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MathGroup Archive 2006

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Re: Re: Demostration


On 16 Oct 2006, at 21:51, Andrzej Kozlowski wrote:

>
> 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

It was supposed to say "in a rather liberal way".

Andrzej Kozlowski


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