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 l, then \[ \sum \limits _x {{x^{l - 2}}} \left [ {\frac {{kl}}{N} < x < \frac {{(k + 1)l}}{N}} \right ] \equiv 0\quad \pmod l\] for all pairs of positive integers $N, k,N \leq 94, 0 \leq k \leq N - 1$." The proof of this theorem is based on a recent paper of Skula and uses computer techniques.