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Re: Special Set of Eleven Consecutive Primes

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  • djbroadhurst
    ... Why stop at 11? Here is the same sort of thing, but with 12 consecutive primes: [2123971, 2123983, 2123999, 2124007, 2124011, 2124013, 2124019, 2124037,
    Message 1 of 5 , Jul 8 8:16 AM
      --- In primenumbers@yahoogroups.com,
      w_sindelar@... wrote:

      > Here is a set of 11 consecutive primes consisting
      > of 8 contiguous subsets of 4 primes. Each subset
      > has 3 distinct differences (gaps) between adjacent primes
      > and 4 distinct rightmost digits.
      > (2213, 2221, 2237, 2239,
      > 2243, 2251, 2267, 2269,
      > 2273, 2281, 2287).

      Why stop at 11? Here is the same sort of thing,
      but with 12 consecutive primes:

      [2123971, 2123983, 2123999, 2124007,
      2124011, 2124013, 2124019, 2124037,
      2124041, 2124043, 2124049, 2124127]

      David
    • Jens Kruse Andersen
      ... The ending digit must go through a cycle of length 4, for example the above 3179 3179 ... If the gap requirement is satisfied modulo 10 for the cycle then
      Message 2 of 5 , Jul 8 9:10 AM
        Bill Sindelar wrote:
        > Here is a set of 11 consecutive primes consisting of 8 contiguous subsets
        > of 4 primes. Each subset has 3 distinct differences (gaps) between
        > adjacent primes and 4 distinct rightmost digits.
        > (2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287).

        > Can theory hint where on the number line one is most likely to find sets
        > like this? Anyone care to try to find a bigger one? Thanks folks.

        The ending digit must go through a cycle of length 4, for example the above
        3179 3179 ...

        If the gap requirement is satisfied modulo 10 for the cycle then
        it will automatically be satisfied for the whole sequence.
        The record setters up to 10^12 except length 19 are all of this type.

        12:
        2123971
        2123983
        2123999
        2124007
        2124011
        2124013
        2124019
        2124037
        2124041
        2124043
        2124049
        2124127

        13:
        66945301
        66945313
        66945349
        66945407
        66945421
        66945443
        66945449
        66945467
        66945491
        66945503
        66945509
        66945517
        66945521

        14:
        66945301
        66945313
        66945349
        66945407
        66945421
        66945443
        66945449
        66945467
        66945491
        66945503
        66945509
        66945517
        66945521
        66945553

        15:
        184171621
        184171643
        184171679
        184171697
        184171711
        184171733
        184171739
        184171747
        184171751
        184171753
        184171759
        184171777
        184171781
        184171843
        184171849

        16:
        2493412841
        2493412847
        2493412849
        2493412853
        2493412891
        2493412937
        2493412939
        2493413023
        2493413051
        2493413057
        2493413059
        2493413063
        2493413071
        2493413077
        2493413099
        2493413113

        17:
        2732087209
        2732087227
        2732087231
        2732087263
        2732087279
        2732087317
        2732087321
        2732087333
        2732087359
        2732087447
        2732087461
        2732087483
        2732087489
        2732087537
        2732087551
        2732087563
        2732087569

        18:
        10206934519
        10206934523
        10206934541
        10206934567
        10206934579
        10206934583
        10206934601
        10206934607
        10206934609
        10206934613
        10206934631
        10206934697
        10206934709
        10206934723
        10206934741
        10206934757
        10206934769
        10206934783

        19:
        554804585227
        554804585233
        554804585299
        554804585311
        554804585347
        554804585393
        554804585449
        554804585471
        554804585507
        554804585513
        554804585579
        554804585651
        554804585737
        554804585743
        554804585779
        554804585831
        554804585837
        554804585873
        554804585899

        20:
        692245399661
        692245399687
        692245399709
        692245399723
        692245399751
        692245399787
        692245399789
        692245399793
        692245399801
        692245399837
        692245399849
        692245399873
        692245399891
        692245399907
        692245399919
        692245399943
        692245399991
        692245399997
        692245400009
        692245400023

        --
        Jens Kruse Andersen
      • w_sindelar@juno.com
        Thank you David, and thank you Jens.You never cease to astound me. In the 9 examples you so quickly calculated, the sets 12, 13, 14 and 15 have the pattern (1,
        Message 3 of 5 , Jul 9 6:50 AM
          Thank you David, and thank you Jens.You never cease to astound me. In the 9 examples you so quickly calculated, the sets 12, 13, 14 and 15 have the pattern (1, 3, 9, 7) for the beginning set of 4 consecutive primes. For sets 16 and 20 the pattern is (1, 7, 9, 3). For set 11 the pattern is (3, 1, 7, 9). For set 19 the pattern is ((7, 3, 9, 1). For set 18 the pattern is (9, 3, 1, 7). For set 17 the pattern is (9, 7, 1, 3).


          There are 24 possible patterns into which the 4 digits 1, 3, 7 and 9 can be arranged, so that each pattern contains distinct digits. The sets 11 to 20 account for six.


          The obvious question comes up. Is it possible to find additional sets 21, 22, 23, 24...N, such that for the sets 11 to N all 24 possible patterns are used in the begining sets of 4 consecutive primes?


          Bill Sindelar



          ____________________________________________________________
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        • djbroadhurst
          ... That was indeed my methodology, adopted in order quickly to attain length 12, with gaps in Z/Z10 given by [2, 6, 8, 4, 2, 6, 8, 4, 2, 6, 8] It s truly
          Message 4 of 5 , Jul 9 3:57 PM
            --- In primenumbers@yahoogroups.com,
            "Jens Kruse Andersen" <jens.k.a@...> wrote:

            > The ending digit must go through a cycle of length 4 ...
            > If the gap requirement is satisfied modulo 10 for the cycle then
            > it will automatically be satisfied for the whole sequence.

            That was indeed my methodology, adopted
            in order quickly to attain length 12,
            with gaps in Z/Z10 given by

            [2, 6, 8, 4, 2, 6, 8, 4, 2, 6, 8]

            It's truly impressive that Jens attained length 20.
            Moreover, he had the foresight to study gaps in Z.
            Videlicit:

            > 19:
            > 554804585227
            > 554804585233
            > 554804585299
            > 554804585311
            > 554804585347
            > 554804585393
            > 554804585449
            > 554804585471
            > 554804585507
            > 554804585513
            > 554804585579
            > 554804585651
            > 554804585737
            > 554804585743
            > 554804585779
            > 554804585831
            > 554804585837
            > 554804585873
            > 554804585899

            where the gaps in Z/Z10 are

            [6, 6, 2, 6, 6, 6, 2, 6, 6, 6, 2, 6, 6, 6, 2, 6, 6, 6]

            but are still OK in Z:

            [6, 66, 12, 36, 46, 56, 22, 36, 6, 66, 72, 86, 6, 36, 52, 6, 36, 26]

            David (well and truly outdone by Jens, as so often before)
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