ALGEBRAIC PROOFS FERMAT'S LAST THEOREM, BEAL'S CONJECTURE

James E Joseph

doi:10.24297/jam.v12i9.130

Abstract

In this paper, the following statememt of Fermat's Last Theorem is proved. If x; y; z are positive integers, _ is an odd prime and z_ = x_ + y_; then x; y; z are all even. Also, in this paper, is proved Beal's conjecture; the equation z_ = x_ + y_ has no solution in relatively prime positive integers x; y; z; with _; _; _ primes at least 3:

Highlights

For other theorems named after Pierre de Fermat, see [1]
The _rst proof agreed upon as successful was released in 1994 by Andrew Wiles formally published in 1995 [2], ]3], after 358 years of e_ort by mathematicians. This unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century
It is among the most notable theorems in the history of mathematics It is known that if x; y; z are relatively prime positive integers, z4 6= x4 + y4[1]: In view of this fact, it is only necessary to prove if x; y; z; are relatively prime positive inte- gers, _ is an is odd prime, z_ = x_ +y_; x; y; z; are each divisible by _: Before and since Wiles paper, many papers and books have been written trying to solve this problem in an elegant algebraic way, but none have suceeded. (See [1], and go to a search engine on the com- puter and search Fermat's Last Theorem)

Summary

Introduction

The following statememt of Fermat's Last Theorem is proved. If x; y; z are positive integers, _ is an odd prime and z_ = x_ + y_; x; y; z are all even. In this paper, is proved Beal's conjecture; the equation z_ = x_ + y_ has no solution in relatively prime positive integers x; y; z; with _; _; _ primes at least 3:

Results

Conclusion