Number Theory


On Fermat's Equation and the General Case

Authors: C Sloane

We discovered a beautiful symmetry to the equation x^n+y^n± z^n , first studied by Fermat, in a dependent variable t = x+y-z and the product (xyz) if we introduce a term we call the symmetric r = x^2+yz-xt-t^2. Once x^n+y^n± z^n is written in terms of powers of t, r and (xyz) we looked at the coefficient vs. exponent abstract space and found Lucas, Fibonacci and Convoluted Fibonacci sequences among other corollaries. We also found that 3 cases of a prime decomposition factor q of x^2+yz gave certain results for Fermat’s Last Theorem which could be eliminated if a forth case could also be solved. Intrigued by this, we then introduce partial congruence representations modulo a prime for this much harder forth case to find the ‘form’ of the solutions modulo q. The form of the solutions leads us to a cubic congruence method that solves the special and general cases. There are several pages and stages of the proofs where computer verification of the results is possible.

Comments: 29 Pages. This paper is with several institutions and this submission is a time-stamp of authorship.

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[v1] 2017-04-06 23:53:45

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