A Result on Odd Powers in Fermat’s Last Theorem

In this work, a partial proof of Fermat’s Last Theorem (FLT) relying on elementary number theory is presented. The main result asserts that when certain natural assumptions are placed on the variables involved in the equation of the statement of FLT, then FLT holds for any prime number greater than 9, and consequently for any positive integer greater than 9. The proof of the main and supporting results is by the method of contradiction. It is first proved that if there is a prime number greater than 9 for which FLT is false under a natural assumption on the variables of the equation of FLT, then there is a set of equations that the variables must satisfy. From this set of equations, it is proved that the variables of the equation of FLT are further constrained by an additional set of equations and inequalities, which ultimately results in a contradiction. The elementary number theoretic methods employed are centered around the theory of greatest common divisors, the binomial theorem, the theory of indices, and the theory of polynomials over the ring of all integers. The algebraic operations involved are those defined on the ring of all integers, and those defined on the field of all rational numbers. The elementary order properties of the set of integers as a subset of the totally ordered field of real numbers are also applied. The cancellation and unique prime power factorization properties of the integers are taken for granted.