2194721,2194723 #16093 2194901,2194903 #16094 2194991,2194993 #16095 2195117,2195119 #16096 2195339,2195341 #16097 2195381,2195383 #16098 2195441,2195443 #16099 2195579,2195581 #16100 2195729,2195731 #16101 2195861,2195863 #16102 2196287,2196289 #16103 2196539,2196541 #16104 2196611,2196613 #16105 2196869,2196871 #16106 2196977,2196979 #16107 2197409,2197411 #16108 2197631,2197633 #16109 2197847,2197849 #16110 2198291,2198293 #16111 2198759,2198761 #16112 2198879,2198881 #16113 2199179,2199181 #16114 2199311,2199313 #16115 2199521,2199523 #16116 2199959,2199961 #16117 2200139,2200141 #16118 2200589,2200591 #16119 2200619,2200621 #16120 2200727,2200729 #16121 2200811,2200813 #16122 2200841,2200843 #16123 2201189,2201191 #16124 2201201,2201203 #16125 2201531,2201533 #16126 2201597,2201599 #16127 2201669,2201671 #16128 2202047,2202049 #16129 2202131,2202133 #16130 2202311,2202313 #16131 2202377,2202379 #16132 2202437,2202439 #16133 2202791,2202793 #16134 2202797,2202799 #16135 2202857,2202859 #16136 2202929,2202931 #16137 2203301,2203303 #16138 2203631,2203633 #16139 2203961,2203963 #16140 2203967,2203969 #16141 2204009,2204011 #16142 2204471,2204473 #16143 2204831,2204833 #16144 2205011,2205013 #16145 2205449,2205451 #16146 2205587,2205589 #16147 2205611,2205613 #16148 2205659,2205661 #16149 2205947,2205949 #16150 2206121,2206123 #16151 2206151,2206153 #16152 2206247,2206249 #16153 2206439,2206441 #16154 2206469,2206471 #16155 2206619,2206621 #16156 2206817,2206819 #16157 2207201,2207203 #16158 2207279,2207281 #16159 2207321,2207323 #16160 2207537,2207539 #16161 2207831,2207833 #16162 2207981,2207983 #16163 2208257,2208259 #16164 2208707,2208709 #16165 2208797,2208799 #16166 2208887,2208889 #16167 2209001,2209003 #16168 2209061,2209063 #16169 2209169,2209171 #16170 2209547,2209549 #16171 2209661,2209663 #16172 2209787,2209789 #16173 2209841,2209843 #16174 2209901,2209903 #16175 2209937,2209939 #16176 2210009,2210011 #16177 2210027,2210029 #16178 2210057,2210059 #16179 2210279,2210281 #16180 2210387,2210389 #16181 2210567,2210569 #16182 2210651,2210653 #16183 2210777,2210779 #16184 2211257,2211259 #16185 2211929,2211931 #16186 2212097,2212099 #16187 2212181,2212183 #16188 2212349,2212351 #16189 2212631,2212633 #16190 2212781,2212783 #16191 2213201,2213203 #16192 2213399,2213401 #16193 2213411,2213413 #16194 2213591,2213593 #16195 2213837,2213839 #16196 2214101,2214103 #16197 2214269,2214271 #16198 2214479,2214481 #16199 2214491,2214493 #16200 2215097,2215099 #16201 2215307,2215309 #16202 2215349,2215351 #16203 2215469,2215471 #16204 2215529,2215531 #16205 2215667,2215669 #16206 2215691,2215693 #16207 2215901,2215903 #16208 2216321,2216323 #16209 2216609,2216611 #16210 2216657,2216659 #16211 2216699,2216701 #16212 2216759,2216761 #16213 2216999,2217001 #16214 2217491,2217493 #16215 2217539,2217541 #16216 2217569,2217571 #16217 2217581,2217583 #16218 2217641,2217643 #16219 2217671,2217673 #16220 2218091,2218093 #16221 2218127,2218129 #16222 2218199,2218201 #16223 2218427,2218429 #16224 2218547,2218549 #16225 2218607,2218609 #16226 2218901,2218903 #16227 2218967,2218969 #16228 2219081,2219083 #16229 2219279,2219281 #16230 2219351,2219353 #16231 2219489,2219491 #16232 2219681,2219683 #16233 2219771,2219773 #16234 2220527,2220529 #16235 2220551,2220553 #16236 2220917,2220919 #16237 2220971,2220973 #16238 2221127,2221129 #16239 2221229,2221231 #16240 2221379,2221381 #16241 2221631,2221633 #16242 2221859,2221861 #16243 2221907,2221909 #16244 2222249,2222251 #16245 2222501,2222503 #16246 2223161,2223163 #16247 2223281,2223283 #16248 2223449,2223451 #16249 2223467,2223469 #16250 2223497,2223499 #16251 2223671,2223673 #16252 2223677,2223679 #16253 2223839,2223841 #16254 2224457,2224459 #16255 2224667,2224669 #16256 2224679,2224681 #16257 2225051,2225053 #16258 2225057,2225059 #16259 2225231,2225233 #16260 2225387,2225389 #16261 2225567,2225569 #16262 2225579,2225581 #16263 2225681,2225683 #16264 2225747,2225749 #16265 2225999,2226001 #16266 2226149,2226151 #16267 2226197,2226199 #16268 2226227,2226229 #16269 2226311,2226313 #16270 2226407,2226409 #16271 2226461,2226463 #16272 2226527,2226529 #16273 2226569,2226571 #16274 2226617,2226619 #16275 2226767,2226769 #16276 2226941,2226943 #16277 2227031,2227033 #16278 2227061,2227063 #16279 2227259,2227261 #16280 2227367,2227369 #16281 2227397,2227399 #16282 2227439,2227441 #16283 2227499,2227501 #16284 2227607,2227609 #16285 2227649,2227651 #16286 2228117,2228119 #16287 2228321,2228323 #16288 2228507,2228509 #16289 2228519,2228521 #16290 2228531,2228533 #16291 2228657,2228659 #16292 2228711,2228713 #16293 2228981,2228983 #16294 2229041,2229043 #16295 2229119,2229121 #16296 2229167,2229169 #16297 2229389,2229391 #16298 2229587,2229589 #16299 2229767,2229769 #16300 2229791,2229793 #16301 2230157,2230159 #16302 2230409,2230411 #16303 2230511,2230513 #16304 2230871,2230873 #16305 2231309,2231311 #16306 2231429,2231431 #16307 2231477,2231479 #16308 2231819,2231821 #16309 2232509,2232511 #16310 2232749,2232751 #16311 2232779,2232781 #16312 2232887,2232889 #16313 2232929,2232931 #16314 2233079,2233081 #16315 2233199,2233201 #16316 2233379,2233381 #16317 2233499,2233501 #16318 2233529,2233531 #16319 2233571,2233573 #16320 2233709,2233711 #16321 2233877,2233879 #16322 2233937,2233939 #16323 2234117,2234119 #16324 2234159,2234161 #16325 2234207,2234209 #16326 2234339,2234341 #16327 2234501,2234503 #16328 2234591,2234593 #16329 2234717,2234719 #16330 2234927,2234929 #16331 2235047,2235049 #16332 2235137,2235139 #16333 2235227,2235229 #16334 2235509,2235511 #16335 2235731,2235733 #16336 2235809,2235811 #16337 2235941,2235943 #16338 2235971,2235973 #16339 2236007,2236009 #16340 2236049,2236051 #16341 2236079,2236081 #16342 2236187,2236189 #16343 2236517,2236519 #16344 2236709,2236711 #16345 2236769,2236771 #16346 2237399,2237401 #16347 2237561,2237563 #16348 2237771,2237773 #16349 2238011,2238013 #16350 2238161,2238163 #16351 2238209,2238211 #16352 2238287,2238289 #16353 2238359,2238361 #16354 2238419,2238421 #16355 2238527,2238529 #16356 2238569,2238571 #16357 2238809,2238811 #16358 2238959,2238961 #16359 2239007,2239009 #16360 2239217,2239219 #16361 2239229,2239231 #16362 2239331,2239333 #16363 2239649,2239651 #16364 2239709,2239711 #16365 2239751,2239753 #16366 2240111,2240113 #16367 2240321,2240323 #16368 2240477,2240479 #16369 2240531,2240533 #16370 2240657,2240659 #16371 2240699,2240701 #16372 2240807,2240809 #16373 2240837,2240839 #16374 2241011,2241013 #16375 2241047,2241049 #16376 2241119,2241121 #16377 2241191,2241193 #16378 2241299,2241301 #16379 2241311,2241313 #16380 2241359,2241361 #16381 2241389,2241391 #16382 2241521,2241523 #16383 2241779,2241781 #16384 2241917,2241919 #16385 2242127,2242129 #16386 2242187,2242189 #16387 2242211,2242213 #16388 2242379,2242381 #16389 2242517,2242519 #16390 2242727,2242729 #16391 2242781,2242783 #16392 2242811,2242813 #16393 2242841,2242843 #16394 2242871,2242873 #16395 2242949,2242951 #16396 2243207,2243209 #16397 2243429,2243431 #16398 2243621,2243623 #16399 2243741,2243743 #16400 2243819,2243821 #16401 2243909,2243911 #16402 2244257,2244259 #16403 2244587,2244589 #16404 2244659,2244661 #16405 2244689,2244691 #16406 2244719,2244721 #16407 2244881,2244883 #16408 2245427,2245429 #16409 2245457,2245459 #16410 2245541,2245543 #16411 2245679,2245681 #16412 2245721,2245723 #16413 2245811,2245813 #16414 2246051,2246053 #16415 2246141,2246143 #16416 2246147,2246149 #16417 2246357,2246359 #16418 2246687,2246689 #16419 2246789,2246791 #16420 2246969,2246971 #16421 2247101,2247103 #16422 2247227,2247229 #16423 2247461,2247463 #16424 2247521,2247523 #16425 2247809,2247811 #16426 2247851,2247853 #16427 2247899,2247901 #16428 2248019,2248021 #16429 2248067,2248069 #16430 2248241,2248243 #16431 2248247,2248249 #16432 2248331,2248333 #16433 2248529,2248531 #16434 2248637,2248639 #16435 2248661,2248663 #16436 2248679,2248681 #16437 2248691,2248693 #16438 2248847,2248849 #16439 2249309,2249311 #16440 2249411,2249413 #16441 2249981,2249983 #16442 2250041,2250043 #16443 2250089,2250091 #16444 2250401,2250403 #16445 2250419,2250421 #16446 2250761,2250763 #16447 2250929,2250931 #16448 2250977,2250979 #16449 2251199,2251201 #16450 2251211,2251213 #16451 2251727,2251729 #16452 2251901,2251903 #16453 2251979,2251981 #16454 2252009,2252011 #16455 2252219,2252221 #16456 2252231,2252233 #16457 2252309,2252311 #16458 2252387,2252389 #16459 2252681,2252683 #16460 2252951,2252953 #16461 2253257,2253259 #16462 2253281,2253283 #16463 2253479,2253481 #16464 2253497,2253499 #16465 2253971,2253973 #16466 2254097,2254099 #16467 2254157,2254159 #16468 2254409,2254411 #16469 2254781,2254783 #16470 2254799,2254801 #16471 2254871,2254873 #16472 2255021,2255023 #16473 2255159,2255161 #16474 2255249,2255251 #16475 2255549,2255551 #16476 2255567,2255569 #16477 2255969,2255971 #16478 2255987,2255989 #16479 2256029,2256031 #16480 2256179,2256181 #16481 2256311,2256313 #16482 2256341,2256343 #16483 2256347,2256349 #16484 2256377,2256379 #16485 2256467,2256469 #16486 2256557,2256559 #16487 2256911,2256913 #16488 2257049,2257051 #16489 2257439,2257441 #16490 2257529,2257531 #16491 2257691,2257693 #16492 2257859,2257861 #16493 2258327,2258329 #16494 2258519,2258521 #16495 2258651,2258653 #16496 2258741,2258743 #16497 2258819,2258821 #16498 2259029,2259031 #16499 2259137,2259139 #16500 2259197,2259199 #16501 2259239,2259241 #16502 2259527,2259529 #16503 2260169,2260171 #16504 2260499,2260501 #16505 2260547,2260549 #16506 2260631,2260633 #16507 2260649,2260651 #16508 2260787,2260789 #16509 2260889,2260891 #16510 2261267,2261269 #16511 2261471,2261473 #16512 2261801,2261803 #16513 2261837,2261839 #16514 2262641,2262643 #16515 2262857,2262859 #16516 2262971,2262973 #16517 2262977,2262979 #16518 2263067,2263069 #16519 2263139,2263141 #16520 2263169,2263171 #16521 2263319,2263321 #16522 2263439,2263441 #16523 2263517,2263519 #16524 2263559,2263561 #16525 2263739,2263741 #16526 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2270171,2270173 #16570 2270249,2270251 #16571 2270267,2270269 #16572 2270309,2270311 #16573 2270339,2270341 #16574 2270447,2270449 #16575 2270549,2270551 #16576 2270687,2270689 #16577 2270771,2270773 #16578 2270837,2270839 #16579 2271161,2271163 #16580 2271221,2271223 #16581 2271341,2271343 #16582 2271497,2271499 #16583 2271551,2271553 #16584 2271569,2271571 #16585 2271749,2271751 #16586 2271881,2271883 #16587 2272199,2272201 #16588 2272217,2272219 #16589 2272451,2272453 #16590 2272547,2272549 #16591 2272727,2272729 #16592 2272859,2272861 #16593 2273069,2273071 #16594 2273309,2273311 #16595 2273501,2273503 #16596 2273567,2273569 #16597 2273669,2273671 #16598 2274269,2274271 #16599 2274287,2274289 #16600 2274407,2274409 #16601 2274521,2274523 #16602 2274689,2274691 #16603 2274761,2274763 #16604 2275199,2275201 #16605 2275391,2275393 #16606 2275529,2275531 #16607 2275769,2275771 #16608 2276231,2276233 #16609 2276399,2276401 #16610 2276429,2276431 #16611 2276999,2277001 #16612 2277551,2277553 #16613 2277617,2277619 #16614 2277659,2277661 #16615 2277731,2277733 #16616 2277809,2277811 #16617 2277857,2277859 #16618 2278019,2278021 #16619 2278091,2278093 #16620 2278139,2278141 #16621 2278259,2278261 #16622 2278301,2278303 #16623 2278517,2278519 #16624 2278691,2278693 #16625 2278811,2278813 #16626 2278961,2278963 #16627 2278979,2278981 #16628 2279117,2279119 #16629 2279351,2279353 #16630 2279489,2279491 #16631 2279567,2279569 #16632 2279741,2279743 #16633 2279897,2279899 #16634 2280071,2280073 #16635 2280167,2280169 #16636 2280401,2280403 #16637 2280671,2280673 #16638 2281001,2281003 #16639 2281229,2281231 #16640 2281379,2281381 #16641 2281661,2281663 #16642 2282321,2282323 #16643 2282381,2282383 #16644 2282897,2282899 #16645 2283137,2283139 #16646 2283317,2283319 #16647 2283497,2283499 #16648 2283581,2283583 #16649 2283707,2283709 #16650 2283731,2283733 #16651 2283887,2283889 #16652 2284211,2284213 #16653 2284277,2284279 #16654 2284367,2284369 #16655 2284487,2284489 #16656 2284871,2284873 #16657 2284949,2284951 #16658 2285069,2285071 #16659 2285159,2285161 #16660 2285219,2285221 #16661 2285249,2285251 #16662 2285357,2285359 #16663 2285399,2285401 #16664 2285639,2285641 #16665 2285741,2285743 #16666 2285861,2285863 #16667 2285891,2285893 #16668 2285951,2285953 #16669 2286197,2286199 #16670 2286377,2286379 #16671 2286797,2286799 #16672 2286881,2286883 #16673 2287247,2287249 #16674 2287289,2287291 #16675 2287421,2287423 #16676 2287529,2287531 #16677 2287667,2287669 #16678 2287991,2287993 #16679 2288057,2288059 #16680 2288261,2288263 #16681 2288747,2288749 #16682 2288771,2288773 #16683 2288807,2288809 #16684 2288831,2288833 #16685 2288927,2288929 #16686 2289149,2289151 #16687 2289179,2289181 #16688 2289431,2289433 #16689 2289641,2289643 #16690 2289647,2289649 #16691 2289839,2289841 #16692 2290031,2290033 #16693 2290037,2290039 #16694 2290151,2290153 #16695 2290571,2290573 #16696 2290829,2290831 #16697 2291351,2291353 #16698 2291477,2291479 #16699 2291657,2291659 #16700 2291747,2291749 #16701 2291801,2291803 #16702 2291909,2291911 #16703 2291999,2292001 #16704 2292359,2292361 #16705 2292461,2292463 #16706 2292947,2292949 #16707 2293139,2293141 #16708 2293301,2293303 #16709 2293391,2293393 #16710 2293481,2293483 #16711 2293631,2293633 #16712 2293727,2293729 #16713 2293799,2293801 #16714 2293829,2293831 #16715 2293847,2293849 #16716 2294009,2294011 #16717 2294051,2294053 #16718 2294057,2294059 #16719 2294249,2294251 #16720 2294309,2294311 #16721 2294429,2294431 #16722 2294489,2294491 #16723 2295077,2295079 #16724 2295479,2295481 #16725 2295539,2295541 #16726 2295719,2295721 #16727 2295911,2295913 #16728 2295947,2295949 #16729 2296079,2296081 #16730 2296517,2296519 #16731 2296727,2296729 #16732 2296781,2296783 #16733 2296871,2296873 #16734 2296907,2296909 #16735 2297039,2297041 #16736 2297369,2297371 #16737 2297591,2297593 #16738 2297717,2297719 #16739 2297747,2297749 #16740 2297759,2297761 #16741 2298011,2298013 #16742 2298071,2298073 #16743 2298209,2298211 #16744 2298311,2298313 #16745 2298377,2298379 #16746 2298389,2298391 #16747 2298761,2298763 #16748 2298839,2298841 #16749 2298869,2298871 #16750 2298887,2298889 #16751 2299481,2299483 #16752 2299601,2299603 #16753 2299937,2299939 #16754 2299949,2299951 #16755 2300201,2300203 #16756 2300267,2300269 #16757 2300279,2300281 #16758 2300609,2300611 #16759 2300951,2300953 #16760 2301029,2301031 #16761 2301197,2301199 #16762 2301281,2301283 #16763 2301491,2301493 #16764 2301569,2301571 #16765 2301599,2301601 #16766 2301707,2301709 #16767 2302217,2302219 #16768 2302301,2302303 #16769 2302379,2302381 #16770 2302451,2302453 #16771 2302679,2302681 #16772 2303531,2303533 #16773 2303591,2303593 #16774 2303597,2303599 #16775 2303627,2303629 #16776 2304017,2304019 #16777 2304317,2304319 #16778 2304689,2304691 #16779 2304791,2304793 #16780 2305109,2305111 #16781 2305337,2305339 #16782 2305361,2305363 #16783 2305409,2305411 #16784 2305481,2305483 #16785 2305607,2305609 #16786 2305649,2305651 #16787 2305967,2305969 #16788 2306039,2306041 #16789 2306327,2306329 #16790 2306387,2306389 #16791 2306567,2306569 #16792 2306639,2306641 #16793 2307161,2307163 #16794 2307449,2307451 #16795 2307467,2307469 #16796 2308001,2308003 #16797 2308049,2308051 #16798 2308181,2308183 #16799 2308529,2308531 #16800 2308679,2308681 #16801 2308721,2308723 #16802 2308841,2308843 #16803 2309231,2309233 #16804 2309339,2309341 #16805 2309759,2309761 #16806 2309891,2309893 #16807 2310221,2310223 #16808 2310479,2310481 #16809 2310491,2310493 #16810 2310701,2310703 #16811 2310731,2310733 #16812 2310767,2310769 #16813 2310899,2310901 #16814 2311409,2311411 #16815 2311469,2311471 #16816 2311667,2311669 #16817 2311739,2311741 #16818 2311817,2311819 #16819 2312201,2312203 #16820 2312747,2312749 #16821 2312897,2312899 #16822 2313161,2313163 #16823 2313347,2313349 #16824 2313401,2313403 #16825 2313431,2313433 #16826 2313539,2313541 #16827 2313599,2313601 #16828 2313629,2313631 #16829 2313767,2313769 #16830 2313797,2313799 #16831 2313929,2313931 #16832 2314061,2314063 #16833 2314589,2314591 #16834 2314721,2314723 #16835 2314841,2314843 #16836 2314931,2314933 #16837 2314997,2314999 #16838 2315057,2315059 #16839 2315231,2315233 #16840 2315657,2315659 #16841 2315771,2315773 #16842 2315981,2315983 #16843 2316329,2316331 #16844 2316371,2316373 #16845 2316449,2316451 #16846 2317121,2317123 #16847 2317169,2317171 #16848 2317499,2317501 #16849 2317787,2317789 #16850 2317811,2317813 #16851 2317919,2317921 #16852 2318189,2318191 #16853 2318387,2318389 #16854 2318597,2318599 #16855 2318609,2318611 #16856 2318807,2318809 #16857 2318819,2318821 #16858 2318957,2318959 #16859 2319179,2319181 #16860 2319407,2319409 #16861 2319431,2319433 #16862 2320361,2320363 #16863 2320397,2320399 #16864 2320649,2320651 #16865 2320697,2320699 #16866 2320739,2320741 #16867 2321087,2321089 #16868 2321147,2321149 #16869 2321381,2321383 #16870 2321507,2321509 #16871 2321747,2321749 #16872 2322077,2322079 #16873 2322107,2322109 #16874 2322119,2322121 #16875 2322401,2322403 #16876 2322491,2322493 #16877 2322569,2322571 #16878 2322629,2322631 #16879 2323001,2323003 #16880 2323037,2323039 #16881 2323229,2323231 #16882 2323259,2323261 #16883 2323367,2323369 #16884 2323379,2323381 #16885 2323421,2323423 #16886 2323457,2323459 #16887 2323691,2323693 #16888 2323817,2323819 #16889 2324351,2324353 #16890 2324501,2324503 #16891 2324681,2324683 #16892 2325317,2325319 #16893 2325437,2325439 #16894 2325509,2325511 #16895 2326019,2326021 #16896 2326097,2326099 #16897 2326211,2326213 #16898 2326277,2326279 #16899 2326367,2326369 #16900 2326481,2326483 #16901 2326661,2326663 #16902 2326769,2326771 #16903 2326991,2326993 #16904 2327027,2327029 #16905 2327051,2327053 #16906 2327399,2327401 #16907 2327597,2327599 #16908 2327639,2327641 #16909 2327651,2327653 #16910 2327681,2327683 #16911 2327711,2327713 #16912 2327849,2327851 #16913 2327867,2327869 #16914 2327909,2327911 #16915 2327951,2327953 #16916 2327987,2327989 #16917 2328281,2328283 #16918 2328617,2328619 #16919 2328761,2328763 #16920 2328827,2328829 #16921 2328971,2328973 #16922 2329337,2329339 #16923 2329469,2329471 #16924 2329517,2329519 #16925 2329667,2329669 #16926 2330099,2330101 #16927 2330201,2330203 #16928 2330387,2330389 #16929 2330687,2330689 #16930 2330927,2330929 #16931 2330957,2330959 #16932 2331377,2331379 #16933 2331419,2331421 #16934 2331647,2331649 #16935 2331689,2331691 #16936 2331779,2331781 #16937 2331869,2331871 #16938 2332397,2332399 #16939 2332511,2332513 #16940 2332661,2332663 #16941 2332829,2332831 #16942 2332931,2332933 #16943 2333081,2333083 #16944 2333237,2333239 #16945 2333321,2333323 #16946 2333531,2333533 #16947 2333867,2333869 #16948 2333951,2333953 #16949 2333999,2334001 #16950 2334257,2334259 #16951 2334401,2334403 #16952 2334767,2334769 #16953 2334779,2334781 #16954 2334947,2334949 #16955 2335217,2335219 #16956 2335241,2335243 #16957 2335367,2335369 #16958 2335547,2335549 #16959 2335637,2335639 #16960 2335691,2335693 #16961 2335967,2335969 #16962 2336207,2336209 #16963 2336309,2336311 #16964 2336471,2336473 #16965 2336861,2336863 #16966 2337089,2337091 #16967 2337149,2337151 #16968 2337317,2337319 #16969 2337479,2337481 #16970 2337539,2337541 #16971 2337869,2337871 #16972 2337899,2337901 #16973 2337911,2337913 #16974 2338079,2338081 #16975 2338151,2338153 #16976 2338541,2338543 #16977 2338871,2338873 #16978 2338949,2338951 #16979 2339039,2339041 #16980 2339369,2339371 #16981 2339609,2339611 #16982 2339669,2339671 #16983 2339681,2339683 #16984 2339927,2339929 #16985 2340251,2340253 #16986 2340257,2340259 #16987 2340419,2340421 #16988 2340491,2340493 #16989 2340659,2340661 #16990 2340719,2340721 #16991 2341217,2341219 #16992 2341301,2341303 #16993 2341457,2341459 #16994 2341817,2341819 #16995 2341979,2341981 #16996 2341991,2341993 #16997 2342027,2342029 #16998 2342099,2342101 #16999 2342189,2342191 #17000 2342201,2342203 #17001 2342237,2342239 #17002 2342399,2342401 #17003 2342537,2342539 #17004 2342609,2342611 #17005 2342771,2342773 #17006 2342777,2342779 #17007 2342981,2342983 #17008 2343239,2343241 #17009 2343359,2343361 #17010 2343527,2343529 #17011 2343611,2343613 #17012 2343641,2343643 #17013 2343791,2343793 #17014 2343881,2343883 #17015 2344259,2344261 #17016 2344469,2344471 #17017 2344649,2344651 #17018 2344709,2344711 #17019 2344751,2344753 #17020 2344787,2344789 #17021 2345039,2345041 #17022 2345129,2345131 #17023 2345459,2345461 #17024 2345477,2345479 #17025 2345657,2345659 #17026 2345729,2345731 #17027 2345807,2345809 #17028 2345867,2345869 #17029 2345921,2345923 #17030 2345969,2345971 #17031 2346269,2346271 #17032 2346347,2346349 #17033 2346521,2346523 #17034 2346779,2346781 #17035 2346857,2346859 #17036 2347151,2347153 #17037 2347271,2347273 #17038 2347337,2347339 #17039 2347439,2347441 #17040 2347451,2347453 #17041 2347559,2347561 #17042 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2354951,2354953 #17086 2355137,2355139 #17087 2355209,2355211 #17088 2355557,2355559 #17089 2355761,2355763 #17090 2355971,2355973 #17091 2356127,2356129 #17092 2356139,2356141 #17093 2356349,2356351 #17094 2356379,2356381 #17095 2356427,2356429 #17096 2356661,2356663 #17097 2356901,2356903 #17098 2357027,2357029 #17099 2357057,2357059 #17100 2357351,2357353 #17101 2357417,2357419 #17102 2357741,2357743 #17103 2357807,2357809 #17104 2357921,2357923 #17105 2357939,2357941 #17106 2358329,2358331 #17107 2358371,2358373 #17108 2358401,2358403 #17109 2358737,2358739 #17110 2358779,2358781 #17111 2358899,2358901 #17112 2358911,2358913 #17113 2359031,2359033 #17114 2359391,2359393 #17115 2359559,2359561 #17116 2359829,2359831 #17117 2360021,2360023 #17118 2360087,2360089 #17119 2360201,2360203 #17120 2360417,2360419 #17121 2360591,2360593 #17122 2360849,2360851 #17123 2361041,2361043 #17124 2361089,2361091 #17125 2361629,2361631 #17126 2361911,2361913 #17127 2361941,2361943 #17128 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2493707,2493709 #17903 2493719,2493721 #17904 2493851,2493853 #17905 2493947,2493949 #17906 2494031,2494033 #17907 2494169,2494171 #17908 2494379,2494381 #17909 2494421,2494423 #17910 2494439,2494441 #17911 2494829,2494831 #17912 2494949,2494951 #17913 2494979,2494981 #17914 2494991,2494993 #17915 2495321,2495323 #17916 2495459,2495461 #17917 2496077,2496079 #17918 2496251,2496253 #17919 2496269,2496271 #17920 2496491,2496493 #17921 2496827,2496829 #17922 2496917,2496919 #17923 2497211,2497213 #17924 2497421,2497423 #17925 2497511,2497513 #17926 2497631,2497633 #17927 2497751,2497753 #17928 2497877,2497879 #17929 2497931,2497933 #17930 2499059,2499061 #17931 2499137,2499139 #17932 2499149,2499151 #17933 2499641,2499643 #17934 2499779,2499781 #17935 2499941,2499943 #17936 2499947,2499949 #17937 2500049,2500051 #17938 2500439,2500441 #17939 2500637,2500639 #17940 2500781,2500783 #17941 2500889,2500891 #17942 2501099,2501101 #17943 2501249,2501251 #17944 2501591,2501593 #17945 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2517131,2517133 #18032 2517269,2517271 #18033 2517467,2517469 #18034 2517677,2517679 #18035 2517701,2517703 #18036 2517797,2517799 #18037 2518067,2518069 #18038 2518079,2518081 #18039 2518289,2518291 #18040 2518559,2518561 #18041 2518727,2518729 #18042 2518949,2518951 #18043 2519087,2519089 #18044 2519159,2519161 #18045 2519639,2519641 #18046 2519729,2519731 #18047 2519939,2519941 #18048 2520149,2520151 #18049 2520431,2520433 #18050 2520611,2520613 #18051 2520629,2520631 #18052 2520779,2520781 #18053 2521019,2521021 #18054 2521037,2521039 #18055 2521271,2521273 #18056 2521499,2521501 #18057 2521721,2521723 #18058 2521919,2521921 #18059 2522057,2522059 #18060 2522087,2522089 #18061 2522657,2522659 #18062 2522669,2522671 #18063 2522759,2522761 #18064 2522789,2522791 #18065 2522909,2522911 #18066 2522981,2522983 #18067 2523011,2523013 #18068 2523041,2523043 #18069 2523149,2523151 #18070 2523161,2523163 #18071 2523527,2523529 #18072 2523641,2523643 #18073 2523707,2523709 #18074 2524031,2524033 #18075 2524199,2524201 #18076 2524217,2524219 #18077 2524259,2524261 #18078 2524349,2524351 #18079 2524469,2524471 #18080 2524649,2524651 #18081 2524679,2524681 #18082 2524859,2524861 #18083 2524937,2524939 #18084 2525177,2525179 #18085 2525189,2525191 #18086 2525219,2525221 #18087 2525267,2525269 #18088 2525387,2525389 #18089 2525669,2525671 #18090 2526299,2526301 #18091 2526581,2526583 #18092 2526647,2526649 #18093 2527097,2527099 #18094 2527277,2527279 #18095 2527451,2527453 #18096 2527559,2527561 #18097 2527961,2527963 #18098 2528231,2528233 #18099 2528627,2528629 #18100 2528819,2528821 #18101 2528831,2528833 #18102 2528861,2528863 #18103 2528891,2528893 #18104 2529227,2529229 #18105 2529251,2529253 #18106 2529347,2529349 #18107 2529419,2529421 #18108 2529689,2529691 #18109 2529911,2529913 #18110 2530109,2530111 #18111 2530139,2530141 #18112 2530457,2530459 #18113 2530571,2530573 #18114 2530961,2530963 #18115 2530991,2530993 #18116 2531099,2531101 #18117 2531369,2531371 #18118 2531609,2531611 #18119 2531687,2531689 #18120 2531699,2531701 #18121 2531831,2531833 #18122 2531981,2531983 #18123 2532107,2532109 #18124 2532197,2532199 #18125 2532401,2532403 #18126 2532449,2532451 #18127 2532707,2532709 #18128 2532989,2532991 #18129 2533007,2533009 #18130 2533031,2533033 #18131 2533301,2533303 #18132 2534039,2534041 #18133 2534267,2534269 #18134 2534501,2534503 #18135 2534561,2534563 #18136 2534879,2534881 #18137 2534951,2534953 #18138 2535017,2535019 #18139 2535101,2535103 #18140 2535107,2535109 #18141 2535161,2535163 #18142 2535917,2535919 #18143 2536241,2536243 #18144 2536307,2536309 #18145 2536361,2536363 #18146 2536379,2536381 #18147 2536559,2536561 #18148 2536577,2536579 #18149 2536799,2536801 #18150 2536811,2536813 #18151 2536907,2536909 #18152 2537081,2537083 #18153 2537111,2537113 #18154 2537459,2537461 #18155 2537501,2537503 #18156 2538059,2538061 #18157 2538101,2538103 #18158 2538299,2538301 #18159 2538449,2538451 #18160 2538509,2538511 #18161 2538617,2538619 #18162 2538707,2538709 #18163 2538749,2538751 #18164 2538917,2538919 #18165 2539319,2539321 #18166 2539349,2539351 #18167 2539529,2539531 #18168 2539571,2539573 #18169 2539631,2539633 #18170 2539961,2539963 #18171 2540177,2540179 #18172 2540201,2540203 #18173 2540339,2540341 #18174 2540441,2540443 #18175 2540537,2540539 #18176 2540687,2540689 #18177 2540981,2540983 #18178 2541527,2541529 #18179 2541701,2541703 #18180 2541941,2541943 #18181 2541947,2541949 #18182 2542049,2542051 #18183 2542481,2542483 #18184 2542511,2542513 #18185 2542607,2542609 #18186 2542619,2542621 #18187 2543111,2543113 #18188 2543141,2543143 #18189 2543237,2543239 #18190 2543459,2543461 #18191 2543507,2543509 #18192 2543621,2543623 #18193 2544131,2544133 #18194 2544161,2544163 #18195 2544209,2544211 #18196 2544299,2544301 #18197 2544359,2544361 #18198 2544629,2544631 #18199 2544767,2544769 #18200 2544791,2544793 #18201 2544809,2544811 #18202 2544929,2544931 #18203 2545451,2545453 #18204 2545679,2545681 #18205 2545757,2545759 #18206 2545769,2545771 #18207 2545889,2545891 #18208 2546177,2546179 #18209 2546231,2546233 #18210 2546237,2546239 #18211 2546561,2546563 #18212 2546657,2546659 #18213 2546669,2546671 #18214 2546837,2546839 #18215 2546909,2546911 #18216 2547029,2547031 #18217 2547581,2547583 #18218 2547971,2547973 #18219 2548277,2548279 #18220 2548289,2548291 #18221 2548499,2548501 #18222 2548571,2548573 #18223 2548751,2548753 #18224 2548769,2548771 #18225 2548877,2548879 #18226 2549291,2549293 #18227 2549357,2549359 #18228 2549381,2549383 #18229 2549429,2549431 #18230 2549621,2549623 #18231 2550179,2550181 #18232 2550467,2550469 #18233 2550857,2550859 #18234 2550971,2550973 #18235 2551097,2551099 #18236 2551247,2551249 #18237 2551499,2551501 #18238 2551979,2551981 #18239 2552111,2552113 #18240 2552117,2552119 #18241 2552357,2552359 #18242 2552621,2552623 #18243 2552651,2552653 #18244 2552657,2552659 #18245 2552777,2552779 #18246 2553149,2553151 #18247 2553431,2553433 #18248 2553539,2553541 #18249 2553599,2553601 #18250 2553869,2553871 #18251 2554247,2554249 #18252 2554271,2554273 #18253 2554337,2554339 #18254 2554397,2554399 #18255 2554457,2554459 #18256 2554787,2554789 #18257 2554829,2554831 #18258 2555009,2555011 #18259 2555129,2555131 #18260 2555171,2555173 #18261 2555261,2555263 #18262 2555549,2555551 #18263 2556161,2556163 #18264 2556791,2556793 #18265 2557169,2557171 #18266 2557199,2557201 #18267 2557277,2557279 #18268 2557367,2557369 #18269 2557517,2557519 #18270 2557601,2557603 #18271 2558009,2558011 #18272 2558249,2558251 #18273 2558321,2558323 #18274 2558471,2558473 #18275 2558531,2558533 #18276 2558951,2558953 #18277 2559041,2559043 #18278 2559077,2559079 #18279 2559287,2559289 #18280 2559437,2559439 #18281 2559617,2559619 #18282 2559827,2559829 #18283 2560169,2560171 #18284 2560211,2560213 #18285 2560601,2560603 #18286 2560637,2560639 #18287 2560739,2560741 #18288 2560847,2560849 #18289 2560937,2560939 #18290 2561021,2561023 #18291 2561231,2561233 #18292 2561261,2561263 #18293 2561267,2561269 #18294 2561387,2561389 #18295 2561549,2561551 #18296 2561651,2561653 #18297 2561681,2561683 #18298 2561729,2561731 #18299 2561759,2561761 #18300 2561927,2561929 #18301 2562029,2562031 #18302 2562251,2562253 #18303 2562347,2562349 #18304 2562431,2562433 #18305 2562557,2562559 #18306 2562611,2562613 #18307 2562689,2562691 #18308 2562941,2562943 #18309 2562977,2562979 #18310 2563007,2563009 #18311 2563151,2563153 #18312 2563367,2563369 #18313 2563907,2563909 #18314 2564249,2564251 #18315 2564321,2564323 #18316 2564327,2564329 #18317 2564519,2564521 #18318 2565047,2565049 #18319 2565149,2565151 #18320 2565347,2565349 #18321 2565389,2565391 #18322 2565461,2565463 #18323 2566019,2566021 #18324 2566049,2566051 #18325 2566127,2566129 #18326 2566139,2566141 #18327 2566259,2566261 #18328 2566517,2566519 #18329 2566589,2566591 #18330 2567111,2567113 #18331 2567177,2567179 #18332 2567351,2567353 #18333 2567447,2567449 #18334 2567531,2567533 #18335 2567819,2567821 #18336 2568029,2568031 #18337 2568119,2568121 #18338 2568497,2568499 #18339 2568701,2568703 #18340 2568869,2568871 #18341 2568911,2568913 #18342 2569421,2569423 #18343 2569739,2569741 #18344 2569751,2569753 #18345 2569937,2569939 #18346 2570201,2570203 #18347 2570219,2570221 #18348 2570369,2570371 #18349 2570387,2570389 #18350 2570429,2570431 #18351 2570507,2570509 #18352 2570537,2570539 #18353 2570609,2570611 #18354 2570849,2570851 #18355 2571071,2571073 #18356 2571449,2571451 #18357 2571551,2571553 #18358 2571731,2571733 #18359 2572079,2572081 #18360 2572091,2572093 #18361 2572121,2572123 #18362 2572397,2572399 #18363 2572487,2572489 #18364 2572517,2572519 #18365 2572649,2572651 #18366 2572679,2572681 #18367 2572697,2572699 #18368 2572937,2572939 #18369 2573057,2573059 #18370 2573099,2573101 #18371 2573357,2573359 #18372 2574029,2574031 #18373 2574149,2574151 #18374 2574179,2574181 #18375 2574587,2574589 #18376 2574851,2574853 #18377 2575019,2575021 #18378 2575061,2575063 #18379 2575091,2575093 #18380 2575799,2575801 #18381 2575817,2575819 #18382 2575877,2575879 #18383 2575919,2575921 #18384 2576219,2576221 #18385 2576261,2576263 #18386 2576549,2576551 #18387 2576591,2576593 #18388 2576597,2576599 #18389 2576729,2576731 #18390 2576771,2576773 #18391 2577077,2577079 #18392 2577437,2577439 #18393 2577569,2577571 #18394 2577917,2577919 #18395 2577941,2577943 #18396 2578109,2578111 #18397 2578349,2578351 #18398 2578391,2578393 #18399 2578451,2578453 #18400 2578517,2578519 #18401 2578757,2578759 #18402 2578799,2578801 #18403 2578817,2578819 #18404 2578991,2578993 #18405 2579177,2579179 #18406 2579387,2579389 #18407 2579651,2579653 #18408 2579807,2579809 #18409 2580167,2580169 #18410 2580287,2580289 #18411 2580419,2580421 #18412 2580467,2580469 #18413 2580509,2580511 #18414 2580647,2580649 #18415 2580659,2580661 #18416 2580671,2580673 #18417 2580689,2580691 #18418 2580701,2580703 #18419 2580839,2580841 #18420 2580857,2580859 #18421 2581079,2581081 #18422 2581121,2581123 #18423 2581211,2581213 #18424 2581391,2581393 #18425 2581559,2581561 #18426 2581697,2581699 #18427 2581769,2581771 #18428 2581919,2581921 #18429 2581961,2581963 #18430 2582087,2582089 #18431 2582159,2582161 #18432 2582399,2582401 #18433 2582441,2582443 #18434 2582609,2582611 #18435 2582729,2582731 #18436 2582771,2582773 #18437 2583017,2583019 #18438 2583101,2583103 #18439 2583179,2583181 #18440 2583239,2583241 #18441 2583389,2583391 #18442 2583461,2583463 #18443 2583629,2583631 #18444 2583767,2583769 #18445 2583857,2583859 #18446 2584007,2584009 #18447 2584031,2584033 #18448 2584079,2584081 #18449 2584367,2584369 #18450 2584787,2584789 #18451 2584889,2584891 #18452 2584949,2584951 #18453 2585111,2585113 #18454 2585351,2585353 #18455 2585507,2585509 #18456 2585837,2585839 #18457 2585951,2585953 #18458 2585969,2585971 #18459 2585981,2585983 #18460 2586137,2586139 #18461 2586167,2586169 #18462 2586341,2586343 #18463 2586377,2586379 #18464 2586629,2586631 #18465 2586719,2586721 #18466 2586767,2586769 #18467 2586797,2586799 #18468 2586821,2586823 #18469 2587127,2587129 #18470 2587157,2587159 #18471 2587421,2587423 #18472 2587547,2587549 #18473 2587967,2587969 #18474 2588357,2588359 #18475 2588669,2588671 #18476 2588819,2588821 #18477 2588897,2588899 #18478 2589227,2589229 #18479 2589341,2589343 #18480 2589551,2589553 #18481 2589677,2589679 #18482 2589701,2589703 #18483 2589791,2589793 #18484 2590031,2590033 #18485 2590127,2590129 #18486 2590769,2590771 #18487 2590871,2590873 #18488 2591261,2591263 #18489 2591681,2591683 #18490 2591747,2591749 #18491 2591777,2591779 #18492 2591819,2591821 #18493 2591969,2591971 #18494 2592521,2592523 #18495 2592587,2592589 #18496 2592647,2592649 #18497 2592761,2592763 #18498 2592851,2592853 #18499 2592899,2592901 #18500 2593247,2593249 #18501 2593361,2593363 #18502 2593397,2593399 #18503 2593607,2593609 #18504 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2635907,2635909 #18763 2637197,2637199 #18764 2637251,2637253 #18765 2637539,2637541 #18766 2637671,2637673 #18767 2637797,2637799 #18768 2637959,2637961 #18769 2638199,2638201 #18770 2638367,2638369 #18771 2638631,2638633 #18772 2638841,2638843 #18773 2639069,2639071 #18774 2639177,2639179 #18775 2639489,2639491 #18776 2639699,2639701 #18777 2639759,2639761 #18778 2639939,2639941 #18779 2639969,2639971 #18780 2640137,2640139 #18781 2640347,2640349 #18782 2640509,2640511 #18783 2640707,2640709 #18784 2640821,2640823 #18785 2640917,2640919 #18786 2641061,2641063 #18787 2641151,2641153 #18788 2641277,2641279 #18789 2641367,2641369 #18790 2641421,2641423 #18791 2641517,2641519 #18792 2641547,2641549 #18793 2641631,2641633 #18794 2641799,2641801 #18795 2641829,2641831 #18796 2641907,2641909 #18797 2642039,2642041 #18798 2642147,2642149 #18799 2642201,2642203 #18800 2642357,2642359 #18801 2642789,2642791 #18802 2643059,2643061 #18803 2643131,2643133 #18804 2643239,2643241 #18805 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2656499,2656501 #18892 2656727,2656729 #18893 2656919,2656921 #18894 2656991,2656993 #18895 2657069,2657071 #18896 2657189,2657191 #18897 2657201,2657203 #18898 2657327,2657329 #18899 2657339,2657341 #18900 2657399,2657401 #18901 2657561,2657563 #18902 2658077,2658079 #18903 2658287,2658289 #18904 2658377,2658379 #18905 2658449,2658451 #18906 2658659,2658661 #18907 2658701,2658703 #18908 2658881,2658883 #18909 2658977,2658979 #18910 2659091,2659093 #18911 2659337,2659339 #18912 2659439,2659441 #18913 2659451,2659453 #18914 2659469,2659471 #18915 2659859,2659861 #18916 2660069,2660071 #18917 2660087,2660089 #18918 2660351,2660353 #18919 2660657,2660659 #18920 2661641,2661643 #18921 2661731,2661733 #18922 2661779,2661781 #18923 2661929,2661931 #18924 2661947,2661949 #18925 2661959,2661961 #18926 2662067,2662069 #18927 2662157,2662159 #18928 2662769,2662771 #18929 2662787,2662789 #18930 2662811,2662813 #18931 2662901,2662903 #18932 2662949,2662951 #18933 2663027,2663029 #18934 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2669201,2669203 #18978 2669267,2669269 #18979 2669339,2669341 #18980 2669417,2669419 #18981 2669507,2669509 #18982 2669801,2669803 #18983 2670719,2670721 #18984 2670791,2670793 #18985 2671169,2671171 #18986 2671181,2671183 #18987 2671367,2671369 #18988 2671649,2671651 #18989 2671661,2671663 #18990 2671679,2671681 #18991 2672387,2672389 #18992 2673059,2673061 #18993 2673257,2673259 #18994 2673329,2673331 #18995 2673521,2673523 #18996 2673527,2673529 #18997 2673761,2673763 #18998 2673791,2673793 #18999 2673857,2673859 #19000 2673989,2673991 #19001 2674121,2674123 #19002 2674367,2674369 #19003 2674541,2674543 #19004 2674547,2674549 #19005 2674577,2674579 #19006 2674979,2674981 #19007 2675009,2675011 #19008 2675087,2675089 #19009 2675129,2675131 #19010 2675147,2675149 #19011 2675201,2675203 #19012 2675327,2675329 #19013 2675831,2675833 #19014 2675921,2675923 #19015 2676239,2676241 #19016 2676407,2676409 #19017 2676491,2676493 #19018 2676719,2676721 #19019 2677097,2677099 #19020 2677151,2677153 #19021 2677289,2677291 #19022 2677817,2677819 #19023 2677901,2677903 #19024 2678147,2678149 #19025 2678321,2678323 #19026 2678381,2678383 #19027 2678549,2678551 #19028 2678747,2678749 #19029 2679239,2679241 #19030 2679401,2679403 #19031 2679461,2679463 #19032 2679491,2679493 #19033 2679497,2679499 #19034 2679617,2679619 #19035 2679641,2679643 #19036 2679869,2679871 #19037 2680607,2680609 #19038 2680877,2680879 #19039 2680967,2680969 #19040 2681387,2681389 #19041 2681681,2681683 #19042 2681879,2681881 #19043 2681927,2681929 #19044 2682011,2682013 #19045 2682101,2682103 #19046 2682269,2682271 #19047 2682299,2682301 #19048 2682539,2682541 #19049 2682947,2682949 #19050 2683061,2683063 #19051 2683097,2683099 #19052 2683319,2683321 #19053 2683361,2683363 #19054 2683517,2683519 #19055 2683589,2683591 #19056 2683781,2683783 #19057 2683787,2683789 #19058 2684021,2684023 #19059 2684081,2684083 #19060 2684177,2684179 #19061 2684807,2684809 #19062 2685017,2685019 #19063 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2692637,2692639 #19107 2692799,2692801 #19108 2693057,2693059 #19109 2693111,2693113 #19110 2693237,2693239 #19111 2693261,2693263 #19112 2693357,2693359 #19113 2693441,2693443 #19114 2693501,2693503 #19115 2693651,2693653 #19116 2693729,2693731 #19117 2693849,2693851 #19118 2693891,2693893 #19119 2694047,2694049 #19120 2694299,2694301 #19121 2694689,2694691 #19122 2694869,2694871 #19123 2695139,2695141 #19124 2695181,2695183 #19125 2695409,2695411 #19126 2695697,2695699 #19127 2695787,2695789 #19128 2695997,2695999 #19129 2696009,2696011 #19130 2696051,2696053 #19131 2696921,2696923 #19132 2696927,2696929 #19133 2697059,2697061 #19134 2697239,2697241 #19135 2697437,2697439 #19136 2697767,2697769 #19137 2697911,2697913 #19138 2697971,2697973 #19139 2698097,2698099 #19140 2698181,2698183 #19141 2698229,2698231 #19142 2698277,2698279 #19143 2698571,2698573 #19144 2698607,2698609 #19145 2698649,2698651 #19146 2698679,2698681 #19147 2699297,2699299 #19148 2699339,2699341 #19149 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2705819,2705821 #19193 2705849,2705851 #19194 2706059,2706061 #19195 2706167,2706169 #19196 2706677,2706679 #19197 2706971,2706973 #19198 2707037,2707039 #19199 2707127,2707129 #19200 2707319,2707321 #19201 2707457,2707459 #19202 2707499,2707501 #19203 2707589,2707591 #19204 2707739,2707741 #19205 2707799,2707801 #19206 2707841,2707843 #19207 2708261,2708263 #19208 2708309,2708311 #19209 2708681,2708683 #19210 2708819,2708821 #19211 2709167,2709169 #19212 2709197,2709199 #19213 2709269,2709271 #19214 2709359,2709361 #19215 2709479,2709481 #19216 2709599,2709601 #19217 2709737,2709739 #19218 2709821,2709823 #19219 2710007,2710009 #19220 2710187,2710189 #19221 2710397,2710399 #19222 2710439,2710441 #19223 2710451,2710453 #19224 2710679,2710681 #19225 2710907,2710909 #19226 2710931,2710933 #19227 2711087,2711089 #19228 2711327,2711329 #19229 2711339,2711341 #19230 2711459,2711461 #19231 2711939,2711941 #19232 2712131,2712133 #19233 2712179,2712181 #19234 2712251,2712253 #19235 2712371,2712373 #19236 2712377,2712379 #19237 2712767,2712769 #19238 2712971,2712973 #19239 2713649,2713651 #19240 2713811,2713813 #19241 2713871,2713873 #19242 2713937,2713939 #19243 2714009,2714011 #19244 2714027,2714029 #19245 2714279,2714281 #19246 2714627,2714629 #19247 2714729,2714731 #19248 2715281,2715283 #19249 2715287,2715289 #19250 2715437,2715439 #19251 2715521,2715523 #19252 2715617,2715619 #19253 2715857,2715859 #19254 2715929,2715931 #19255 2715959,2715961 #19256 2716157,2716159 #19257 2716451,2716453 #19258 2716541,2716543 #19259 2716709,2716711 #19260 2716997,2716999 #19261 2717087,2717089 #19262 2717129,2717131 #19263 2717147,2717149 #19264 2717249,2717251 #19265 2717291,2717293 #19266 2717411,2717413 #19267 2717651,2717653 #19268 2717711,2717713 #19269 2717831,2717833 #19270 2718059,2718061 #19271 2718101,2718103 #19272 2718137,2718139 #19273 2718227,2718229 #19274 2718557,2718559 #19275 2718671,2718673 #19276 2718839,2718841 #19277 2718887,2718889 #19278 2718971,2718973 #19279 2719139,2719141 #19280 2719151,2719153 #19281 2719391,2719393 #19282 2719529,2719531 #19283 2719667,2719669 #19284 2720147,2720149 #19285 2720189,2720191 #19286 2720297,2720299 #19287 2720381,2720383 #19288 2720897,2720899 #19289 2721317,2721319 #19290 2721419,2721421 #19291 2721449,2721451 #19292 2721869,2721871 #19293 2722061,2722063 #19294 2722469,2722471 #19295 2722799,2722801 #19296 2722877,2722879 #19297 2723339,2723341 #19298 2723351,2723353 #19299 2723549,2723551 #19300 2723561,2723563 #19301 2723717,2723719 #19302 2723759,2723761 #19303 2723837,2723839 #19304 2723879,2723881 #19305 2723909,2723911 #19306 2724119,2724121 #19307 2724479,2724481 #19308 2724719,2724721 #19309 2725001,2725003 #19310 2725367,2725369 #19311 2725451,2725453 #19312 2725517,2725519 #19313 2725691,2725693 #19314 2725781,2725783 #19315 2725817,2725819 #19316 2726387,2726389 #19317 2726741,2726743 #19318 2726819,2726821 #19319 2726837,2726839 #19320 2727119,2727121 #19321 2727299,2727301 #19322 2727311,2727313 #19323 2727449,2727451 #19324 2727839,2727841 #19325 2728169,2728171 #19326 2728259,2728261 #19327 2728541,2728543 #19328 2728547,2728549 #19329 2728751,2728753 #19330 2728769,2728771 #19331 2728931,2728933 #19332 2729099,2729101 #19333 2729117,2729119 #19334 2729381,2729383 #19335 2729591,2729593 #19336 2729651,2729653 #19337 2729957,2729959 #19338 2730179,2730181 #19339 2730239,2730241 #19340 2730569,2730571 #19341 2730599,2730601 #19342 2730989,2730991 #19343 2731061,2731063 #19344 2731187,2731189 #19345 2731241,2731243 #19346 2731277,2731279 #19347 2731427,2731429 #19348 2731607,2731609 #19349 2731667,2731669 #19350 2731691,2731693 #19351 2731697,2731699 #19352 2731901,2731903 #19353 2731907,2731909 #19354 2732207,2732209 #19355 2732381,2732383 #19356 2732489,2732491 #19357 2732501,2732503 #19358 2732537,2732539 #19359 2732579,2732581 #19360 2732759,2732761 #19361 2733041,2733043 #19362 2733257,2733259 #19363 2733329,2733331 #19364 2733371,2733373 #19365 2733461,2733463 #19366 2733539,2733541 #19367 2733779,2733781 #19368 2734007,2734009 #19369 2734097,2734099 #19370 2734607,2734609 #19371 2734817,2734819 #19372 2734967,2734969 #19373 2735021,2735023 #19374 2735189,2735191 #19375 2735267,2735269 #19376 2735279,2735281 #19377 2735441,2735443 #19378 2735609,2735611 #19379 2735921,2735923 #19380 2736497,2736499 #19381 2736581,2736583 #19382 2736689,2736691 #19383 2737127,2737129 #19384 2737169,2737171 #19385 2737211,2737213 #19386 2737409,2737411 #19387 2737487,2737489 #19388 2737871,2737873 #19389 2737979,2737981 #19390 2738117,2738119 #19391 2738387,2738389 #19392 2738621,2738623 #19393 2738651,2738653 #19394 2739239,2739241 #19395 2739281,2739283 #19396 2739419,2739421 #19397 2739557,2739559 #19398 2739719,2739721 #19399 2739731,2739733 #19400 2740037,2740039 #19401 2740139,2740141 #19402 2740187,2740189 #19403 2740217,2740219 #19404 2740511,2740513 #19405 2740601,2740603 #19406 2741351,2741353 #19407 2741579,2741581 #19408 2741657,2741659 #19409 2741729,2741731 #19410 2741927,2741929 #19411 2741939,2741941 #19412 2742029,2742031 #19413 2742161,2742163 #19414 2742197,2742199 #19415 2742407,2742409 #19416 2742461,2742463 #19417 2742671,2742673 #19418 2742737,2742739 #19419 2742809,2742811 #19420 2742917,2742919 #19421 2742977,2742979 #19422 2742989,2742991 #19423 2743547,2743549 #19424 2743709,2743711 #19425 2743859,2743861 #19426 2743931,2743933 #19427 2744081,2744083 #19428 2744447,2744449 #19429 2744591,2744593 #19430 2745047,2745049 #19431 2745371,2745373 #19432 2745569,2745571 #19433 2745929,2745931 #19434 2746031,2746033 #19435 2746199,2746201 #19436 2746421,2746423 #19437 2746481,2746483 #19438 2746607,2746609 #19439 2746661,2746663 #19440 2746787,2746789 #19441 2747021,2747023 #19442 2747117,2747119 #19443 2747177,2747179 #19444 2747321,2747323 #19445 2747357,2747359 #19446 2747447,2747449 #19447 2747711,2747713 #19448 2748059,2748061 #19449 2748131,2748133 #19450 2748281,2748283 #19451 2748467,2748469 #19452 2748827,2748829 #19453 2748857,2748859 #19454 2748971,2748973 #19455 2749067,2749069 #19456 2749301,2749303 #19457 2749361,2749363 #19458 2749709,2749711 #19459 2749847,2749849 #19460 2749919,2749921 #19461 2750159,2750161 #19462 2750261,2750263 #19463 2750399,2750401 #19464 2750771,2750773 #19465 2750789,2750791 #19466 2750981,2750983 #19467 2751101,2751103 #19468 2751251,2751253 #19469 2751479,2751481 #19470 2751809,2751811 #19471 2751821,2751823 #19472 2752049,2752051 #19473 2752151,2752153 #19474 2752199,2752201 #19475 2752229,2752231 #19476 2752637,2752639 #19477 2752667,2752669 #19478 2752721,2752723 #19479 2752877,2752879 #19480 2753129,2753131 #19481 2753549,2753551 #19482 2753939,2753941 #19483 2753999,2754001 #19484 2754047,2754049 #19485 2755031,2755033 #19486 2755199,2755201 #19487 2755211,2755213 #19488 2755301,2755303 #19489 2755661,2755663 #19490 2755859,2755861 #19491 2756009,2756011 #19492 2756069,2756071 #19493 2756099,2756101 #19494 2756267,2756269 #19495 2756519,2756521 #19496 2756561,2756563 #19497 2756627,2756629 #19498 2756681,2756683 #19499 2757119,2757121 #19500 2757191,2757193 #19501 2757257,2757259 #19502 2757317,2757319 #19503 2757659,2757661 #19504 2758241,2758243 #19505 2758517,2758519 #19506 2758529,2758531 #19507 2758541,2758543 #19508 2758631,2758633 #19509 2758841,2758843 #19510 2759171,2759173 #19511 2759291,2759293 #19512 2759297,2759299 #19513 2759411,2759413 #19514 2759441,2759443 #19515 2759459,2759461 #19516 2759819,2759821 #19517 2759879,2759881 #19518 2760221,2760223 #19519 2760629,2760631 #19520 2760671,2760673 #19521 2760761,2760763 #19522 2760881,2760883 #19523 2761007,2761009 #19524 2761091,2761093 #19525 2761151,2761153 #19526 2761181,2761183 #19527 2761277,2761279 #19528 2761301,2761303 #19529 2761721,2761723 #19530 2761727,2761729 #19531 2761901,2761903 #19532 2762027,2762029 #19533 2762117,2762119 #19534 2762171,2762173 #19535 2762759,2762761 #19536 2762777,2762779 #19537 2762939,2762941 #19538 2762951,2762953 #19539 2763587,2763589 #19540 2763599,2763601 #19541 2763659,2763661 #19542 2763779,2763781 #19543 2763881,2763883 #19544 2764121,2764123 #19545 2764127,2764129 #19546 2764649,2764651 #19547 2764787,2764789 #19548 2764871,2764873 #19549 2764901,2764903 #19550 2765207,2765209 #19551 2765297,2765299 #19552 2765471,2765473 #19553 2765837,2765839 #19554 2765927,2765929 #19555 2766329,2766331 #19556 2766581,2766583 #19557 2766677,2766679 #19558 2766791,2766793 #19559 2767067,2767069 #19560 2767229,2767231 #19561 2767319,2767321 #19562 2767361,2767363 #19563 2767409,2767411 #19564 2767571,2767573 #19565 2768069,2768071 #19566 2768177,2768179 #19567 2768189,2768191 #19568 2768201,2768203 #19569 2768417,2768419 #19570 2768429,2768431 #19571 2768609,2768611 #19572 2768681,2768683 #19573 2768789,2768791 #19574 2768957,2768959 #19575 2769257,2769259 #19576 2769551,2769553 #19577 2769617,2769619 #19578 2769887,2769889 #19579 2770091,2770093 #19580 2770169,2770171 #19581 2770589,2770591 #19582 2770769,2770771 #19583 2770841,2770843 #19584 2770991,2770993 #19585 2771141,2771143 #19586 2771177,2771179 #19587 2771381,2771383 #19588 2771609,2771611 #19589 2771861,2771863 #19590 2771957,2771959 #19591 2772017,2772019 #19592 2772191,2772193 #19593 2772569,2772571 #19594 2772629,2772631 #19595 2772827,2772829 #19596 2772881,2772883 #19597 2773019,2773021 #19598 2773079,2773081 #19599 2773319,2773321 #19600 2773679,2773681 #19601 2773697,2773699 #19602 2773817,2773819 #19603 2773919,2773921 #19604 2773997,2773999 #19605 2774141,2774143 #19606 2774309,2774311 #19607 2774477,2774479 #19608 2774501,2774503 #19609 2774729,2774731 #19610 2774867,2774869 #19611 2775041,2775043 #19612 2775161,2775163 #19613 2775389,2775391 #19614 2775611,2775613 #19615 2775737,2775739 #19616 2775989,2775991 #19617 2776001,2776003 #19618 2776061,2776063 #19619 2776181,2776183 #19620 2776649,2776651 #19621 2776799,2776801 #19622 2776841,2776843 #19623 2776979,2776981 #19624 2777111,2777113 #19625 2777141,2777143 #19626 2777231,2777233 #19627 2777309,2777311 #19628 2777837,2777839 #19629 2778107,2778109 #19630 2778341,2778343 #19631 2778647,2778649 #19632 2778827,2778829 #19633 2778911,2778913 #19634 2779487,2779489 #19635 2779631,2779633 #19636 2779769,2779771 #19637 2779781,2779783 #19638 2780177,2780179 #19639 2780207,2780209 #19640 2780597,2780599 #19641 2780621,2780623 #19642 2780777,2780779 #19643 2781017,2781019 #19644 2781059,2781061 #19645 2781209,2781211 #19646 2781347,2781349 #19647 2781377,2781379 #19648 2782061,2782063 #19649 2782097,2782099 #19650 2782691,2782693 #19651 2782859,2782861 #19652 2782937,2782939 #19653 2782991,2782993 #19654 2783321,2783323 #19655 2783579,2783581 #19656 2783657,2783659 #19657 2783687,2783689 #19658 2783771,2783773 #19659 2784167,2784169 #19660 2784281,2784283 #19661 2784347,2784349 #19662 2784371,2784373 #19663 2784569,2784571 #19664 2784911,2784913 #19665 2785019,2785021 #19666 2785031,2785033 #19667 2785511,2785513 #19668 2785577,2785579 #19669 2785589,2785591 #19670 2785631,2785633 #19671 2785901,2785903 #19672 2785961,2785963 #19673 2786081,2786083 #19674 2786219,2786221 #19675 2786429,2786431 #19676 2786477,2786479 #19677 2786741,2786743 #19678 2787017,2787019 #19679 2787119,2787121 #19680 2787227,2787229 #19681 2787329,2787331 #19682 2787479,2787481 #19683 2787527,2787529 #19684 2787557,2787559 #19685 2787749,2787751 #19686 2787767,2787769 #19687 2788529,2788531 #19688 2788781,2788783 #19689 2788829,2788831 #19690 2789117,2789119 #19691 2789327,2789329 #19692 2789489,2789491 #19693 2789627,2789629 #19694 2790101,2790103 #19695 2790251,2790253 #19696 2790257,2790259 #19697 2790449,2790451 #19698 2790479,2790481 #19699 2790647,2790649 #19700 2790857,2790859 #19701 2791037,2791039 #19702 2791091,2791093 #19703 2791121,2791123 #19704 2791181,2791183 #19705 2791559,2791561 #19706 2791637,2791639 #19707 2791697,2791699 #19708 2791967,2791969 #19709 2791979,2791981 #19710 2792087,2792089 #19711 2792159,2792161 #19712 2792171,2792173 #19713 2792189,2792191 #19714 2792399,2792401 #19715 2792429,2792431 #19716 2792747,2792749 #19717 2792771,2792773 #19718 2792831,2792833 #19719 2792861,2792863 #19720 2792987,2792989 #19721 2793071,2793073 #19722 2793101,2793103 #19723 2793179,2793181 #19724 2793731,2793733 #19725 2793809,2793811 #19726 2793941,2793943 #19727 2794217,2794219 #19728 2794241,2794243 #19729 2794301,2794303 #19730 2794397,2794399 #19731 2794541,2794543 #19732 2794787,2794789 #19733 2795267,2795269 #19734 2795321,2795323 #19735 2795381,2795383 #19736 2795561,2795563 #19737 2796221,2796223 #19738 2796527,2796529 #19739 2796707,2796709 #19740 2797211,2797213 #19741 2797337,2797339 #19742 2797439,2797441 #19743 2798141,2798143 #19744 2798459,2798461 #19745 2798597,2798599 #19746 2798639,2798641 #19747 2798867,2798869 #19748 2798921,2798923 #19749 2799131,2799133 #19750 2799149,2799151 #19751 2799449,2799451 #19752 2799497,2799499 #19753 2799749,2799751 #19754 2799791,2799793 #19755 2799911,2799913 #19756 2799989,2799991 #19757 2800001,2800003 #19758 2800139,2800141 #19759 2800247,2800249 #19760 2800331,2800333 #19761 2800781,2800783 #19762 2800949,2800951 #19763 2800979,2800981 #19764 2801219,2801221 #19765 2801441,2801443 #19766 2801597,2801599 #19767 2801801,2801803 #19768 2802011,2802013 #19769 2802089,2802091 #19770 2802311,2802313 #19771 2802599,2802601 #19772 2802641,2802643 #19773 2802857,2802859 #19774 2802929,2802931 #19775 2803067,2803069 #19776 2803121,2803123 #19777 2803571,2803573 #19778 2803637,2803639 #19779 2803649,2803651 #19780 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
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Introduction to twin primes and Brun's constant computation
Source : http://numbers.computation.free.fr/Constants/Primes/twin.html
Introduction to twin primes and Brun's constant computation
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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
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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):
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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
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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
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or which is more convenient to sieve twin primes
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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
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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
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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
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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:
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twin primes. So that (don't forget to count the two couples (3,5) and (5,7)!)
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And, for the same couples, up to 1012:
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which produces
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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
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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
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(1) |
or
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(2) |
Note that C2 is the twin prime constant and is defined by
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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).
Remark 4 The function Li2(n) occuring in (1) may be related to the logarithmic integral Li(n) by the trivial relation
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2.2.1 Generalizations
In fact, Hardy and Littlewood made a more general conjecture on the primes separated by a gap of d. A natural generalization of the twin primes is to search for primes distant of d=2k (which should be infinite for any d according to Polignac's conjecture). The case d=2 is the twin primes set, d=4 forms the cousin primes set, d=6 is the sexy primes set, ...
If we denote pd(n) the number of primes p £ n such as p+d is also prime (observe that here p and p+d may not be consecutive), Hardy-Littlewood's conjecture states (in [7]) that for d ³ 2:
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with Rd ³ 1 being the rational number
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The first values of the function Rd are
d | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |
Rd | 1 | 1 | 2 | 1 | 4/3 | 2 | 6/5 | 1 | 2 | 4/3 |
According to this conjecture the density of twin primes is equivalent to the density of cousin primes. For example, the exact computed values up to 1012 are:
p2(1012) =1870585220
p4(1012) =1870585458,
which can be compared to the predicted value 1870559867 by the conjecture. Marek Wolf has studied the function
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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
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when n becomes large and the sum takes all integers in account. We know since Euler, for instance, that
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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
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and we have the beautiful results that
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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
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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
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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])
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and thanks to this relation, the extrapolated value
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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
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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.
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
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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
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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]):
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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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On 29 Jul 2002, 14:41.
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L'union fait la force des mathématiciens LE MONDE SCIENCE ET TECHNO | 24.06.2013 à 15h40 | En savoir plus sur http://www.lemonde.fr/sciences/article/2013/06/24/l-union-fait-la-force-des-mathematiciens_3435624_1650684.html#64byvQVhFuDDXMoA.99
L'union fait la force des mathématiciens
LE MONDE SCIENCE ET TECHNO | |
"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
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Nombres premiers jumeaux
Nombres premiers jumeaux
En mathématiques, deux nombres premiers jumeaux sont deux nombres premiers qui ne diffèrent que de 2. Hormis pour le couple (2, 3), cet écart entre nombres premiers de 2 est le plus petit possible. Les plus petits nombres premiers jumeaux sont 3 et 5, 5 et 7, 11 et 13.
Au 25 décembre 2011, les plus grands nombres premiers jumeaux connus, découverts dans le cadre du projet de calcul distribué PrimeGrid, sont 3 756 801 695 685 × 2666 669 ± 1 ; ils possèdent 200 700 chiffres en écriture décimale.
Selon la conjecture des nombres premiers jumeaux, il existe une infinité de nombres premiers jumeaux ; les observations numériques et des raisonnements heuristiquesjustifient la conjecture, mais aucune démonstration n'en a encore été faite.
Sommaire
[masquer]
Définition[modifier | modifier le code]
Soient p et q deux nombres premiers. On dit que (p, q) forme un couple de nombres premiers jumeaux si q = p + 2.
Liste des premiers nombres premiers jumeaux[modifier | modifier le code]
L'ensemble des nombres premiers jumeaux jusqu'à 1000 :
(3, 5) | (5, 7) | (11, 13) | (17, 19) | (29, 31) |
(41, 43) | (59, 61) | (71, 73) | (101, 103) | (107, 109) |
(137, 139) | (149, 151) | (179, 181) | (191, 193) | (197, 199) |
(227, 229) | (239, 241) | (269, 271) | (281, 283) | (311, 313) |
(347, 349) | (419 , 421) | (431 , 433) | (461 , 463) | (521 , 523) |
(569 , 571) | (599 , 601) | (617 , 619) | (641 , 643) | (659 , 661) |
(809 , 811) | (821 , 823) | (827 , 829) | (857 , 859) | (881 , 883) |
Quelques propriétés[modifier | modifier le code]
- Le couple (2, 3) est le seul couple de nombres premiers consécutifs.
- Si l'on omet le couple (2, 3), la plus petite distance possible entre deux nombres premiers est 2 ; deux nombres premiers jumeaux sont ainsi deux nombres impairsconsécutifs.
- Tout couple de nombres premiers jumeaux, à l'exception du couple (3, 5), est de la forme (6n – 1, 6n + 1) pour un certain entier n. En effet, tout triplet d'entiers consécutifs comporte au moins un multiple de 2 (éventuellement deux) et un seul multiple de 3 ; l'entier qui se trouve entre les deux nombres premiers jumeaux est à la fois ce multiple de 2 et ce multiple de 3, car cela ne peut pas être l'un des nombres premiers.
- Pour tout entier m ≥ 2, le couple (m, m + 2) est constitué de nombres premiers jumeaux si et seulement si 4[(m - 1)! + 1] + m est divisible par m(m + 2). Cette caractérisation des nombres premiers jumeaux, remarquée par P. A. Clement en 19491, résulte du théorème de Wilson.
- Alors que la série des inverses des nombres premiers est divergente, la série des inverses de nombres premiers jumeaux est convergente (vers un nombre appelé constante de Brun). Cette propriété fut démontrée par Viggo Brun en 19192.
Records[modifier | modifier le code]
Le 15 janvier 2007, les deux projets de calcul distribué Twin Prime Search et PrimeGrid ont découvert le plus grand couple de nombres premiers jumeaux connu à l'époque, de 58 711 chiffres en écriture décimale. Le découvreur était le Français Éric Vautier3.
Le 25 décembre 2011, le couple record3 est 3 756 801 695 685 × 2666 669 ± 1 ; les deux nombres possèdent 200 700 chiffres.
Conjecture des nombres premiers jumeaux[modifier | modifier le code]
La conjecture des nombres premiers jumeaux affirme qu'il existe une infinité de nombres premiers jumeaux:
Il existe une infinité de nombres premiers p tels que p + 2 soit aussi premier.
Cette conjecture partage avec l'hypothèse de Riemann et la conjecture de Goldbach le numéro 8 des problèmes de Hilbert, énoncés par ce dernier en 1900. Bien que la plupart des chercheurs en théorie des nombres pensent que cette conjecture est vraie, elle n'a jamais été démontrée. Ils se basent sur des observations numériques et des raisonnements heuristiques utilisant la distribution probabiliste des nombres premiers.
En 1849, Alphonse de Polignac émit une conjecture plus générale : la conjecture de Polignac :
Tout nombre pair est égal à la différence de deux nombres premiers consécutifs d'une infinité de manières.
dont le cas n = 2 correspond à la conjecture des nombres premiers jumeaux.
Il existe également une version plus forte de cette conjecture : la première conjecture de Hardy-Littlewood (cf. infra), qui fournit une loi de distribution des nombres premiers jumeaux et qui s'inspire du théorème des nombres premiers.
La conjecture des nombres premiers jumeaux est un cas particulier de la conjecture de Schinzel.
Résultats partiels[modifier | modifier le code]
En 1940, Paul Erdős démontra l'existence d'une constante positive c < 1 pour laquelle l'ensemble des nombres premiers p tels que p' – p < c ln(p) est infini, où p' désigne le nombre premier suivant immédiatement p.
Ce résultat fut plusieurs fois amélioré ; en 1986, Helmut Maier montra que c peut être choisi inférieur à 1/4. En 2005, Daniel Goldston, János Pintz et Cem Yıldırım démontrèrent que c peut être choisi arbitrairement petit.
Par ailleurs, en 1966, Chen Jingrun démontra l'existence d'une infinité de « nombres premiers de Chen », c'est-à-dire de nombres premiers p tels que p + 2 soit premier ou semi-premier (un nombre semi-premier est le produit de deux nombres premiers). Son approche est celle de la théorie des cribles, qu'il a utilisée pour traiter de façon similaire la conjecture des nombres premiers jumeaux et la conjecture de Goldbach (voir Théorème de Chen).
À partir de 2009, à la suite de la découverte d'une optimisation du crible d'Eratosthène, Zhang Yitang établit qu'il existe une infinité de nombres premiers consécutifs dont l'écart est inférieur à 70 000 000, résultat qui constitue une forme faible de la conjecture des nombres premiers jumeaux. Début 2013, le projet Polymath, un projet de mathématiques collaboratives mené par Tim Gowers et Terence Tao, a proposé de réduire progressivement cet écart N = 70 millions pour le faire tendre vers 2 : en septembre 2013 l'écart a été réduit à N = 4 6804,5. En novembre 2013, une amélioration significative de ces résultats est annoncée indépendamment par James Maynard (en) et Terence Tao6 : non seulement l'écart entre deux nombres premiers consécutifs est inférieur ou égal à 600 infiniment souvent, mais un résultat équivalent est valable pour m nombres premiers consécutifs, quel que soit m ≥ 2. Une nouvelle amélioration est annoncée par le projet Polymath (section Polymath8) début 2014 : d'une part, l'écart serait inférieur à 270 infiniment souvent, d'autre part, en admettant une version généralisée de la conjecture d'Elliott-Halberstam, l'écart serait alors inférieur ou égal à 67.
Le résultat de Zhang a été publié dans les Annals of Mathematics8. Dans un premier temps il a été difficile de trouver des relecteurs acceptant d'évaluer le travail9.
La conjecture de Hardy-Littlewood[modifier | modifier le code]
Il existe aussi une généralisation de la conjecture des nombres premiers jumeaux, connue sous le nom de première10 conjecture de Hardy-Littlewood, en rapport avec la distribution des premiers jumeaux, par analogie avec le théorème des nombres premiers. Soit π2(x) le nombre de nombres premiers p ≤ x tels que p + 2 soit aussi premier.
On note C2 le nombre obtenu de la façon suivante :
(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]
- (en) P. A. Clement, « Congruences for sets of primes », American Mathematical Monthly, vol. 56, , p. 23-25 (lire en ligne [archive])
- Viggo Brun, « La série 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + 1/19 + 1/29 + 1/31 + 1/41 + 1/43 + 1/59 + 1/61 + ... où les dénominateurs sont « nombres premiers jumeaux » est convergente ou finie », Bulletin des Sciences Mathématiques, vol. 43, , p. 100-104 et 124-128
- (en) « Twin Primes » [archive], sur Top Twenty
- L'union fait la force des mathématiciens [archive], Le Monde, 24/06/2013.
- (en) Bounded gaps between primes [archive].
- (en) Polymath8b: Bounded intervals with many primes, after Maynard [archive], sur le blog de Terence Tao.
- (en) Annonce de ce résultat [archive] sur le blog de Gil Kalai (en)
- Y. Zhang, « Bounded gaps between primes », Annals of Mathematics, 179 (2014), p. 1121–1174
- John Friedlander, « Prime Numbers: A Much Needed Gap Is Finally Found », Notices of the American Mathematical Society, vol. 62, no 6, (lire en ligne [archive]).
- Il existe une seconde conjecture de Hardy-Littlewood
- suite A005597 de l'OEIS des décimales de cette constante.
- (en) Eric W. Weisstein, « Twin Primes Constant [archive] », MathWorld
- François Le Lionnais, Les nombres remarquables, Hermann, 1983, p. 30 [archive]
- (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
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- (en) Twin Primes (Chris Caldwell)
- (en) Introduction to Twin Primes and Brun's Constant (Xavier Gourdon, Pascal Sebah)
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- (en) http://www.twinprimesearch.org
- (en) Projet Polymath 8 : le projet Polymath correspondant.
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Combinatorial Number Theory: Proceedings of the 'Integers Conference 2005 ... Par Ronald L. Graham,Bruce M. Landman
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A Short Biography of A.N. Kolmogorov
Source : http://homepages.cwi.nl/~paulv/KOLMOGOROV.BIOGRAPHY.html
A Short Biography of A.N. Kolmogorov
(``Andrei Nikolaevich Kolmogorov,'' CWI Quarterly, 1(1988), pp. 3-18.)
by Paul M.B. Vitanyi, CWI and University of Amsterdam
Andrei Nikolaevich Kolmogorov, born 25 April 1903 in Tambov, Russia, died 20 October 1987 in Moscow. He was perhaps the foremost contemporary Soviet mathematician and counts as one of the great mathematicians of this century. His many creative and fundamental contributions to a vast variety of mathematical fields are so wide-ranging that I cannot even attempt to treat them either completely or in any detail.
For now let me mention a non-exhaustive list of areas he enriched by his fundamental research: The theory of trigonometric series, measure theory, set theory, the theory of integration, constructive logic (intuitionism), topology, approximation theory, probability theory, the theory of random processes, information theory, mathematical statistics, dynamical systems, automata theory, theory of algorithms, mathematical linguistics, turbulence theory, celestial mechanics, differential equations, Hilbert's 13th problem, ballistics, and applications of mathematics to problems of biology, geology, and the crystallization of metals.
In over 300 research papers, textbooks and monographs, Kolmogorov covered almost every area of mathematics except number theory. In all of these areas even his short contributions did not just study an isolated question, but in contrast exposed fundamental insights and deep relations, and started whole new fields of investigations.
Apart from his penetrating work in Mathematics and the Sciences, he devoted much of his time to improving the teaching of mathematics in secondary schools in the Soviet Union, and in providing special schools for the mathematically gifted - which were very successful. Famous are also his efforts to capture in quantitative form some aspects of Russian poetry, especially that of Pushkin. It is told that ``it was fascinating to hear him lecture on this, whether one understood Russian or not.'' In 1942 Kolmogorov married Anna Dmitriyevna Egorov. He did not have children of his own.
Apart from being acknowledged without question in science, Kolmogorov was also blessed with social recognition. The USSR conferred to him seven orders of Lenin, the Order of the October Revolution, and also the high title of Hero of Socialist Labour; he gained Lenin prizes and State prizes. He occupies the first place among all Soviet mathematicians in the number of foreign academies and scientific societies that have elected him as member. These number over twenty, among them the Royal Netherlands Academy of Sciences (1963), the London Royal Society (1964), the USA National Society (1967), the Paris Academy of Sciences (1968), the Polish Academy of Sciences, the Rumanian Academy of Sciences (1956), the German Academy of Sciences Leopoldina (1959), the American Academy of Sciences and Arts in Boston (1959).
He was presented with honorary doctorates from the universities of Paris, Berlin, Warsaw, Stockholm, etc. He was elected a honorary member of the Moscow, London, Indian, and Calcutta Mathematical Societies, of the London Royal Statistical Society, the International Statistical Institute, and the American Meteorological Society. In 1963 he was awarded the International Bolzano prize.
Let me state here that I do not claim any personal relations with Kolmogorov. These remarks are based on second hand information, and primarily on sources in the Russian Mathematical Surveys, other references, and to a much lesser extent on personal communications. My credentials for writing about Kolmogorov's achievements are founded solely on my interests in that excellent notion we call``Kolmogorov complexity''. Since Kolmogorov was a man of many aspects, it is a pleasure to share some of these with the reader. This writeup was originally published as an obituary: P.M.B. Vitanyi, Andrei Nikolaevich Kolmogorov, CWI Quarterly, 1(1988), pp. 3-18. See also references in Section 1.6 of M. Li and P.M.B. Vitányi, An Introduction to Kolmogorov Complexity and its Applications, Springer-Verlag, New York, 1993 (xx + 546 pp). (This is Section 1.13 in the Second Edition of 1997.)
Early Years: 1903-1933
Kolmogorov was born on 25 April 1903 in the town of Tambov, where his mother Mariya Yakovlevna Kolmogorova had been delayed on her way from the Crimea. She died in childbed, and the responsibility to bring up the child was taken over by her sister Vera Yakovlevna Kolmogorova, ``an independent woman who held high social ideals. She passed this over to her nephew, raising him in the sense of responsibility, independence of opinion, intolerance towards idleness and poorly performed tasks, and the desire to understand and not just to memorize.''
K. treated her as his mother until her death in 1950 at Komarovka (his dacha) at the age of 87. From his mother's side K. was of aristocratic stock, his grandfather Yakov Stephanovitch Kolmogorov was a district head of the nobles in Uglich.
He spent his early years (before the revolution of 1917) at the family estate. The sources are less clear about his father. Apparently, K.'s father was the son of a clergyman, and was himself an agronomist with highly specialized training, what they called at the time ``a learned agronomist.''
K. started to work already at an early age (but presumably after the revolution); and before he became a student at Moscow University, he worked for some time as a railway conductor. He arrived at the University in autumn 1920, with already a fair knowledge of mathematics, gleaned from a book called ``New Ideas in Mathematics.'' Students at the time received grants that had little material value, but at the second course received, in addition, a ration of 16 kilos of baked bread and a kilo of fat. Hence, K. lost little time to check the minimum requirements for moving to the second course (lecture attendance being noncompulsory).
Conditions were generally harsh, and lecture rooms cold and unheated in the winter of 1920/1921. The following (unattributed) lines describe it:
``That grim year, nineteen twenty-one,
the scientific march began
Of Moscow University.
Though I was not then very old,
Though sheepskin coats enveloped me,
I still recall that beastly cold.''
For some time K. was interested in Russian history as well as mathematics. He did serious scientific research on XV-XVI century manuscripts concerning agrarian relations in ancient Novgorod. In the twenties he made a hypothesis on the way the upper Pinega was settled, and this conjecture was later confirmed by an expedition to that area.
In this early post-October revolution period the mathematical life in Moscow was dominated by ``young Luzitania'' (1920-1923) and ``post-Luzitania'' (1923-1927), a nickname for the school of real function theory headed by N.N. Luzin. This legendary personality apparently created either enthusiastic admiration or, in their struggle for independence, one-sided negation in his pupils. Among the first subjects in mathematics K. took were set theory, projective geometry, and theory of analytic functions. In 1921-1922 he obtained his first independent mathematical result (the existence of Fourier-Lebesgue series with arbitrarily slowly decreasing Fourier coefficients), and he became a pupil of N.N. Luzin. During this time he was also approached by P.S. Urysohn, who tried to interest him in topological problems. Since K. had obtained some results on the descriptive theory of functions, work that did not fit into Luzin's plans, Urysohn brought him into contact with P.S. Alexandrov whose research interests were better related to this topic. However, at about this time K. constructed a Fourier series divergent everywhere, a result that attracted international attention, and brought him for the time being in Luzin's orbit again. For this reason K.'s initial contacts with Aleksandrov stayed very limited at the time.
K. got interested in mathematical logic, and in 1925 published a paper in Mathematicheskii Sbornik on the law of the excluded middle, which has been a continuous source for later work in mathematical logic. This was the first Soviet publication on mathematical logic containing (very substantial) new results, and the first systematic research in the world on intuitionistic logic. K. anticipated to a large extentA. Heyting 's formalization of intuitionistic reasoning, and made a more definite correlation between classical and intuitionistic mathematics. K. defined an operation for `embedding' one logical theory in another. Using this - historically the first such operation, now called the `Kolmogorov operation' - to embed classical logic in intuitionistic logic, he proved that application of the law of the excluded middle in itself cannot lead to a contradiction. In 1932 K. published a second paper on intuitionistic logic, in which for the first time a semantics was proposed (for this logic), free from the philosophical aims of intuitionism. This paper made it possible to treat intuitionistic logic as constructive logic.
His interest in probability theory originated in 1924. His first steps in this area were performed jointly with A.Y. Khinchin. In 1928 he succeeded in finding necessary and sufficient conditions for the strong law of large numbers to hold, and proved the law of the iterated logarithm for sums of independent random variables, under very general conditions on the summands. In "A general theory of measure and the calculus of probabilities", 1929, he put forward a first draft of an axiom system for probability theory based on the theory of measure and the theory of functions of a real variable. Such a theory had been first suggested by E. Borel in 1909, was further developed by Lomnicki in 1923, and received its so successful final form with K.'s classic treatment of 1933. Much important work on probability theory had already been done without benefit of foundations, but this little book ``Foundations of the Calculus of Probabilities,'' published in German in 1933, immediately became the definitive formulation of the subject. This determined not only a new stage in the development of probability theory as a branch of mathematics, but also gave the necessary basis for the creation of the theory of random processes - the subject of his 1931 paper below. It was here that the basic theorems on infinite-dimensional distributions, now the logical foundations for the rigorous construction of the theory of random functions and sequences of random variables, were first formulated. The involved ideas lie at the heart of the modern theory of random processes; they form essential concepts in the very idea of control theory, and play a vital role in K.'s later synthesis of information theory and ergodic theory. K.'s many contributions in the theory of probability and statistics made him generally acknowledged as the foremost representative of this discipline.
In 1931 K.'s paper ``Analytical methods in probability theory'' appeared, in which he laid the foundations for the modern theory of Markov processes. According to Gnedenko: "In the history of probability theory it is difficult to find other works that changed the established points of view and basic trends in research work in such a decisive way. In fact, this work could be considered as the beginning of a new stage in the development of the whole theory".
The theory had a few forerunners: A.A. Markov, Poincaré and Bashelier, Fokker, Planck, Smolukhovski and Chapman. Their particular equations for individual problems in physics, informally obtained, followed as special cases in K.'s theory. A long series of subsequent publications followed, by K. and his followers, among which a paper by K. dealing with one of the basic problems of mathematical statistics, where he introduces his famous criterion (Kolmogorov's test) for using the empirical distribution function of observed random variables to test the validity of an hypothesis about their true distribution. In general K.'s ideas on probability and statistics have led to numerous theoretic developments, and to numerous applications in present-day physical sciences.
After graduation in 1925, K. stretched his stay at the University for four more years as a research student, but finally in 1928-1929 stricter control on the number of years a student had for research was enforced. An unprecedented number of 70 students finished in 1929, including K. This raised the problem of where to continue his research. Aleksandrov was instrumental in securing for K. the single available vacancy in 1929 at the Institute of Mathematics and Mechanics of Moscow University, against heavy competition.
Youth: 1929-1940
From 1930-1940 K. published more than sixty papers on probability theory, projective geometry, mathematical statistics, the theory of functions of a real variable, topology, mathematical logic, mathematical biology, philosophy and the history of mathematics. In 1931 K. was made professor at Moscow University, and from 1937 held the chair of theory of probability. From this time dates the life long friendship between K. and Aleksandrov. Says Aleksandrov: ``in 1979 this friendship [with K.] celebrated its fiftieth anniversary and over the whole of this half century there was not only never any breach in it, there was also never any quarrel, in all this time there was never any misunderstanding between us on any question, no matter how important for our lives and our philosophy; even when our opinions on one of these questions differed, we showed complete understanding and sympathy for the views of each other.''
Says K.: ``for me these 53 years of close and indissoluble friendship were the reason why all my life was on the whole full of happiness, and the basis of that happiness was the unceasing thoughtfulness on the part of Aleksandrov.''
K. describes how this friendship started in 1929 during a sailing trip on the Volga. At that time, the ``Society for Proletarian Tourism and Excursions'' offered active vacations: one obtained a boat and camping equipment at one city on the Volga which could be handed in at other cities downstream. K., already experienced in boating, decided to organize such a trip, and asked (besides two others) Aleksandrov to join. The young men bought the then popular ``Jungsturm'' suits for all of the crew. By way of books they took along only a steamboat timetable and a copy of the Odyssey (and also manuscripts to work on and a folding writing desk). They started out at June 16, and covered 1300 kilometers before handing in the boat at Samar downstream. K. and Aleksandrov then proceeded together to the Caucasus by steamer. After some more wandering, they set up residence in an unused cell of a monastery on a small peninsula in Lake Savan. Whiling their time away at secluded bays, in between swimming and sun bathing they also managed to get some work done: K. in the shadows on integration theory and analytic description of Markov processes in continuous time, and Aleksandrov dressed only in dark glasses and white panama hat in the burning sun on his Topology book with Hopf. They stayed in these idyllic surroundings for about three weeks, then set off partly on foot, partly by other means of transport, and eventually climbed the Alagez mountain (4100m). They wound up at Tiflis, from where Aleksandrov proceeded alone to a prearranged appointment with a group of mathematicians. K. continued hiking and mountain climbing. (By this time it was August.) Later, they joined up again at Gagra, on the Black Sea, and spent some more time there, sunbathing and swimming and doing mathematics. At about this time they decided to share a house together.
After returning to Moscow they forthwith rented the first in a series of houses in the nearby vacation village of Klyaz'm, and moved in together with K.'s aunt Vera Yakovlevna. A short time later, Masha Barbanova, who had been K.'s nanny at the family estate near Jaroslavl' before the revolution, joined as housekeeper. In 1935 they acquired (initially part of) an old manor house at Komarovka, with room for a large library and several guests. This `house at Komarovka' became a meeting place for mathematicians. One of them said ``It is just like Oberwolfach (a mathematical institute in the Black Forest), except that here Kolmogorov buys all the drinks.''
It is perhaps instructive to see a glimpse of the mathematicians' country life:
``As a rule,'' says K., ``of the seven days a week, four were spent in Komarovka, one of which was devoted entirely to physical recreation - skiing, rowing, long excursions on foot (these long walks covered on average about 30 kilometers, rising to 50; on sunny March days we went out on skis wearing nothing but shorts, for as much as four hours on a stretch. On the other days, morning exercise was compulsory, supplemented in the winter by a 10 kilometer ski run ... Especially did we love swimming in the river just as it began to melt ... I swam only short distances in icy water but Aleksandrov swam much further. It was I however who skied naked for considerably longer distances.'' As P. Halmos, visiting K. in Moscow in 1965 tells it:
``Kolmogorov [had] five rooms [apartment in the University]. ... stacks of reprints in one corner, a collection of theatrical masks somewhere, and a couple of skis somewhere else. "Is this where you work? I asked. "No, no", he said: "I work out at the dacha; I am here only three days a week."'' (At the celebration meeting of K's seventieth birthday, a skiing trip was organized where K. clad only in shorts outskied every other participant.)
In 1930-1931 K. and Aleksandrov were mainly abroad. The year 1930 they spent both at Gottingen. Here K. had contacts with R. Courant on limit theorems, with H. Weyl on intuitionistic logic, and with E. Landau on function theory. K. relates the story that he solved a problem Landau much liked to be solved, and wrote it up in detail. Landau being very pleased told everybody about the success and invited a paper on the subject, but, to his embarrassment, K. discovered a few weeks later exactly the same result with the same proof by Besicovitch in Fundamenta Mathematicae. The summer both K. and Aleksandrov visited Caratheodory at Munich (measure theory), and were invited to stay with Frechet on the Mediterranean (to work on probability theory in K.'s case). The journey there involved hiking through Bavaria, staying with Frechet for about a month, visiting P.S. Urysohn's grave in Normandy, and continuing on to Paris. Aleksandrov left Paris by the end of September for Goettingen, and K. stayed on until December, and had some meetings with Borel and P. Lévy, especially the last. While K. returned to Goettingen, Aleksandrov spent the spring 1931 semester in the U.S.A.
As another highlight of this period the article "Mathematics" for the second edition of the Great Soviet Encyclopaedia is often mentioned. Another area he turned to at the time was topology. Simultaneously with the U.S. topologist J.W. Alexander and independently of him, K. discovered the notion of cohomology and founded the theory of cohomological operations. The work of K. and his school on the deep connections between topology, the theory of ordinary differential equations, celestial mechanics and the theory of dynamical systems, determined to a considerable extent its present state.
At the end of the thirties, K.'s attention was drawn to the mechanics of turbulence. In the hands of K. and his school the theory of turbulence obtained an accurate mathematical form as an applied chapter in the theory of measure of function spaces. With great physical intuition, in two short papers in 1941, K. posited in concise mathematical form ideas about the structure of the small-scale components of turbulent motion of fluids and gasses, latent in earlier experimental work, particularly by G.I. Taylor. These hypotheses imply many qualitative results that are widely applicable - what goes on, for instance, within the turbulence that occurs in the wake of a jet aircraft. Some of the quantitative relations arising have the character of new laws of nature - like K.'s law of "2/3": in each developed turbulent flow the mean square difference of the velocities at two points is proportional to the 2/3rd power of their distance (if the distance is not too small or not too large). K. made also quantitative predictions on the basis of his theories, that were later confirmed by experiments, e.g., the stratified structure of the ocean, an effect known as "pancakes". K.'s 1941 contributions to the theory of turbulence are perhaps the most important ones in the long and unfinished history of the theory of turbulence.
Middle Years: 1940-1960
He was interested in every branch of science, he and his pupils wrote about crystal growth, about geometry of the interaction of plants, and also made significant contributions to ``birth and death'' processes and to genetics. One of these papers brought him to a head-on confrontation with Lysenko. In a courageous stand in emphasizing scientific truth, in a paper published in 1940 in the "Genetics" section of Dokl. Akad. Nauk SSSR , K. showed that the material gathered by followers of Stalin's proteg e Academician Lysenko, contrary to opinion, supported Mendel's laws. Another joint work (with Piscounov andPetrovsky) treated the rate of advance of an advantageous gene in a linear environment, (a topic studied independently by R.A. Fisher, for whom K. had high regard). This was later adapted to describe spreading of epidemics of innovations, and rumours.
The theory of smoothing and prediction of stationary time-series is usually associated with the name of Norbert Wiener but in fact it was developed simultaneously by Wiener and K. during the second world war.
In the post-war period K. turned again to turbulence, and made small improvements on laws he discovered before, that were experimentally verified as well. Topics in the vast range of classical mechanics, ergodic theory, function theory, information theory and the theory of algorithms belong to this period. He managed to find links between totally unconnected fields, and published a small number of papers, but quite fundamental ones, on each topic. In his work on dynamical systems one can distinguish two periods. In 1953-1954 he made a seminal contribution to the fundamental problem of classical mechanics, identified fifty years earlier by H. Poincare in his study of the motion of planets around the sun. Neglecting all but one planet one deals with an ``integrable'' problem that is well understood. However, the small effects associated with gravitational interaction between the planets introduces a profound qualitative change related to the fact that the equations are now ``nonintegrable.'' In attacking this problem, K.'s great achievement was to develop a general theory of Hamiltonian systems under small perturbations, which has several practical applications, among others in the study of magnetic fields and plasma physics. This work also spawned, together with improvements of K.'s pupil Arnol'd and by Moser, what is now known as the study of ``KAM-tori.'' Subsequent computational studies aptly confirm K.'s insights and have opened up the enormously fruitful field of ``chaos in dynamical systems,'' which is currently attracting much attention. These studies lead, for example, to better weather forecasting.
At this time he also started to work on the theory of automata and the theory of algorithms. Together with his pupil Uspenskii he formulated the important notion of Kolmogorov-Uspenskii machine. He supported the up and coming field of cybernetics (theory of computation) against heavy initial antagonism (in the USSR). Many USSR computer scientists are K.'s pupils or pupils of K.'s pupils.
The second period from 1955-1959 consisted in applications from information theory to the ergodic theory of dynamical systems. He introduced the fruitful idea of informational (entropic) characteristics in the study of metric spaces and of dynamical systems. Together with Arnol'd, K. settled in 1956-1957 Hilbert's 13th problem, disproving the conjectured outcome, by showing that a continuous function in any number of variables can be represented as a composition of continuous functions of a single variable and addition. The ideas of introducing entropic characteristics in the theory of dynamical systems opened up a large new area. Another important concept, that of a quasi-regular system (now called K-system), plays a very important role in the analysis of classical dynamic systems with strong stochastic properties, such as in physics, biology and chemistry. In the years 1958-1959 K. applied ergodic theory to phenomena of the type of turbulence, which had a great influence on subsequent work.
Later Years: 1960-1987
While in previous years K. used concepts of information theory in mathematical sciences, now it was the turn of information theory to be reconstructed using the theory of algorithms, incidentally closing the circle of his research by giving logico-algorithmic foundations to the theory of probability. Algorithmic information theory, or " Kolmogorov complexity theory", originated with the discovery of universal descriptions of finite objects, and a recursively invariant approach to the concepts of complexity of description, randomness and a priori probability. Historically, it is firmly rooted in R. von Mises' notion of random infinite sequences ( Kollektivs ), proposed from 1919 onwards as foundation for the theory of probability in the spirit of a physical theory (according to the program outlined in D. Hilbert's 6th problem), using the frequency interpretation of probability. In 1940 A. Church proposed an algorithmic version of von Mises random sequences, but the results were not yet satisfactory.
In his 1933 booklet K. had in some sense executed Hilbert's suggestion in his 6th problem: "To treat (in the same manner as geometry) by means of axioms, those physical sciences in which mathematics plays an important part; in the first rank are the theory of probability ..", in 1963 K. observes: "This theory [K's 1933 set theoretic axiomatic approach] was so successful, that the problem of finding the basis of real applications of the results of the mathematical theory of probability became rather secondary to many investigators. ..[However] the basis for the applicability of the results of the mathematical theory of probability to real 'random' phenomena must depend in some form on the frequency concept of probability , the unavoidable nature of which has been established by von Mises in a spirited manner."
However, von Mises based his approach on axiomatically postulated infinite random sequences, representing repetitious independent trials with a limiting frequency. To this K. objects: "The frequency concept based on the notion of limiting frequency as the number of trials increases to infinity, does not contribute anything to substantiate the application of the results of probability theory to real practical problems where we always have to deal with a finite number of trials."
Following a four decades long controversy on von Mises' intended notion of an infinite random sequence, in a 1965 paper K. used the theory of algorithms to describe the complexity of a finite object as the length of the smallest description (algorithm to reconstruct it). This would seem to make the definition depend on the algorithmic method used. However, it turns out that there are optimal and universal methods for which the complexities of the objects described are asymptotically optimal. Although there are many optimal methods, the corresponding complexities differ by no more than an additive constant. It is natural to call a finite object random if it has no description of complexity less than it has itself. It is seductive to define an random infinite sequence as one of which the growth of complexity if the initial segments with the length is sufficiently fast, thus relating to von Mises' earlier approach. Due to unavoidable oscillations of the complexity of prefixes as function of their length this did not work out. However, P. Martin-Loef, a Swedish mathematician visiting K. in Moscow in 1964-1965, was able to show that under appropriate axiomatic definitions of randomness, one can prove once and for all that the thus defined sequences satisfy all effective tests for randomness, and have measure one in the set of all such infinite sequences. This rigorously defined an appropriate class, intuitively satisfactory as well, to qualify as von Mises' Kollektivs. Later it was shown by L.A. Levin, P. Gacs and G.J. Chaitin that one can refine the notion of complexity by defining it relative to a set of admissible descriptions. If admissible descriptions are restricted such that no description is a proper prefix of any other description, then an infinite sequence is Martin-L of random if and only if each of its finite initial sequences has a complexity that equals (up to a fixed constant) its length.
With the advent of electronic computers in the 1950's, a new emphasis on computer algorithms, and a maturing general recursive function theory, ideas tantamount to Kolmogorov complexity came to many people's minds, because ``when the time is ripe for certain things, these things appear in different places in the manner of violets coming to light in early spring,'' in the phrase of Wolfgang Bolyai in another famous context. Thus, R. Solomonoff in Cambridge, Massachusetts, had formulated the same ideas already in 1960. and had published his truly innovative work on the subject already in 1964 in `Information and Control'.
According to Solomonoff his work got far more attention after K. started to refer to it from 1968 onward, even though the attribution ``Kolmogorov'' complexity seems to have stuck. Says K.: ``I came to similar conclusions before becoming aware of Solomonoff's work, in 1963-1964.''
Yet a third independent inventor entered slightly later, Gregory Chaitin who was an 18 year old undergraduate in New York when he submitted a very similar set of inventions for publication end 1965 for publication in `J. Assoc. Comp. Mach.' (published in 1966 and 1969, the last paper containing the definition of Kolmogorov complexity and results thereof, while the 1966 paper extends C.E. Shannon's non-invariant notion of state-symbol measure for the complexity of Turing machines). Says Chaitin: ``this definition [of Kolmogorov complexity] was independently proposed about 1965 by A.N. Kolmogorov and me ... Both Kolmogorov and I were then unaware of related proposals made in 1960 by Ray Solomonoff.''
One of the last papers of K. was on the topic of algorithmic information theory - a paper together with Uspenskii published in 1987. For a comprehensive introduction and a survey of the astonishing range of applications of Kolmogorov complexity, see M. Li and P.M.B. Vitányi, An Introduction to Kolmogorov Complexity and its Applications, Springer-Verlag, New York, 1993 (xx + 546 pp).
As a Teacher
K.'s pedagogical activities began in 1922, when he became teacher at the experimental model school of the People's Commissariat for Education. He taught there until 1925. From 1925 till 1929 he was instructor at the University. Passing on knowledge and scientific ideas was very important for K. His interests in this subject ranged over the full scale from earliest education to higher education, and occupied much of his time. He actively took part in organizing mathematical Olympiads in schools and gave talks to school children. Thus he wrote a booklet on the topic ``Mathematics as a Profession'', which circulated in tens of thousands of copies. He put special emphasis on selection of mathematically gifted adolescents, since even the nonmathematicians will need such training in their later career
According to K., by 14-15 years about half of the pupils have come to the conclusion that mathematics and physics will be of little use to them. In recognition of that fact a special simplified program should be followed by such pupils. ``The mechanically understood principles of uniformity of schools providing general education, which excludes schools with a more detailed study of individual subjects, has outlived itself. As applied to mathematics it has already been destroyed by the creation of schools giving special training to computer operators and computer programmers.'' And: ``At 14-16 everything changes. At this age interest in mathematics usually becomes apparent, which quickly and painlessly leads the student to concentrated work and then to the real research work of the young scientist (at 18-20 years). ... For the beginners, the young people entering science for the first time, it is important to be convinced as soon as possible that they are capable of doing something original, their very own. When offering a subject for research to a graduate or a research student, the supervisor must not think only about the objective importance, or urgency of the subject, but also whether the work on the subject will stimulate the development of the young scientist, and whether it is within his powers to carry out, and at the same time demand the maximal effort of which he is capable.'' The ability to offer the students exactly what is most important and ripe in the development of science, and avoid pursuing dead-ends, and what is at the same time in their powers to accomplish is very characteristic for K.
The number of Kolmogorov's research students who have obtained their Ph.D. exceeds sixty. He was instrumental in substantial transformation (in the Soviet Union) of the very character of university education in mathematics, in particular the organization of practical work in mathematics, and updating the contents of mathematics. He also engaged in the search for new contents of mathematics in secondary schools, the founding of mathematical boarding schools, gave cycles of lectures for teachers on the structure of modern mathematics, and so on. Finally, he created an author's collective, and took part himself in writing textbooks on geometry, algebra and analysis for 6th through 10th grades. At the mathematical boarding school No. 18 at the University of Moscow, otherwise known as the ``Kolmogorov school'', he gave for years lessons up to 26 hours a week, and wrote accompanying syllabi. He also gave lectures to the students on music, art and literature. He felt that intellectual development must be evenly balanced. The former pupils of this school are very successful and systematically take the first places in All-Union and International Mathematical Olympiads.
In 1964 K. became head of the mathematical section of a joint syllabus committee of the USSR Academy of Sciences and that of Pedagogical Sciences. K. also organized a Statistical Laboratory at the University of Moscow, and succeeded in upgrading the budding library by obtaining large funds, and also international literature through partial use of money he received as part of the international Bolzano prize. In 1972 on K.'s initiative a compulsory course in mathematical logic was introduced for the first time in the Department of Mechanics and Mathematics at Moscow State University. He wrote the syllabus (which was still followed in 1983) and was the first to teach it.
According to V.I. Arnol'd, ``K. never explained anything, just posed problems, and didn't chew them over. He gave the student complete independence and never forced one to do anything, always waiting to hear from the student something remarkable. He stood out from the other professors I met by his complete respect for the personality of the student. I remember only one case where he interfered with my work: in 1959 he asked me to omit from the paper on self-maps of the circle the section on applications to heartbeats, adding "That is not one of the classical problems one ought to work on". The application to the theory of heartbeats was published by L. Glass 25 years later, while I had to concentrate my efforts on the celestial-mechanical applications of the same theory.''
L.S. Pontryagin relates: ``Kolmogorov gave me an interesting task..: to study [some problems in] locally compact algebraic fields in which multiplication is not necessarily commutative... A week later I reported to Aleksandrov that I had solved it in the case of commutative fields. Directly afterwards the three of us, Aleksandrov, Kolmogorov and I, met in Aleksandrov's flat. With a shade of ironical doubt, Kolmogorov said: "Well now, Lev Semenovich, I hear you have already solved my problem, let's hear you." Kolmogorov declared my very first statement to be false, but I immediately refuted him. Then he said: "Yes, it seems that the problem turned out to be much easier than I supposed." None of the rest of my answer aroused doubt. For the case of the noncommutative field the problem was immeasurably more difficult. It took me a whole year to work it out.'' It is also said that K. was one of the very few non-political mathematicians in the Soviet Union with yet real power. He quietly helped talented people with otherwise unfashionable views.
K.'s pupils included in the early years: Millionshchikov (later Vice-President of the USSR Academy of Sciences), Mal'tsev, Nikol'skii, Gnedenko, Gel'fand, Bavli and Verchenko. The subjects ranged from theoretical geophysics, mathematical logic, functional analysis, probability theory, function theory. During and after the war: Shilov, Fage, Sevast'yanov, Sirazhdinov, Pinsker, Prikhorov, Barenblatt, Bol'shev, Dobrushin, Medvedev, Mikhalevich, Uspenskii, Borovkov, Zolotarev, Alekseev, Belyaev, Mehhalkin, Epokhin, Rozanov, Sinai, Tikhomirov, Shiryaev, Arnol'd, Bassalygo, and Ofman. Later also Prokhorov, L.A. Levin, Kozlov, Zhurbenko, Abramov, and Bulinskii. His pupils include a number of well-known foreign mathematicians, among who the Swede P. Martin-L of. Pupils who became member of the USSR Academy of Sciences: A.I. Mal'tsev (algebra, mathematical logic), S.M. Nikol'skii (function theory), A.M. Obukhov (physics of the atmosphere), I.M. Gel'fand (functional analysis), Yu.V. Prokhorov (probability theory); and corresponding member: L.N. Bol'shev (mathematical statistics), A.A. Borovokov (probability theory, mathematical statistics), A.S. Monin (oceanology), and V.I. Arnol'd. The Ukrainian Academy of Sciences: B.V. Gnedenko (probability theory, history of mathematics), V.M. Mikhalevich (cybernetics), etc.
Scientific Career
K. entered Moscow University in 1920, graduated in 1925, and got his (equivalent of) Ph.D. in 1929, when he also got a position on the faculty. In 1931 K. became professor at Moscow University, and from 1933-1939 he also became Director of the Scientific Research Institute of Mathematics at the Moscow State University. Apparently, he was involved with the scientific research of all graduate students at the institute, not only his own. Most of them mention the unforgettable hikes on Sundays when K. invited all his own students (graduates and undergraduates) as well as students from other supervisors. These 40 km walks in the environment of Bolshevo, Klyaz'm, later Komarovka, are remembered as intellectually stimulating and culturally wide ranging experiences, ending when he and Aleksandrov treated the whole company to dinner in their dacha. In 1939 K. was elected as an Academician of the All-Union Academy of Sciences and as Academician-Secretary of the Physics-Mathematical Section. He also did enormous work as head of the mathematics editorial board of the Publishing House of Foreign Literature and as editor of the mathematics section of the Great Soviet Encyclopaedia. During the second world war K. engaged in the war effort by solving problems in ballistics and began research on problems of quality control of mass industrial production. From 1964 to 1966, and from 1976 till at least 1983 K. has been President of the Moscow Mathematical Society; from 1946 to 1954 and from 1983 on Editor-in-chief of Uspekhi Math. Nauk (Russian Mathematical Surveys). At the University of Moscow, K. held from 1938 to 1966 the chair of probability theory. From 1966 till 1976 he was the head of the Interdepartmental Laboratory of Statistical Methods, and from 1976 to 1980 he held the chair of mathematical statistics, which he organized. From 1980 on K. held the chair of mathematical logic. From 1951 to 1953 he was Director of the Institute of Mathematics and Mechanics of the Moscow State University; from 1954 to 1956 and from 1978 to at least 1983 the head of the mathematics section of the Faculty of Mechanics and Mathematics. From 1954 to 1958 he was Dean of the Faculty of Mechanics and Mathematics of the University.
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