A New Criterion for the First Case of Fermat's Last Theorem

It is shown that if the first case of Fermat's last theorem fails for an odd prime /, then the sums of reciprocals modulo /, s(k, N) = £ 1/7 (kl/N < j < (k + 1)//A0 are congruent to zero mod/ for all integers N and k with I < N < 46 and 0 < k < N 1 . This is equivalent to Bi_{(k/N) B¡_x =0 (mod/), where B„ and B„(x) are the «th Bernoulli number and polynomial, respectively. The work can be considered as a result on Rummer's system of congruences.