Re: Re: Demostration

*To*: mathgroup at smc.vnet.net*Subject*: [mg70473] Re: [mg70411] Re: [mg70369] Demostration*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Tue, 17 Oct 2006 02:58:55 -0400 (EDT)*References*: <200610140707.DAA07983@smc.vnet.net> <200610160633.CAA27618@smc.vnet.net> <D005FF9A-FE9C-4738-A4BF-73969FA8A98E@mimuw.edu.pl>

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

**References**:**Demostration***From:*"Miguel" <mibelair@hotmail.com>

**Re: Demostration***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>