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Re: [PrimeNumbers] Largest Ordinary Prime?

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  • Andy Steward
    Firstly, congratulations to Paul, Paul, David and Bouk. Secondly, DAMN. I had this great idea while I was on holiday about ten days ago. It appears to be much
    Message 1 of 22 , Oct 3, 2002
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      Firstly, congratulations to Paul, Paul, David and Bouk.

      Secondly, DAMN. I had this great idea while I was on holiday
      about ten days ago. It appears to be much the the same idea that
      you guys had a long while earlier ;-)

      I used my experience in trying to prove Gigantic GRUs to model
      how many digits of prime factors I would expect to extract from
      a cyclotomic factor of a given size given the amount of ECM work
      I have done on such factors to date (about 7-10 GHz hours on each
      composite on average, though some I've just started and a few have
      had thousands of hours).

      I then generated a list of promising composite exponents c such that
      2^c-1 gave a decent chance of being useful in proving 2^a - 2^b +/- 1
      where c=a-b. 64680 came out at 19.85%, so your 30% indicates how much
      more work you did on it (and a bit of luck). I reckon the best few
      exponents of around the same size are: 65520, 75600, 69300, 83160,
      73920, 70560 then 64680.

      Another thought I had was to use 2^c-1 where it's a Mersenne prime
      ... or c=2p, 3p etc where 2^p-1 is a Mersenne prime: diminishing
      returns but worth a try for small multiples.

      Ah well, I'll stick to the GRUs.

      Good luck,
      Andy
    • David Broadhurst
      ... Then recall that Paul U restricted his search of 2^a-2^b-1 to b
      Message 2 of 22 , Oct 4, 2002
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        Andy:

        > I reckon the best few exponents of around
        > the same size are: 65520, 75600, 69300,
        > 83160, 73920, 70560 then 64680.

        Then recall that Paul U restricted his
        search of 2^a-2^b-1 to b<=40.

        So that gives 280 chances for those 7
        values of c=a-b, whereas
        ln(N) ~ ln(2^70000) = 50,000,
        giving you odds of order 100:1 against.

        It's neat that one of your 280 showed up,
        with b=15, and even neater that Bouk's eagle
        eyes spotted it in Henri's pages.

        PS: How's the latest GRU cooking?

        Remember that Phil has just offered
        double the usual extra bits,
        if/when you need them :-)

        Also Paul L was holding NFS cycles
        in reserve for 2^15*(2^64680-1)-1
        which we didn't need. So maybe
        you could do something really
        impressive with another 150 digits
        from him, in some strategic place :-?

        Best wishes

        David
      • paulunderwooduk
        ... b
        Message 3 of 22 , Oct 4, 2002
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          > > I reckon the best few exponents of around
          > > the same size are: 65520, 75600, 69300,
          > > 83160, 73920, 70560 then 64680.
          >
          > Then recall that Paul U restricted his
          > search of 2^a-2^b-1 to b<=40.
          >

          b<44 actually. Also a<100000.

          Bouk suggests having a look at 118080.

          I will find minimal PrPs for all the suggestions at the end of the
          month; also for 2^a-2^b+1. ;-)

          Paul
        • Andy Steward
          ... Did you mean 110880? I get an expected factorisation of 18.4% for that one, but only 9.7% for 118080. By my model, 110880 is the most promising exponent in
          Message 4 of 22 , Oct 5, 2002
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            Paul Underwood wrote:

            > Bouk suggests having a look at 118080.

            Did you mean 110880? I get an expected factorisation of
            18.4% for that one, but only 9.7% for 118080. By my model,
            110880 is the most promising exponent in (85680,750000]

            Here's a list of exponents where each entry has a higher
            expected factorisation yield than any larger:

            Exponent, factors, expectation
            1680,40,100.00%
            1740,24,96.33%
            1890,32,95.48%
            2100,36,94.98%
            2520,48,91.05%
            2640,40,88.53%
            2940,36,88.41%
            3360,48,86.53%
            3780,48,82.88%
            3960,48,80.44%
            4200,48,79.05%
            4620,48,78.67%
            5040,60,78.06%
            5460,48,71.49%
            5880,48,69.92%
            6120,48,69.12%
            7560,64,68.14%
            7920,60,63.67%
            9240,64,62.61%
            10080,72,61.54%
            10920,64,56.45%
            12600,72,53.33%
            13860,72,52.73%
            15120,80,51.39%
            16380,72,46.94%
            18480,80,46.48%
            20160,84,43.47%
            21840,80,41.16%
            27720,96,39.70%
            30240,96,36.29%
            32760,96,35.11%
            36960,96,31.72%
            37800,96,30.69%
            40320,96,28.96%
            41580,96,28.95%
            42840,96,28.73%
            55440,120,28.00%
            65520,120,24.39%
            75600,120,21.10%
            83160,128,21.05%
            85680,120,19.43%
            110880,144,18.40%
            131040,144,15.89%
            138600,144,15.06%
            166320,160,14.15%
            171360,144,12.41%
            196560,160,12.14%
            221760,168,11.37%
            240240,160,10.17%
            277200,180,9.96%
            332640,192,8.95%
            360360,192,8.35%
            393120,192,7.64%
            415800,192,7.26%
            443520,192,6.75%
            471240,192,6.43%
            480480,192,6.31%
            498960,200,6.29%
            554400,216,6.21%
            556920,192,5.47%
            720720,240,5.39%
            739200,192,4.09%
            742560,192,4.08%
            748440,192,3.93%
            749700,162,3.33%

            HTH,
            Andy
          • Bouk de Water
            ... 110880 it should be indeed. Moreover because phi(36960,2) is a prp. 36960 * 2 = 73920 2^73920-1 is already BLS 36960 * 3 = 110880 2^110880-1 is about 1300
            Message 5 of 22 , Oct 5, 2002
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              --- Andy Steward <aads@...> wrote:
              > Paul Underwood wrote:
              >
              > > Bouk suggests having a look at 118080.
              >
              > Did you mean 110880? I get an expected factorisation of
              > 18.4% for that one, but only 9.7% for 118080. By my model,
              > 110880 is the most promising exponent in (85680,750000]

              110880 it should be indeed. Moreover because phi(36960,2) is a prp.

              36960 * 2 = 73920

              2^73920-1 is already BLS

              36960 * 3 = 110880

              2^110880-1 is about 1300 digits short for Konyagin-Pomarance.

              Bouk.

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            • Phil Carmody
              ... Not yet... ;-) Phil ===== First rule of Factor Club - you do not talk about Factor Club. Second rule of Factor Club - you DO NOT talk about Factor Club.
              Message 6 of 22 , Oct 5, 2002
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                --- Bouk de Water <bdewater@...> wrote:
                > Moreover because phi(36960,2) is a prp.
                ^^^
                > 36960 * 2 = 73920
                >
                > 2^73920-1 is already BLS

                Not yet... ;-)

                Phil



                =====
                First rule of Factor Club - you do not talk about Factor Club.
                Second rule of Factor Club - you DO NOT talk about Factor Club.
                Third rule of Factor Club - when the cofactor is prime, or you've trial-
                divided up to the square root of the number, the factoring is over.

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              • David Broadhurst
                ... True, oh sage. But 2312 digits would be a mere bagatelle for primo. More significantly what a would make 2^a*(2^73920-1)-1 PrP? On average you have to
                Message 7 of 22 , Oct 5, 2002
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                  Phil:

                  > Not yet... ;-)

                  True, oh sage.

                  But 2312 digits would be
                  a mere bagatelle for primo.

                  More significantly what "a" would make
                  2^a*(2^73920-1)-1
                  PrP?

                  On average you have to wait a long time
                  and by then 2^a gives you most of BLS anyway,
                  thus eroding you factorization work.

                  It was Paul U's a=15 that was really notable in

                  2^15*(2^64680-1)-1

                  David
                • Andrey Kulsha
                  ... [snip] ... To be precise, 239 phi-factors and 5.895144%: http://groups.yahoo.com/group/primenumbers/files/Factors/m720720.zip Best, Andrey [Non-text
                  Message 8 of 22 , Oct 13, 2002
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                    Andy Steward wrote:

                    > Here's a list of exponents where each entry has a higher
                    > expected factorisation yield than any larger:
                    >
                    > Exponent, factors, expectation
                    [snip]
                    > 720720,240,5.39%

                    To be precise, 239 phi-factors and 5.895144%:

                    http://groups.yahoo.com/group/primenumbers/files/Factors/m720720.zip

                    Best,

                    Andrey


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