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Re: what is this problem all about...

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  • Adam
    Not to say that this makes it easy, but I thought the Catalan conjecture was resolved about a year and a half ago. A Google search found
    Message 1 of 4 , Oct 7, 2004
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      Not to say that this makes it "easy," but I thought the Catalan
      conjecture was resolved about a year and a half ago. A Google
      search found http://www.ams.org/bull/2004-41-01/S0273-0979-03-00993-
      5/S0273-0979-03-00993-5.pdf as an AMS Bulletin summarizing the
      history.

      On the other hand I tested out to .... oh shoot, I cut and pasted
      something over the cut and pasted number that I had searched up to
      but it is started with a 2 and many digits, like 2 million or 2
      billion. Darn darn darn. Oh well, not any more solutions for a
      long long time.

      Adam

      --- In primenumbers@yahoogroups.com, mikeoakes2@a... wrote:
      > I wrote
      > >Perhaps someone would like to use Pari to settle this question?
      >
      > I asked David Broadhurst for help on this and, guru that he is, he
      has come up with the goods. To quote his email:-
      > >
      > >MordellWeilRank(EllipticCurve([0,3,1,2,0]));
      > >
      > >Magma V2.11-6
      > >Sun Oct 3 2004 11:26:59 on modular [Seed = 2793930133]
      > > -------------------------------------
      > >
      > > 1
      > >
      > > Total time: 0.190 seconds, Total memory usage: 3.50MB
      >
      > So, the rank is indeed 1, and all rational solutions are of the
      form p_n = n*P_1.
      > Now "all" that remains is to show that the denominator of the x
      and y coordinates of P_n are never 1, for n > 6.
      >
      > Ideas, anyone?
      >
      > [This part is going to be /hard/: as David pointed out, the
      Catalan problem (a^2=b^3+1) is of the same type, and has so far
      resisted all attempts at a solution.]
      >
      > -Mike Oakes
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