A NEW CRITERION FOR THE FIRST CASE OF FERMAT LAST THEOREM

It is shown that if the first case of Fermat's last theorem fails for an odd prime l, then the sums of reciprocals module l, s(k, N) = Sigma 1/j (kl/N < j < (k + 1)l/N) are congruent to zero modl for all intege rs N and k with 1 less than or equal to N less than or equal to 46 and 0 less than or equal to k less than or equal to N - 1. This is equiva lent to B-l-1(k/N) - B-l-1 = 0 (modl), where B-n and B-n(x) are the nt h Bernoulli number and polynomial, respectively. The work can be consi dered as a result an Kummer's system of congruences.