Small Modular Solutions to Fermat’s Last Theorem

Abstract

The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x , y , z to the equation x n + y n = z n for n > 2 . There are none. Surprisingly, there are infinitely many solutions if the problem is recast in terms of modular arithmetic. Over a hundred years ago Issai Schur was able to show that for any n there is always a sufficiently large prime p 0 such that for all primes p ≥ p 0 the congruence x n + y n ≡ z n ( mod p ) has a non-trivial solution. Schur’s argument wasnon-constructive, and there is no systematic method available at present to construct specific examples for small primes. We offer a simple method for constructing all possible solutions to a large class of congruences of this type.