Re: Re: Demostration
- To: mathgroup at smc.vnet.net
- Subject: [mg70472] Re: [mg70411] Re: [mg70369] Demostration
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Tue, 17 Oct 2006 02:58:53 -0400 (EDT)
- References: <200610140707.DAA07983@smc.vnet.net> <200610160633.CAA27618@smc.vnet.net>
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
- References:
- Demostration
- From: "Miguel" <mibelair@hotmail.com>
- Re: Demostration
- From: Andrzej Kozlowski <akoz@mimuw.edu.pl>
- Demostration