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Old 2009-12-11, 05:53   #5
CRGreathouse's Avatar
Aug 2006

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Originally Posted by flouran View Post
I think that the incorporation of certain ingredients of Matiyasevich's proof and a variant of FLT can be used to prove that x^10+y^10+z^10 = t^4 does not have any solutions in positive integers.
I think not.

1. It's not obvious how Matiyasevich's theorem (Hilbert X, Robinson-Davis-Putnum-Matiyasevich, etc.) applies; it doesn't show that there are no solutions, only that proving that you've found all solutions is hard in the general case.
2. This isn't like the general case. Diophantine equations with 9 variables are known to be universal, but only with ridiculously high degrees (~10^45 as I recall). You have a degree-10 equation with only four variables.
3. Wiles' theorem doesn't seem well-equipped for the additive explosion on the LHS. Three terms is vastly different from two terms.

Also, there aren't many mathematicians in the world capable of extending his proof, and (to my knowledge) none here/
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