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Re: [PrimeNumbers] Re: composite trinomials

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  • Chris Caldwell
    ... Thanks--I was verysuspicious myself and had written to x37, no answer--but now none is needed. CC
    Message 1 of 11 , Jul 1 5:58 PM
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      On Mon, 1 Jul 2002, djbroadhurst wrote:

      > Paul Underwood noted:
      > > 9999 2^65536-2^256+1 19729 x37 02 prime #0207
      > > 9999 2^262144-2^512+1 78914 x37 02 #0207
      > > which look improbable and a BLS no-hoper.
      > Yes, this submitter is mistaken in her/his
      > proving abilities:
      > 2^65536-2^256+1 has factors: 4933
      > 2^262144-2^512+1 has factors: 233
      > Thanks for vigilance, Paul.
      > David

      Thanks--I was verysuspicious myself and had written to x37,
      no answer--but now none is needed.

      CC
    • Bouk de Water
      ... Hi Folks! Actually, the first one doesn t look too bad at all. 65536-256 = 65280. 65280 has 72 divisors, which means N-1 also has 72 cyclotomic factors,
      Message 2 of 11 , Jul 2 6:32 AM
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        --- paulunderwooduk <paulunderwood@...> wrote:
        > Hi,
        > I have just seen these:
        > 9999 2^65536-2^256+1 19729 x37 02 prime #0207
        > 9999 2^262144-2^512+1 78914 x37 02 #0207
        > which look improbable and a BLS no-hoper.

        Hi Folks!

        Actually, the first one doesn't look too bad at all.

        65536-256 = 65280.

        65280 has 72 divisors, which means N-1 also has 72 cyclotomic factors, not
        counting Aurifeullians that is. Might be worth a look. But provable or not it's
        still not archivable. But a nice challenge....

        Bouk.





        > PFGW Version 20020515.Win_Dev (Beta software, 'caveat utilitor')
        >
        > PRP: 2^65536-2^256+1 65000/65535
        > Done.
        > PFGW Version 20020515.Win_Dev (Beta software, 'caveat utilitor')
        >
        > PRP: 2^262144-2^512+1 262143/262143
        > Done.
        > Not even a PRP.
        > Paul
        >
        >
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      • Bouk de Water
        ... Sorry, BOTH are not prp. I thought Paul only meant the largest one. Bouk. __________________________________________________ Do You Yahoo!? Sign up for SBC
        Message 3 of 11 , Jul 2 6:47 AM
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          --- paulunderwooduk <paulunderwood@...> wrote:
          > Hi,
          > I have just seen these:
          > 9999 2^65536-2^256+1 19729 x37 02 prime #0207
          > 9999 2^262144-2^512+1 78914 x37 02 #0207
          > which look improbable and a BLS no-hoper.
          > PFGW Version 20020515.Win_Dev (Beta software, 'caveat utilitor')

          Sorry, BOTH are not prp. I thought Paul only meant the largest one.

          Bouk.

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        • Bouk de Water
          I have a yahoo mailbox myself and sometimes I get messages from other members of the group many hours, sometimes more than a day late. 6 people had already
          Message 4 of 11 , Jul 2 6:55 AM
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            I have a yahoo mailbox myself and sometimes I get messages from other members
            of the group many hours, sometimes more than a day late.

            6 people had already discussed the composite sumbmitted numbers by x37 and I
            had only received Paul's announcement. Do more people have this problem? They
            are quite eager to tell me how to do something about my hairloss or shrink my
            ass in twenty days but sending through messages is obviously not making them
            money.

            Bouk.

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          • paulunderwooduk
            ... I should watch my grammar! If you want a challange, dislodge some of my gigantic PRP trinomials at Henri s site:
            Message 5 of 11 , Jul 2 10:22 AM
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              Bouk wrote:
              > --- I wrote:
              > > Hi,
              > > I have just seen these:
              > > 9999 2^65536-2^256+1 19729 x37 02 prime #0207
              > > 9999 2^262144-2^512+1 78914 x37 02 #0207
              > > which look improbable and a BLS no-hoper.
              > > PFGW Version 20020515.Win_Dev (Beta software, 'caveat utilitor')
              >
              > Sorry, BOTH are not prp. I thought Paul only meant the largest one.
              >
              > Bouk.
              I should watch my grammar!
              If you want a challange, dislodge some of my gigantic PRP trinomials
              at Henri's site:
              http://www.primenumbers.net/prptop/prptop.php
              If you can't do that I can generate a challanging one if you like!
              Paul
            • Bouk de Water
              ... Actually I did browse them for proofs. One or two could be proven with a large ECM effort. This one is a good very good one to try: 2^64695-2^15-1 with
              Message 6 of 11 , Jul 3 3:05 AM
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                > If you want a challange, dislodge some of my gigantic PRP trinomials
                > at Henri's site:
                > http://www.primenumbers.net/prptop/prptop.php

                Actually I did browse them for proofs. One or two could be proven with a large
                ECM effort.

                This one is a good very good one to try:

                2^64695-2^15-1 with 19476 digits.

                N+1 = 2^15*(2^64680-1)

                (2^64680-1) has 96 cyclotomic divisors.

                T 64680={ 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 20, 21, 22, 24, 28, 30,
                33, 35, 40, 42, 44, 49, 55, 56, 60, 66, 70, 77, 84, 88, 98, 105, 110, 120, 132,
                140, 147, 154, 165, 168, 196, 210, 220, 231, 245, 264, 280, 294, 308, 330, 385,
                392, 420, 440, 462, 490, 539, 588, 616, 660, 735, 770, 840, 924, 980, 1078,
                1155, 1176, 1320, 1470, 1540, 1617, 1848, 1960, 2156, 2310, 2695, 2940, 3080,
                3234, 4312, 4620, 5390, 5880, 6468, 8085, 9240, 10780, 12936, 16170, 21560,
                32340, 64680 } [96]

                So: (2^64680-1) =
                phi(1,2)*phi(2,2)*phi(3,2)*phi(4,2)*.....*phi(21560,2)*phi(32340,2)*phi(64680,2)

                There are aurifeuillian factors (L and M) as well when for phi(n,2) n=4*k and
                k=odd.

                L=2^h-2^k+1, M=2^h+2^k+1, h=2k-1. Take gcd's with phi(4*k,2) and L or M.

                E.g. phi(2156,2) can be divided in a L and M part as 2156 = 4*539

                L: 10781.81929.90317512080398683509507180285854441.P83
                M:
                2136469147429.111206916097779728932051224808777.1297662995123479965752936319854262257.P46

                There is great deal of work already done in the cunninghamproject. I would
                expect factorization to be more than 25% done. If one reaches 30% David
                Broadhurst can do his KP magic.

                Bouk de Water.





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              • Bouk de Water
                And for die-hards: 2^118843-2^43+1 with 35776 digits. N-1: 2^43*(2^118800-1) T 118800={ 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 18, 20, 22, 24, 25, 27, 30,
                Message 7 of 11 , Jul 3 3:13 AM
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                  And for die-hards:

                  2^118843-2^43+1 with 35776 digits.

                  N-1: 2^43*(2^118800-1)

                  T 118800={ 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 18, 20, 22, 24, 25, 27,
                  30, 33, 36, 40, 44, 45, 48, 50, 54, 55, 60, 66, 72, 75, 80, 88, 90, 99, 100,
                  108, 110, 120, 132, 135, 144, 150, 165, 176, 180, 198, 200, 216, 220, 225, 240,
                  264, 270, 275, 297, 300, 330, 360, 396, 400, 432, 440, 450, 495, 528, 540, 550,
                  594, 600, 660, 675, 720, 792, 825, 880, 900, 990, 1080, 1100, 1188, 1200, 1320,
                  1350, 1485, 1584, 1650, 1800, 1980, 2160, 2200, 2376, 2475, 2640, 2700, 2970,
                  3300, 3600, 3960, 4400, 4752, 4950, 5400, 5940, 6600, 7425, 7920, 9900, 10800,
                  11880, 13200, 14850, 19800, 23760, 29700, 39600, 59400, 118800 } [120]

                  But that one is extremely hard. Don't try this at home, folks!

                  Bouk.



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