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MathGroup Archive 2006

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Re: Re: Demostration


On 16 Oct 2006, at 15:33, Andrzej Kozlowski wrote:

>
>
> I do not think it is possible to use Mathematica to prove that these
> are all the solutions. The reason is that while there is a general
> theorem whcih states that the Diophantine equation y^2==f[x] has at
> most a finite number of solutions if f[x] is a polynomial of degree
>> =3, with integer coefficients and with distinct zeros, no method is
> known for determining the solutions or the number of solutions except
> in special cases. Since you have stated that in this case there are
> precisely 10 solutions, I assume this must be one of them, and there
> is some way to prove it which is not known to me (this is not my area
> and I do not follow recent development in it). But in any case, even
> if a way to prove this is known in this case, no such general
> algorithm exists and therefore it can't be known to Mathematica.
>
> Andrzej Kozlowski
> Tokyo, Japan
>


The expression "recent developments" above should be interpreted in a  
rather s way. All solutions to the equation y^2==x^3+k for -100<=k<=k  
were already known in 1954; see O. Hemer "Notes on the Diophantine  
equation y^2-k=x^3. Ark. Mat. 3. pp. 67-77, 1954.

Andrzej Kozlowski


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