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Re: [PrimeNumbers] a^b+b^a is PRP!

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  • Bouk de
    If I am correct none of the prp s has more than 3000 decimal digits which makes them all provable by Titanix in a sensible amount of time. 9000 digits would
    Message 1 of 13 , Jun 2, 2001
      If I am correct none of the prp's has more than 3000
      decimal digits which makes them all provable by
      Titanix in a sensible amount of time.

      9000 digits would indeed be way too large. Titanix can
      handle such numbers but to prove them would take ages.
      Real ages ;-)

      Bouk.

      --- Nathan Russell <nrussell@...> wrote:
      > On Fri, 1 Jun 2001 19:13:07 -0700 (PDT), Bouk wrote:
      >
      > >
      > >--- Andrey Kulsha <Andrey_601@...> wrote:
      > >> Hello!
      > >>
      > >> This is a complete list of prp's of the form
      > >> a^b+b^a, where
      > >> 1<a<b<1001:
      > >>
      > >>
      > >> Does anybody want to prove some of them prime via
      > >> Titanix?
      > >
      > >You might want to try that yourself. You'd be
      > >surprised at the speed at which Titanix can prove
      > >primes with less than 2000 digits. There are a few
      > >larger than that that will take a little more time.
      >
      > I'd have to say (based on my limited knowledge) a
      > LOT of time.
      >
      > A rough mental estimate using logs seems to indicate
      > that some of his
      > numbers are in the range of nine thousand digits
      > (though I could be
      > off by a fair amount there). I didn't know such
      > numbers were even
      > close to being proveable at present.
      >
      > Of course, I might be in error on that.
      >
      > Nathan
      >
      > 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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    • Andrey Kulsha
      Hello! ... Yes, I know: 2057-digit prime took less than 4 days on Duron-700 with Titanix 2.0.4b, so on Athlon-1200 all these prp s can be proven in 2 weeks or
      Message 2 of 13 , Jun 2, 2001
        Hello!

        Bouk de wrote:

        >You might want to try that yourself. You'd
        >be surprised at the speed at which Titanix
        >can prove primes with less than 2000 digits.
        >There are a few larger than that that will
        >take a little more time.

        Yes, I know: 2057-digit prime took less than 4 days on
        Duron-700 with Titanix 2.0.4b, so on Athlon-1200 all these
        prp's can be proven in 2 weeks or so...

        But my computer has AMD K6-233 processor... :-(

        So I think it will be better if someone with faster machine
        prove them prime. Of course, he will be a 100% prover.

        Thanks,

        Andrey
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      • Christ van Willegen
        Hi, ... 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
        Message 3 of 13 , Jun 2, 2001
          Hi,

          > Bouk de wrote:
          >
          > >You might want to try that yourself. You'd
          > >be surprised at the speed at which Titanix
          > >can prove primes with less than 2000 digits.
          > >There are a few larger than that that will
          > >take a little more time.
          >
          > Yes, I know: 2057-digit prime took less than 4 days on
          > Duron-700 with Titanix 2.0.4b, so on Athlon-1200 all these
          > prp's can be proven in 2 weeks or so...
          >
          > But my computer has AMD K6-233 processor... :-(
          >
          > So I think it will be better if someone with faster machine
          > prove them prime. Of course, he will be a 100% prover.

          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.

          Just send me them off-list. I don't know if you have any time
          for coordinating this search, but the list doesn't look overly
          long, so perhaps you'd be able to distribute things
          yourself.

          Regards,

          Christ van Willegen
        • 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 4 of 13 , Jun 2, 2001
            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 5 of 13 , Jun 2, 2001
              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 6 of 13 , Jun 2, 2001
                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 7 of 13 , Jun 2, 2001
                  > 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 8 of 13 , Jun 2, 2001
                    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 9 of 13 , Jun 2, 2001
                      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 10 of 13 , Jun 4, 2001
                        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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