Generalization of the Voronoi congruences and applications

The Voronoi congruences involving the Bernoulli numbers became of interest to mathematicians for the search of irregular primes. Kummer had namely shown that Fermat's last theorem holds for all the exponents that are regular primes. In this talk, we generalize the Voronoi congruences to the odd indices, extend them and sharpen them to the next modulus. 
We derive a novel congruence for the residue of the Fermat quotient and also recover congruences of E. Lehmer.