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Consecutive congruent primes

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  • Jens Kruse Andersen
    Last month there was a thread about 3 consecutive small primes not having independent values modulo 3: http://groups.yahoo.com/group/primenumbers/message/17962
    Message 1 of 1 , May 23, 2006
      Last month there was a thread about 3 consecutive small
      primes not having independent values modulo 3:
      http://groups.yahoo.com/group/primenumbers/message/17962
      The value modulo 3 (or 6) changes significantly more than half the time.
      This skew is amplified for many consecutive primes,
      and it's also present for modulo 4.

      Below are tables of the starting prime in the first run of exactly n
      consecutive primes with the same value modulo 4, 6, 10 and 12.
      "Exactly" means it isn't part of a longer run.
      The tables for 4 and 6 also list the first alternating run starting
      with each of the 2 possible residues.
      Congruent runs were searched to 10^13, alternating runs only to 10^12.
      0 means no occurrence below that.

      Note that alternating long runs come much earlier.
      There are much longer alternating runs despite the lower search limit.
      This 21-run especially stands out:
      forprime(p=809,941,print1(p%6" "))
      5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5

      Run 1 (mod 4) 3 (mod 4) 1,3,..(mod 4) 3,1,..(mod 4)
      --- --------- --------- ------------- -------------
      1 5 3 97 211
      2 13 7 17 11
      3 89 739 773 3
      4 389 199 281 23
      5 2593 883 317 47
      6 12401 13127 409 167
      7 77069 463 8009 131
      8 262897 36551 6229 2011
      9 11593 39607 233 17903
      10 373649 183091 23633 25771
      11 766261 4468903 228601 34499
      12 3358169 6419299 1013 65867
      13 12204889 241603 9341 49523
      14 18256561 11739307 325217 90659
      15 23048897 9177431 521 115979
      16 12270077 95949311 51749 819583
      17 297387757 105639091 505049 1865719
      18 310523021 341118307 3773941 1391087
      19 297779117 1800380579 5414081 2264839
      20 3670889597 727334879 2556713 25123303
      21 5344989829 9449915743 17123893 142003427
      22 1481666377 1786054147 2569529 20855239
      23 2572421893 22964264027 15090641 38907623
      24 1113443017 54870713243 49008077 18246451
      25 121117598053 79263248027 855234581 6160043
      26 84676452781 454648144571 1756321181 1557431471
      27 790457451349 722204126767 43679609 152226383
      28 3498519134533 1749300591127 198572029 1450303451
      29 689101181569 5070807638111 701575297 7152451231
      30 3289884073409 8858854801319 7197633617 5552898499
      31 0 6425403612031 12661191689 6639843979
      32 0 113391385603 103272232741 61233611783
      33 3278744415797 0 170951814773 9005520203
      34 0 0 626798928989 99052377023
      35 0 0 521204950769 380268915347
      36 0 0 148638667361 946617613831
      37 0 0 438921167921 0
      38 0 0 0 929291719511

      The 32-run at 113391385603 was the first congruent run above 26.

      Run 1 (mod 6) 5 (mod 6) 1,5,..(mod 6) 5,1,..(mod 6)
      --- --------- --------- ------------- -------------
      1 7 5 157 53
      2 31 23 79 29
      3 151 47 67 239
      4 3049 251 37 137
      5 7351 1889 1039 449
      6 1741 7793 757 179
      7 19471 43451 5569 5
      8 118801 243161 2719 389
      9 498259 726893 277 89
      10 148531 759821 15667 2213
      11 406951 2280857 11149 128903
      12 2513803 1820111 10369 31469
      13 2339041 10141499 8527 6761
      14 89089369 40727657 113341 729269
      15 51662593 19725473 780823 80447
      16 73451737 136209239 151909 1303931
      17 232301497 744771077 43777 1485353
      18 450988159 400414121 2964553 7406639
      19 1558562197 1057859471 4397803 1457333
      20 2506152301 489144599 175573 1295219
      21 1444257673 13160911739 6510079 809
      22 28265029657 766319189 3954889 87902999
      23 24061965043 38451670931 153544819 121930481
      24 87996684091 119618704427 96050953 153576737
      25 43553959717 21549657539 15186319 590121509
      26 502429570231 141116164769 296080717 858510551
      27 1820249525317 140432294381 98380549 3354061163
      28 1892672756731 437339303279 131125681 77011289
      29 4236406530997 1871100711071 2720227693 11048169689
      30 2155866992887 3258583681877 52881047647 5696814287
      31 1552841185921 5611314737339 25183752283 1572386903
      32 0 0 4136299357 27799525007
      33 0 0 95832732277 288413159
      34 0 0 191765532499 62585146739
      35 0 0 114058236679 348989218973
      36 0 0 290614512109 405541876307
      37 0 0 0 0
      38 0 0 143014298809 0

      OEIS has congruent mod 4 and mod 6 sequences with primes to 2^31.
      I will submit extensions.

      The same value mod 12 means the same value both mod 4 and mod 6.

      Run 1 (mod 12) 5 (mod 12) 7 (mod 12) 11 (mod 12)
      --- ---------- ---------- ---------- -----------
      1 13 5 7 11
      2 661 509 619 467
      3 8317 4397 199 1499
      4 12829 42509 32443 16763
      5 586153 657197 407023 260339
      6 1081417 647417 180799 2003387
      7 10793941 1248869 4338787 7722419
      8 7790917 13175609 84885631 20221283
      9 682829881 234946997 472798219 927161471
      10 1921572157 1039154933 1786054267 4284484931
      11 370861009 7114719473 6024282871 7355362139
      12 5637496849 183420597029 64791932287 84805717127
      13 289391626057 32021552837 592175010019 478527373859
      14 469257742237 1237381737257 6265824724519 2046207697631
      15 628337233501 5760582040217 7816088451907 7302359785151
      16 0 9194779588901 0 0
      17 0 2904797643617 0 0

      Most of the above mod 12 values are in OEIS, computed earlier
      by Giovanni Resta for the large primes.
      In http://www.primepuzzles.net/puzzles/puzz_016.htm he also computed
      mod 10 values, i.e. same ending digit in decimal. My computation agrees.

      Run 1 (mod 10) 3 (mod 10) 7 (mod 10) 9 (mod 10)
      --- ---------- ---------- ---------- ----------
      1 11 3 7 19
      2 181 283 337 139
      3 4831 6793 1627 3089
      4 22501 22963 57427 18839
      5 216401 752023 192637 123229
      6 2229971 2707163 776257 2134519
      7 3873011 58339093 15328637 12130109
      8 91335901 44923183 70275277 23884639
      9 36539311 961129823 244650317 363289219
      10 196943081 1147752443 4075366567 9568590299
      11 14293856441 6879806623 452942827 24037796539
      12 363373386721 131145172583 73712513057 130426565719
      13 381206903941 177746482483 319931193737 405033487139
      14 154351758091 795537219143 2618698284817 3553144754209
      15 0 4028596340953 0 4010803176619
      16 0 6987191424553 0 0
      17 0 0 0 0

      The 16-run could be added to Phil's
      http://fatphil.org/maths/trivia/terminal.html

      --
      Jens Kruse Andersen
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