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Known prime gaps

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  • Jens Kruse Andersen
    Polignac s conjecture says all even prime gap sizes occur infinitely many times. So far the only known way to prove existence of a gap size is to find an
    Message 1 of 11 , May 6, 2007
      Polignac's conjecture says all even prime gap sizes occur infinitely many
      times. So far the only known way to prove existence of a gap size is to find
      an occurrence.

      Thomas R. Nicely maintains tables of first known occurrence prime gaps at
      http://www.trnicely.net/gaps/gaplist.html
      For each gap size the smallest known consecutive primes or prp's with that
      gap are listed.

      Torbjörn Alm has searched first known occurrence gaps for a long time with a
      sieve by me, using modular equations to ensure unusually many small factors
      in wanted gaps. Small prp tests are made by the GMP library, and large by
      PrimeForm/GW.

      There is now a proven occurrence of all 10000 even gaps up to 20000.
      Marcel Martin's Primo proved the large majority of the 20000 gap ends.

      In addition, there is now either a proven or prp occurrence of all even gaps
      up to 30000, and currently of 30046 even gaps in total (and 1 odd!).
      Torbjörn found the listed occurrence of 21274 of them. Others had previously
      found larger primes for some of the gap sizes. It is not recorded who was
      the first to find an occurrence of a gap.

      The Top-20 Prime Gaps at
      http://hjem.get2net.dk/jka/math/primegaps/gaps20.htm lists the best gaps in
      different categories.
      The merit of the gap from p1 to p2 is defined as (p2-p1)/log p1, where log
      p1 is the average gap size in that vicinity.
      This year Torbjörn has found the 3 largest known gaps with merit above 20.
      The best is a gap of 114554 between 2227-digit primes. The merit is 22.34.

      --
      Jens Kruse Andersen
    • Mike Oakes
      ... gaps in ... where log ... above 20. ... 22.34. ... Jens: Congratulations on adding yet another /very/ nice page to your site. The only sticking point I
      Message 2 of 11 , May 7, 2007
        --- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
        <jens.k.a@...> wrote:
        >
        > The Top-20 Prime Gaps at
        > http://hjem.get2net.dk/jka/math/primegaps/gaps20.htm lists the best
        gaps in
        > different categories.
        > The merit of the gap from p1 to p2 is defined as (p2-p1)/log p1,
        where log
        > p1 is the average gap size in that vicinity.
        > This year Torbjörn has found the 3 largest known gaps with merit
        above 20.
        > The best is a gap of 114554 between 2227-digit primes. The merit is
        22.34.
        >

        Jens: Congratulations on adding yet another /very/ nice page to your
        site.

        The only "sticking point" I had when reading it was this:-
        <<
        A basic expression is here defined as maximum 25 characters, all
        taken from 0123456789+-*/^( ). Primorial and factorial are not
        allowed since they can be used to ensure many small factors, and the
        idea of the basic expression record is partly to avoid special prime
        gap methods.
        >>

        A rule that excludes the 2 characters ! and # seems weird.

        Is not Pierre's
        PRP38007 = 50491*(87811#)/6 - 657714
        every bit as transparent and explicit as Milton's
        PRP14173 = 10^14173 - 51197
        ?

        You would seem to be in danger of being thought to be penalizing
        users of the humbler n*p#+k construct viz a viz your own more
        sophisticated methods.

        Just my 4c :-)

        -Mike Oakes
      • SWagler@aol.com
        All, Years ago I plotted a frequency distribution of prime gaps from 2 to some small limit and the curve always looked similar to the curve for black body
        Message 3 of 11 , May 7, 2007
          All,

          Years ago I plotted a frequency distribution of prime gaps from 2 to some small limit and the curve always looked similar to the curve for black body radiation. Has anyone done this for limits large or small? Are there theoretical reasons to account for this?

          Steve Wagler



          -----Original Message-----
          From: jens.k.a@...
          To: primenumbers@yahoogroups.com
          Sent: Sun, 6 May 2007 5:14 PM
          Subject: [PrimeNumbers] Known prime gaps


          Polignac's conjecture says all even prime gap sizes occur infinitely many
          times. So far the only known way to prove existence of a gap size is to find
          an occurrence.

          Thomas R. Nicely maintains tables of first known occurrence prime gaps at
          http://www.trnicely.net/gaps/gaplist.html
          For each gap size the smallest known consecutive primes or prp's with that
          gap are listed.

          Torbjörn Alm has searched first known occurrence gaps for a long time with a
          sieve by me, using modular equations to ensure unusually many small factors
          in wanted gaps. Small prp tests are made by the GMP library, and large by
          PrimeForm/GW.

          There is now a proven occurrence of all 10000 even gaps up to 20000.
          Marcel Martin's Primo proved the large majority of the 20000 gap ends.

          In addition, there is now either a proven or prp occurrence of all even gaps
          up to 30000, and currently of 30046 even gaps in total (and 1 odd!).
          Torbjörn found the listed occurrence of 21274 of them. Others had previously
          found larger primes for some of the gap sizes. It is not recorded who was
          the first to find an occurrence of a gap.

          The Top-20 Prime Gaps at
          http://hjem.get2net.dk/jka/math/primegaps/gaps20.htm lists the best gaps in
          different categories.
          The merit of the gap from p1 to p2 is defined as (p2-p1)/log p1, where log
          p1 is the average gap size in that vicinity.
          This year Torbjörn has found the 3 largest known gaps with merit above 20.
          The best is a gap of 114554 between 2227-digit primes. The merit is 22.34.

          --
          Jens Kruse Andersen
          ___
          .

          ________________________________________________________________________
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          [Non-text portions of this message have been removed]
        • Jens Kruse Andersen
          ... Thanks. It s actually an old page originally started by Paul Leyland: http://hjem.get2net.dk/jka/math/primegaps/leylandgaps20.htm The only new page is
          Message 4 of 11 , May 7, 2007
            Mike Oakes wrote:
            > Jens: Congratulations on adding yet another /very/ nice page to your
            > site.

            Thanks. It's actually an old page originally started by Paul Leyland:
            http://hjem.get2net.dk/jka/math/primegaps/leylandgaps20.htm

            The only new page is "Ormiston Tuples" (consecutive primes containing the
            same digits): http://hjem.get2net.dk/jka/math/ormiston_tuples.htm
            The largest known number of such primes is 9:
            26460346024426922096587598498580390201381951306930145595901871467050710000
            + (7839, 7893, 7983, 8379, 8397, 8739, 8793, 8937, 8973)

            > The only "sticking point" I had when reading it was this:-
            > <<
            > A basic expression is here defined as maximum 25 characters, all
            > taken from 0123456789+-*/^( ). Primorial and factorial are not
            > allowed since they can be used to ensure many small factors, and the
            > idea of the basic expression record is partly to avoid special prime
            > gap methods.
            > >>
            >
            > A rule that excludes the 2 characters ! and # seems weird.

            They are excluded for what they represent: An expression designed to have
            specific values (in this case all 0's) modulo a lot of numbers. I want to
            show the best gap which reaches merit above 10 or 20 without a
            construction that increases the chance of large gaps. It doesn't matter
            whether a single character has been defined to express the construction.
            I am not considering to permit ! and #, but I might permit other things if
            they don't allow modular constructions. As far as I know, nobody has
            searched large prime gaps expressed with other operators or functions than
            +-*/^#!, so it's not relevant now. I'm not spending time considering a lot
            of hypothetical functions which are never used for prime gaps. It's only a
            small part of the site.

            > Is not Pierre's
            > PRP38007 = 50491*(87811#)/6 - 657714
            > every bit as transparent and explicit as Milton's
            > PRP14173 = 10^14173 - 51197
            > ?

            Yes, but transparent and explicit are not considerations - and explicit
            decimal expansions are on subpages (unlike Nicely's site which has to list
            far more gaps).

            > You would seem to be in danger of being thought to be penalizing
            > users of the humbler n*p#+k construct viz a viz your own more
            > sophisticated methods.

            Really? The rule for "basic expressions" only applies to two gaps at the
            site. None of them are currently by me and I have no plans to search for
            replacements. I have been a bit harsh in public on Milton's gap
            announcements, so I don't think people will suspect me of doing him a
            favour over Pierre.
            By the way, Pierre's n*p#/6+k has better gap performance on average (but not
            in worst case) than n*p#+k. I don't know whether Pierre was first to notice
            this, and I haven't analyzed whether 6 is the optimal divisor to exclude
            from the primorial.

            --
            Jens Kruse Andersen
          • Mike Oakes
            ... have ... Yet, is it not true that your own Chinese Remainder Theorem technique is equally designed to do precisely just that ! (I obviously can t sway you,
            Message 5 of 11 , May 7, 2007
              --- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
              <jens.k.a@...> wrote:
              >
              >> A rule that excludes the 2 characters ! and # seems weird.
              >
              > They are excluded for what they represent: An expression designed to
              have
              > specific values (in this case all 0's) modulo a lot of numbers.

              Yet, is it not true that your own Chinese Remainder Theorem technique
              is equally designed to do precisely just that !

              (I obviously can't sway you, so I'll shut up now.)

              -Mike Oakes
            • Mike Oakes
              ... Sorry, I m probably being stupid and missing your point, which is: you want to give extra credit where /no/ such technique is employed, don t you? My bad.
              Message 6 of 11 , May 7, 2007
                --- In primenumbers@yahoogroups.com, "Mike Oakes" <mikeoakes2@...>
                wrote:

                > Yet, is it not true that your own Chinese Remainder Theorem technique
                > is equally designed to do precisely just that !
                >

                Sorry, I'm probably being stupid and missing your point, which is: you
                want to give extra credit where /no/ such technique is employed, don't
                you?
                My bad.

                Mike
              • Jens Kruse Andersen
                ... Yes. Credit for doing something harder but more natural . I recall an old Guinness edition which in addition to the official world record listed fastest
                Message 7 of 11 , May 7, 2007
                  Mike Oakes wrote:
                  > Sorry, I'm probably being stupid and missing your point, which is: you
                  > want to give extra credit where /no/ such technique is employed, don't
                  > you?

                  Yes. Credit for doing something harder but more "natural".
                  I recall an old Guinness edition which in addition to the official world
                  record listed "fastest 100m at sea level" (there is less air resistance in
                  thin air at altitude).
                  The other table comment, "Largest gap with proven end points",
                  is also extra credit.
                  Finding large merits with no modular technique requires a huge
                  number of attempts.
                  Such gaps get "unfair" competition from "artificial" modular constructions.
                  Without the basic expression listing, they would have no entry for
                  merit above 20, and until recently no entry for merit above 10.

                  --
                  Jens Kruse Andersen
                • Phil Carmody
                  ... Prime gaps are far from smoothly distributed. Gaps divisible by 3 are more likely than ones not divisible by 3. As 30 becomes small, gaps divisible by 30
                  Message 8 of 11 , May 8, 2007
                    --- SWagler@... wrote:
                    > All,
                    >
                    > Years ago I plotted a frequency distribution of prime gaps from 2 to some
                    > small limit and the curve always looked similar to the curve for black body
                    > radiation. Has anyone done this for limits large or small? Are there
                    > theoretical reasons to account for this?

                    Prime gaps are far from smoothly distributed.
                    Gaps divisible by 3 are more likely than ones not divisible by 3.
                    As 30 becomes small, gaps divisible by 30 also become more popular.
                    As always this can be explained by looking at small primes.

                    Phil

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                  • Andrey Kulsha
                    ... A good illustration: http://ieeta.pt/~tos/gaps.html All the best, Andrey [Non-text portions of this message have been removed]
                    Message 9 of 11 , May 8, 2007
                      > > Years ago I plotted a frequency distribution of prime gaps from 2 to some
                      > > small limit and the curve always looked similar to the curve for black body
                      > > radiation. Has anyone done this for limits large or small? Are there
                      > > theoretical reasons to account for this?
                      >
                      > Prime gaps are far from smoothly distributed.
                      > Gaps divisible by 3 are more likely than ones not divisible by 3.
                      > As 30 becomes small, gaps divisible by 30 also become more popular.
                      > As always this can be explained by looking at small primes.

                      A good illustration: http://ieeta.pt/~tos/gaps.html

                      All the best,

                      Andrey

                      [Non-text portions of this message have been removed]
                    • Phil Carmody
                      ... Except it doesn t venture into the 30 becomes small region. I can t remember where 30 takes over on from 30 as the most likely gap. I presume that s
                      Message 10 of 11 , May 8, 2007
                        --- Andrey Kulsha <Andrey_601@...> wrote:
                        > > > Years ago I plotted a frequency distribution of prime gaps from 2 to some
                        > > > small limit and the curve always looked similar to the curve for black
                        > body
                        > > > radiation. Has anyone done this for limits large or small? Are there
                        > > > theoretical reasons to account for this?
                        > >
                        > > Prime gaps are far from smoothly distributed.
                        > > Gaps divisible by 3 are more likely than ones not divisible by 3.
                        > > As 30 becomes small, gaps divisible by 30 also become more popular.
                        > > As always this can be explained by looking at small primes.
                        >
                        > A good illustration: http://ieeta.pt/~tos/gaps.html

                        Except it doesn't venture into the '30 becomes small' region. I can't remember
                        where 30 takes over on from 30 as the most likely gap. I presume that's touched
                        on somewhere on the prime pages or on mathworld, and if it isn't on both,
                        something should be done about that!

                        Phil

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                      • Jens Kruse Andersen
                        ... They are the first two hits on http://www.google.com/search?hl=en&q=%22jumping+champion%22 -- Jens Kruse Andersen
                        Message 11 of 11 , May 8, 2007
                          Phil Carmody wrote:
                          >> A good illustration: http://ieeta.pt/~tos/gaps.html
                          >
                          > Except it doesn't venture into the '30 becomes small' region. I can't
                          > remember
                          > where 30 takes over on from 30 as the most likely gap. I presume that's
                          > touched
                          > on somewhere on the prime pages or on mathworld, and if it isn't on both,
                          > something should be done about that!

                          They are the first two hits on
                          http://www.google.com/search?hl=en&q=%22jumping+champion%22

                          --
                          Jens Kruse Andersen
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