## Re: what is this problem all about...

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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
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.

> 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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