A Special Extension of Wieferich's Criterion

Abstract

The following theorem is proved in this paper: If the first case of Fermat's Last Theorem does not hold for sufficiently large prime 1, then E x1-2 kl < x < (k X I) =_ 0 (modl1) x NN for all pairs of positive integers N, k, N < 94, 0 < k < NI. The proof of this theorem is based on a recent paper of Skula and uses computer techniques. 0. INTRODUCTION The first case of Fermat's Last Theorem states that for each odd prime I the equation Xi +y1 + zi = 0 has no integral solution x, y, z with I t xyz. One of many methods investigating this problem was introduced by A. Wieferich. This method is connected with the Fermat quotients ql (a), al-l q, (a) = I defined for each integer a such that a is not divisible by 1. Let us assume in this paragraph that / is an odd prime which does not satisfy the first case of Fermat's Last Theorem. In 1909, Wieferich [7] published the following important result: q, (2) =0 (mod1) . Many mathematicians have extended this Wieferich criterion. The latest result is due to A. Granville and B. Monagan [1] and states q1 (p) _ 0 (mod 1) for each prime p such that p < 89. These considerations have been generalized by L. Skula. He studied the sums s(k, N) = E 1-2 (kl <x<(k + 1)1) s(k, N) = EX<X