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Re: composite trinomials

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  • jim_fougeron
    Note, One should not waste time on prp tests or even proof of numbers which look fishy without first trying at least a little bit of factoring. Within a few
    Message 1 of 11 , Jul 1 4:52 PM
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      Note,

      One should not waste time on prp tests or even proof of numbers
      which look "fishy" without first trying at least a little bit
      of factoring. Within a few seconds (most of that was my typing),
      I found this out:

      C:\prime>pfgw -f -q2^65536-2^256+1
      PFGW Version 20020515.Win_Dev (Beta software, 'caveat utilitor')

      trial factoring to 5980328
      2^65536-2^256+1 has factors: 4933

      C:\prime>pfgw -f -q2^262144-2^512+1
      PFGW Version 20020515.Win_Dev (Beta software, 'caveat utilitor')

      trial factoring to 26465419
      2^262144-2^512+1 has factors: 233


      So it is very easy to see that neither of these could possibly be
      prime.

      Jim.
      --- In primenumbers@y..., "paulunderwooduk" <paulunderwood@m...>
      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')
      >
      > 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
    • paulunderwooduk
      David and Jim use the -f switch. I usually forget! I tend to use sieves before OpenPFGW and so I do not use -f. :) Paul
      Message 2 of 11 , Jul 1 4:58 PM
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        David and Jim use the -f switch. I usually forget! I tend to use
        sieves before OpenPFGW and so I do not use -f. :)
        Paul
      • Chris Caldwell
        ... Thanks--I was verysuspicious myself and had written to x37, no answer--but now none is needed. CC
        Message 3 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 4 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
            >
            >
            > Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com
            > The Prime Pages : http://www.primepages.org
            >
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            >
            >


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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 5 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 6 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 7 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 8 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 9 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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