A note on the Voronoi congruences and the residue of the Fermat quotient

Abstract

Given an odd prime p, we prove a congruence on the p-residue of the Fermat quotient q_p(a) in base a with 0<a<p, which arises from a generalization of the Voronoi congruences and from some other congruences on sums and weighted sums of divided Bernoulli numbers. As an application in the base 2 case, we retrieve a congruence for the first generalized harmonic number of order (p-1)/2, a generalization originally due to Z-H. Sun of a classical congruence known since long for the harmonic number of order (p-1)/2 as a special case of the Lerch formula. We find a sharpening of the Voronoi congruences which is different from the one of W. Johnson and which is more computationally efficient. We prove an additional related congruence, which specialized to base 2 allows to retrieve several congruences that were originally due to E. Lehmer.