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Re: a^b+b^a is PRP!

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  • d.broadhurst@open.ac.uk
    ... Phil the mod might sleep easier on his bus if combined efforts resulted in 5 ECPP primes with more than 1905 digits. Then the illegal part of Chris
    Message 1 of 13 , Jun 2, 2001
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      Christ van Willegen wrote:
      > Select a few large ones, so that I
      > will be busy for a week or so.
      Phil the mod might sleep easier on his bus
      if combined efforts resulted in 5 ECPP primes
      with more than 1905 digits. Then the illegal
      part of Chris Caldwell's database will disappear.

      ECPP top-16:

      (348^1223-1)/347 3106
      (30^1789-1)/29 2642
      (2^7757-1)/233....361 2303
      (((((1361^3+....+894)^3+3636 2285
      (2^7331-1)/458072843161 2196
      Phi(4274,10) 2136
      (2^7039-1)/(125...721) 2074
      100^1013-99^1013 2026
      U(9677) 2023
      V(20460) 2007
      4915416*10^1999+19 2006
      4915416*10^1999+17 2006
      10^1999+7321 2000
      Phi(3927,10) 1920
      (10^1918-7)/3 1918
      'css_descramble.c.gz'*256^211+99 1905

      Indeed, folk out there might already have 2k+ digits
      Tx proofs that they have forgotten to post. If so,
      just submit the prime with the comment ECPP.
    • d.broadhurst@open.ac.uk
      ... I forgot to add: if you do not have a code with Titanix in it, ask Chris Caldwell for a c? code, so Marcel get recognized. (c is the first first letter of
      Message 2 of 13 , Jun 2, 2001
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        I wrote:

        > Indeed, folk out there might already have 2k+ digits
        > Tx proofs that they have forgotten to post. If so,
        > just submit the prime with the comment ECPP.

        I forgot to add: if you do not have a code with Titanix
        in it, ask Chris Caldwell for a c? code, so Marcel
        get recognized. (c is the first first letter of Titanix :-)

        David
      • Phil Carmody
        ... Mod? _Mod_? Never has a rocker been so offended! (Yeah, yeah, I know what you meant.) Anyway, I was just curious - I noticed that 3 of the entries in the
        Message 3 of 13 , Jun 2, 2001
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          On Sat, 02 June 2001, d.broadhurst@... wrote:
          > Christ van Willegen wrote:
          > > Select a few large ones, so that I
          > > will be busy for a week or so.
          > Phil the mod might sleep easier on his bus

          Mod? _Mod_? Never has a rocker been so offended!
          (Yeah, yeah, I know what you meant.)

          Anyway, I was just curious - I noticed that 3 of the entries in the a^b+b^a list had a=b+1. Are there any witty decompositions of N-1 which could provide a BLS proof to this case?
          A was able to find the factor b by hand, but are there any others?

          Dues to mystical magical exploding computers I am without Mathematica presently, and unable to do symbolic mathematics!

          (Anyone know of a free symbolic maths tool?)

          I threw the 80/81 number at ECM, and split >33% of it, but that could be pure coincidence!

          Hmmm, back to work :-(

          Phil

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        • Paul Leyland
          ... List deleted. ... I posted to this very forum exactly the same list on Thursday 22nd February under the Subject: Primes and strong pseudoprimes of the
          Message 4 of 13 , Jun 2, 2001
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            > This is a complete list of prp's of the form a^b+b^a, where
            > 1<a<b<1001:

            List deleted.

            > Does anybody want to prove some of them prime via Titanix?

            I posted to this very forum exactly the same list on Thursday 22nd
            February under the Subject: "Primes and strong pseudoprimes of the form
            x^y+y^x". I can repost if wished, but assume that readers know how to
            examine the list archives. That post also included the pair (1015,384).
            Somewhere, still not found but probably on a backup tape, the list
            continues to about 1500 or so.

            Many of the primes were proved by me years ago, and I had two in the
            prime record tables until they were re-catalogued as uninteresting.

            I'd be interested in seeing some of them proved prime --- the ones I
            never got around to completing!


            Paul
          • d.broadhurst@open.ac.uk
            ... Hmm... a^(a-1)+(a-1)^a doesn t look circle cutting (cyclotomic) from here. I think cyclotomy is always a^n-b^n (and if you want the full Monty, set
            Message 5 of 13 , Jun 2, 2001
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              Phil Carmody wrote:

              > I noticed that 3 of the entries in the
              > a^b+b^a list had a=b+1.

              Hmm... a^(a-1)+(a-1)^a doesn't look circle cutting
              (cyclotomic) from here. I think cyclotomy is always
              a^n-b^n (and if you want the full Monty, set
              a=(1+sqrt(5))/2 and b=(1-sqrt(5))/2)
              and divide by a-b=sqrt(5) to get a well known integer)

              David
            • Andrey Kulsha
              Hello! ... for me ... Well, I may prove the numbers which have
              Message 6 of 13 , Jun 2, 2001
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                Hello!

                Christ wan Willegen wrote:

                >I have no prior Titanix experience, but I think it's time
                for me
                >to give it a try...
                >
                >Andrey, I am willing to use my Athlon-800 for a few days
                >to try to prove them. Select a few large ones, so that I
                >will be busy for a week or so. You yourself (or someone
                >else?) can perhaps take the smaller ones.

                Well, I may prove the numbers which have <=1000 digits.

                But:

                Paul Leyland wrote:

                >Many of the primes were proved by me years ago, and I had
                two in the
                >prime record tables until they were re-catalogued as
                uninteresting.

                Paul, please send us a list of primes you have proven, and
                we'll prove all remaining ones.

                I will prove less than 1000 digit numbers, Christ wan
                Willegen, for example, will prove numbers with 1001..2000
                digits, and someone third (maybe Paul Leyland) will prove
                numbers with >2000 digits.

                Note that 1000-digit prime 289^406+406^289 was proven by
                Paul, and then independently by me as the smallest titanic
                prime of such a kind. The primes 342^343+343^342,
                111^322+322^111, 122^333+333^122, 8^69+69^8 was also proven
                by me nearly 5 months ago; 365^444+444^365 was proven by me
                and Marcel Martin (he found a step with record polynomial
                degree of 100) nearly 3 months ago.

                Waiting for comments.

                Best wishes,

                Andrey
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              • Andrey Kulsha
                Hello! OK, I ll prove all remaining prps with less than 1200 digits, i.e.: 8^519+519^8, 20^471+471^20, 5^1036+1036^5, 56^477+477^56, 98^435+435^98,
                Message 7 of 13 , Jun 4, 2001
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                  Hello!

                  OK, I'll prove all remaining prps with less than 1200
                  digits, i.e.:

                  8^519+519^8,
                  20^471+471^20,
                  5^1036+1036^5,
                  56^477+477^56,
                  98^435+435^98,
                  21^782+782^21,
                  32^717+717^32,
                  365^444+444^365,
                  423^436+436^423,
                  34^773+773^34.

                  Best wishes,

                  Andrey
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