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Janos Pintz survey + new results on prime gaps

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  • WarrenS
    http://arxiv.org/abs/1305.6289 Very readable and nice. Also claims a result I had conjectured here before (I had outlined a proof using Zhang, but never
    Message 1 of 5 , Jul 26 5:36 PM
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      http://arxiv.org/abs/1305.6289

      Very readable and nice.
      Also claims a result I had conjectured here before (I had
      outlined a proof using Zhang, but never checked that the proof plan could
      really be carried through)
      is indeed true: there are infinitely many "de Polignac numbers."

      One of the most impressive claims in Pintz's paper is this. Let a "near twin"
      prime p be a prime p such that p,q are two primes with p<q<p+70000000.

      Then: the near-twin primes contain arbitrarily long arithmetic progressions.
    • WarrenS
      ... --So those of you who keep finding long APs with prime entries only, might instead seek long APs with twin-prime entries only (a task which ought to be
      Message 2 of 5 , Jul 27 3:58 AM
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        > http://arxiv.org/abs/1305.6289

        > One of the most impressive claims in Pintz's paper is this. Let a "near twin"
        > prime p be a prime p such that p,q are two primes with p<q<p+70000000.
        >
        > Then: the near-twin primes contain arbitrarily long arithmetic progressions.

        --So those of you who keep finding long APs with prime entries only, might
        instead seek long APs with twin-prime entries only (a task which
        ought to be accomplishable using similar methods).
      • Jens Kruse Andersen
        ... Paul Jobling found 10 twin primes in AP in 2000: https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0001&L=NMBRTHRY&F=&S=&P=8002
        Message 3 of 5 , Jul 27 6:40 AM
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          WarrenS wrote:
          > seek long APs with twin-prime entries only

          Paul Jobling found 10 twin primes in AP in 2000:
          https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0001&L=NMBRTHRY&F=&S=&P=8002
          http://www.primepuzzles.net/puzzles/puzz_121.htm
          I don't know whether his record has been broken.

          --
          Jerns Kruse Andersen
        • WarrenS
          ... --oh. I was about to mention the following twin-prime arithmetic progressions which however are shorter than Jobling s: start=3, gap=2, #terms=2 start=5,
          Message 4 of 5 , Jul 27 7:59 AM
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            --- In primenumbers@yahoogroups.com, "Jens Kruse Andersen" <jens.k.a@...> wrote:
            >
            > WarrenS wrote:
            > > seek long APs with twin-prime entries only
            >
            > Paul Jobling found 10 twin primes in AP in 2000:
            > https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0001&L=NMBRTHRY&F=&S=&P=8002
            > http://www.primepuzzles.net/puzzles/puzz_121.htm
            > I don't know whether his record has been broken.
            >
            > --
            > Jerns Kruse Andersen

            --oh. I was about to mention the following twin-prime arithmetic progressions
            which however are shorter than Jobling's:

            start=3, gap=2, #terms=2
            start=5, gap=6, #terms=3
            start=5, gap=12, #terms=4
            start=11, gap=13222650, #terms=5
            start=17, gap=6930, #terms=5
            start=11, gap=569460150, #terms=6
            start=311, gap=1667820, #terms=6
            start=1049, gap=643896330, #terms=7
            start=191339, gap=16170, #terms=7
            start=451277, gap=15248310, #terms=8
            start=1767419, gap=5897430, #terms=8
            start=3005291, gap=1517670, #terms=8

            which I found in about 30 minutes using a very stupid search.

            Jobling finds 10 APs with 10 twin primes each,
            and conjectures there is a maximum length of such an AP.
            But Pintz's theorem (if believe him) now disproves Jobling's conjecture
            at least if the definition of "twin" prime is no longer (p, p+2) but rather
            (p, p+SuitableAbsoluteConstant).

            I do not think it would be too difficult to break Jobling's "10" record,
            if anybody were willing to put in the effort. Jobling's search took 40 days
            on a 233 MHz machine. Finding one of length 11 ought to be only
            about 50 times harder using his method, but I would think there might be
            better search methods partly based on sieving.
          • WarrenS
            ... --and after a few hours: start=14471111, gap=156791250, #terms=9 it looks like my stupid search method actually is comparable in efficiency to Jobling s
            Message 5 of 5 , Jul 27 9:30 AM
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              > > Paul Jobling found 10 twin primes in AP in 2000:
              > > https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0001&L=NMBRTHRY&F=&S=&P=8002
              > > http://www.primepuzzles.net/puzzles/puzz_121.htm
              > --oh. I was about to mention the following twin-prime arithmetic progressions
              > which however are shorter than Jobling's:
              >
              > start=3, gap=2, #terms=2
              > start=5, gap=6, #terms=3
              > start=5, gap=12, #terms=4
              > start=11, gap=13222650, #terms=5
              > start=17, gap=6930, #terms=5
              > start=11, gap=569460150, #terms=6
              > start=311, gap=1667820, #terms=6
              > start=1049, gap=643896330, #terms=7
              > start=191339, gap=16170, #terms=7
              > start=451277, gap=15248310, #terms=8
              > start=1767419, gap=5897430, #terms=8
              > start=3005291, gap=1517670, #terms=8
              >
              > which I found in about 30 minutes using a very stupid search.

              --and after a few hours:
              start=14471111, gap=156791250, #terms=9


              it looks like my stupid search method actually is comparable in efficiency to Jobling's search method. (Rather to my surprise, since his seemed less stupid on the face of it.)
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