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From: Leonid Lenov on 14 Jan 2010 09:45
On Jan 14, 2:21 pm, Andrew Usher <k_over_hb...(a)yahoo.com> wrote:
> 4 and 2 are units but 1+sqrt(-3) is not! You have two non-units
> multiplying to give a unit, which violates that principle.

I noticed that as I said in my previous post.
From: The Pumpster on 14 Jan 2010 15:15
On Jan 13, 10:22 pm, Andrew Usher <k_over_hb...(a)yahoo.com> wrote:
> On Jan 13, 6:59 pm, The Pumpster <pumpledumplek...(a)gmail.com> wrote:
>
> > The remaining cases give easy contradictions modulo 9.
>
> > I guess nobody wants to invoke Coates-Wiles :)
>
> I prefer my method, it generalises more. The fact the unique
> factorisation can fail in Z[sqrt(-3)] only for 'even' numbers can be
> used in other proofs; it also applies to some others such as (I
> believe) Z[sqrt(-5)].
>
> Andrew Usher

Well, if one is interested in generalizations, all the methods
discussed here are pretty likely to fail if the corresponding curve
has reasonably high rank. For a Mordell curve like this one,
additionally,
if the class group has nontrivial 3-part, corresponding computations
become less pleasant.

In practice, given an "arbitrary" model of an elliptic curve over Q,
to find the integral points one would apply
lower bounds for linear forms in logs, together
with lattice basis reduction.
This is programmed into, e.g., Magma (I think it's in Sage)
where the following

E := EllipticCurve([0, -3]);
P:=IntegralPoints(E);
P;

entered into

http://magma.maths.usyd.edu.au/calc/

will give one the desired answer. I think, for reasons that are
not clear to me, the code in Magma works via elliptic logarithms
instead of complex logs and hence is not strictly speaking an
algorithm (in practical terms, it is likely to choke on a curve
with high rank).

de P
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