A new criterion for the first case of Fermat’s last theorem

Abstract

It is shown that if the first case of Fermat’s last theorem fails for an odd prime l, then the sums of reciprocals modulo l, s ( k , N ) = ∑ 1 / j ( k l / N > j > ( k + 1 ) l / N ) s(k,N) = \sum 1/j\;(kl/N > j > (k + 1)l/N) are congruent to zero mod l \bmod \;l for all integers N and k with 1 ≤ N ≤ 46 1 \leq N \leq 46 and 0 ≤ k ≤ N − 1 0 \leq k \leq N - 1 . This is equivalent to B l − 1 ( k / N ) − B l − 1 ≡ 0 ( mod l ) {B_{l - 1}}(k/N) - {B_{l - 1}} \equiv 0 \pmod l , where B n {B_n} and B n ( x ) {B_n}(x) are the nth Bernoulli number and polynomial, respectively. The work can be considered as a result on Kummer’s system of congruences.