## 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.

**Download:** **PDF**

### Submission history

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

**Unique-IP document downloads:** 20 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary.
In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution.
Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

**Add your own feedback and questions here:**

*You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.*

*
*