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Re: [PrimeNumbers] Prime centuries again (pattern repetition)

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  • Jens Kruse Andersen
    ... 1671800 is the first. 2637800 is second. The first 20 cases (measured by the larger century) of a repeated non-consecutive century: {390500, 480800} + {3,
    Message 1 of 3 , May 16, 2011
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      Rupert Wood wrote:
      > Thanks for your contributions. I should have been more detailed in my
      > original comments. Jens' terms (average etc) are clear to me. Amongst all
      > this - what are the first two (not necessarily consecutive) void centuries?
      > I know that 1671800 is a void century, not even sure if it is the first one.

      1671800 is the first. 2637800 is second.

      The first 20 cases (measured by the larger century) of a
      repeated non-consecutive century:
      {390500, 480800} + {3, 27, 39, 53, 63, 67, 91}
      {351100, 566800} + {21, 33, 51, 57, 63, 85, 97}
      {78900, 578700} + {1, 19, 29, 41, 63, 65, 83, 93}
      {323600, 606200} + {23, 41, 47, 51, 63, 87}
      {326700, 645900} + {1, 7, 37, 41, 63, 65, 71}
      {619400, 745400} + {63}
      {655700, 842600} + {17, 23, 27, 57, 81, 87, 91}
      {437300, 855800} + {21, 51, 57, 63, 85}
      {854700, 876900} + {13, 29, 47, 63, 77, 93}
      {811300, 915400} + {37, 51, 63}
      {561400, 920500} + {9, 19, 39, 61, 73, 83}
      {515400, 956400} + {1, 29, 63, 65, 93}
      {452400, 1023600} + {1, 43, 53, 63, 65}
      {805600, 1049300} + {33, 39, 63, 97}
      {247100, 1092500} + {41, 63}
      {1037900, 1127600} + {3, 17, 29, 41, 57, 63, 67, 81, 93}
      {69700, 1137100} + {9, 37, 39, 61, 63, 73}
      {567100, 1233100} + {1, 7, 21, 43, 63, 65, 71, 85}
      {1107200, 1242500} + {3, 17, 63, 67, 81}
      {358500, 1248600} + {31, 41, 63, 95}

      There are many others before the first case with void centuries:
      {1671800, 2637800} + {}

      Century is a base-10 concept and primes are base-independent.
      The first prime gap above 100 is 112 from 370261 to 370373.
      A maximal prime gap is larger than all gaps between smaller primes.
      The first 75 maximal gaps are here:
      http://users.cybercity.dk/~dsl522332/math/primegaps/maximal.htm

      According to data at http://www.trnicely.net/gaps/gaplist.html
      the first 5 void millenia (meaning 1000X to 1000X+999) are
      in the following gaps found by Bertil Nyman:

      gap gap start
      1050 13893290219203981
      1100 28907866922785967
      1122 31068473876462989
      1062 42079111699533971
      1062 42135469324106969

      I guess there are earlier cases of repeated non-consecutive
      millenia containing primes but I haven't looked.

      Phil Carmody wrote:
      > It appears you've only attacked this from an angle of finding
      > the easiest pattern to detect, rather than an exhaustive one.
      > As primes tend towards sparseness, this is clearly the optimal
      > approach. However, with that view-point, the denser patterns
      > are the more interesting ones - I wonder what remains yet
      > undiscovered?

      An exhaustive century sounds infeasible to me.
      I once computed a non-trivial occurrence (no primes below 100)
      of all exhaustive admissible patterns of width at most 60, but
      didn't publish the results. The mix of early and late hits was
      around as expected, except for a surprisingly late hit on
      {0, 4, 10, 18, 22, 24, 28, 30, 40, 42, 52, 54, 58, 60}
      If you still have a tuplet finder then perhaps you can confirm
      it wasn't an error on my part.

      I found another type of result in
      http://tech.groups.yahoo.com/group/primenumbers/message/17821:

      > n=446863043340173267 and n=534402442999154537 both give
      > consecutive primes:
      > n + (0,2,56,62,80,110,146,150,152,170,230,234,252,264,276,
      > 290,294,296,300,302,344)

      21 consecutive primes in a repeated pattern is still the record
      as far as I know. The pattern is obviously far from exhaustive.

      Raanan Chermoni & Jaroslaw Wroblewski found the far harder
      exhaustive 19-tuplets:
      630134041802574490482213901 or 656632460108426841186109951 +
      {0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76}

      --
      Jens Kruse Andersen
    • woodhodgson@xtra.co.nz
      Thanks again Jens - a veritable mine of information there. I freely admit that looking at base-10 stuff is a bit artificial. Nevertheless, I quite like - for
      Message 2 of 3 , May 17, 2011
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        Thanks again Jens - a veritable mine of information there. I freely admit that looking at base-10 stuff is a bit artificial.

        Nevertheless, I quite like - for instance - the set {1663,1667,1669,1693,1697,1699}, which has two consecutive identical decadal "triples" with no intervening primes, and the "midpoint" of the array, 1681, is the square of a prime. A rough calculation on a spreadsheet showed me (I think) that if the triples pattern is {3,7,9} then there is no other instance of this (with the midpoint being any square of an integer) below 10E7.

        --- In primenumbers@yahoogroups.com, "Jens Kruse Andersen" <jens.k.a@...> wrote:
        >
        > Rupert Wood wrote:
        > > Thanks for your contributions. I should have been more detailed in my
        > > original comments. Jens' terms (average etc) are clear to me. Amongst all
        > > this - what are the first two (not necessarily consecutive) void centuries?
        > > I know that 1671800 is a void century, not even sure if it is the first one.
        >
        > 1671800 is the first. 2637800 is second.
        >
        > The first 20 cases (measured by the larger century) of a
        > repeated non-consecutive century:
        > {390500, 480800} + {3, 27, 39, 53, 63, 67, 91}
        > {351100, 566800} + {21, 33, 51, 57, 63, 85, 97}
        > {78900, 578700} + {1, 19, 29, 41, 63, 65, 83, 93}
        > {323600, 606200} + {23, 41, 47, 51, 63, 87}
        > {326700, 645900} + {1, 7, 37, 41, 63, 65, 71}
        > {619400, 745400} + {63}
        > {655700, 842600} + {17, 23, 27, 57, 81, 87, 91}
        > {437300, 855800} + {21, 51, 57, 63, 85}
        > {854700, 876900} + {13, 29, 47, 63, 77, 93}
        > {811300, 915400} + {37, 51, 63}
        > {561400, 920500} + {9, 19, 39, 61, 73, 83}
        > {515400, 956400} + {1, 29, 63, 65, 93}
        > {452400, 1023600} + {1, 43, 53, 63, 65}
        > {805600, 1049300} + {33, 39, 63, 97}
        > {247100, 1092500} + {41, 63}
        > {1037900, 1127600} + {3, 17, 29, 41, 57, 63, 67, 81, 93}
        > {69700, 1137100} + {9, 37, 39, 61, 63, 73}
        > {567100, 1233100} + {1, 7, 21, 43, 63, 65, 71, 85}
        > {1107200, 1242500} + {3, 17, 63, 67, 81}
        > {358500, 1248600} + {31, 41, 63, 95}
        >
        > There are many others before the first case with void centuries:
        > {1671800, 2637800} + {}
        >
        > Century is a base-10 concept and primes are base-independent.
        > The first prime gap above 100 is 112 from 370261 to 370373.
        > A maximal prime gap is larger than all gaps between smaller primes.
        > The first 75 maximal gaps are here:
        > http://users.cybercity.dk/~dsl522332/math/primegaps/maximal.htm
        >
        > According to data at http://www.trnicely.net/gaps/gaplist.html
        > the first 5 void millenia (meaning 1000X to 1000X+999) are
        > in the following gaps found by Bertil Nyman:
        >
        > gap gap start
        > 1050 13893290219203981
        > 1100 28907866922785967
        > 1122 31068473876462989
        > 1062 42079111699533971
        > 1062 42135469324106969
        >
        > I guess there are earlier cases of repeated non-consecutive
        > millenia containing primes but I haven't looked.
        >
        > Phil Carmody wrote:
        > > It appears you've only attacked this from an angle of finding
        > > the easiest pattern to detect, rather than an exhaustive one.
        > > As primes tend towards sparseness, this is clearly the optimal
        > > approach. However, with that view-point, the denser patterns
        > > are the more interesting ones - I wonder what remains yet
        > > undiscovered?
        >
        > An exhaustive century sounds infeasible to me.
        > I once computed a non-trivial occurrence (no primes below 100)
        > of all exhaustive admissible patterns of width at most 60, but
        > didn't publish the results. The mix of early and late hits was
        > around as expected, except for a surprisingly late hit on
        > {0, 4, 10, 18, 22, 24, 28, 30, 40, 42, 52, 54, 58, 60}
        > If you still have a tuplet finder then perhaps you can confirm
        > it wasn't an error on my part.
        >
        > I found another type of result in
        > http://tech.groups.yahoo.com/group/primenumbers/message/17821:
        >
        > > n=446863043340173267 and n=534402442999154537 both give
        > > consecutive primes:
        > > n + (0,2,56,62,80,110,146,150,152,170,230,234,252,264,276,
        > > 290,294,296,300,302,344)
        >
        > 21 consecutive primes in a repeated pattern is still the record
        > as far as I know. The pattern is obviously far from exhaustive.
        >
        > Raanan Chermoni & Jaroslaw Wroblewski found the far harder
        > exhaustive 19-tuplets:
        > 630134041802574490482213901 or 656632460108426841186109951 +
        > {0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76}
        >
        > --
        > Jens Kruse Andersen
        >
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