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        riddles >> putnam exam (pure math) >> Fermat's last for generalised integers
        
(Message started by: JocK on Jun 24^th, 2005, 12:15pm)

Title: Fermat's last for generalised integers
Post by JocK on Jun 24^th, 2005, 12:15pm

Does the equation x^3 + y^3 = z^3 have a solution in non-zero Eisenstein integers?

The similar question for Gaussian integers still seems to be unanswered: as far as I know it has not been proven that non-zero Gaussian integer solutions fail to exist, but also no example of such a solution is yet known.

So, I was wondering: "How much further do we need to go along the road of generalised integers before solutions do appear?"

The first reasonable step beyond Gaussian integers could be the Eisenstein integers that fill the complex plane denser than the Gaussian integers.

(The next step: Hamiltonian integers? :) )

Title: Re: Fermat's last for generalised integers
Post by Barukh on Jun 25^th, 2005, 9:28am

Sources tell that your first quesition was answered in negative by Gauss. I haven't seen the proof though.