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

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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 1 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 2 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 3 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 4 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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