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Re: composite trinomials

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  • djbroadhurst
    ... 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,
    Message 1 of 11 , Jul 1, 2002
      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
    • 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 2 of 11 , Jul 1, 2002
        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 3 of 11 , Jul 1, 2002
          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 4 of 11 , Jul 1, 2002
            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 5 of 11 , Jul 2, 2002
              --- 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
              >
              >
              >
              > Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
              >
              >


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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 6 of 11 , Jul 2, 2002
                --- 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 7 of 11 , Jul 2, 2002
                  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 8 of 11 , Jul 2, 2002
                    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 9 of 11 , Jul 3, 2002
                      > 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 10 of 11 , Jul 3, 2002
                        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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