Congruences involving Bernoulli numbers and Fermat-Euler quotients. (Q1609530)

The classical Voronoï congruence for Bernoulli numbers \(B_m\) (in the even suffix notation) expresses the residue mod \(N\) of \((b^m-1)B_m/m\) as a sum of multiples of \((m-1)\)st powers of \(bj\,(j=1,\dots,N-1)\), provided \(b\) and \(N\) are coprime. As a nice supplement to this result the author finds an analogous expression for the residue mod \(p^e\) of \((p^{m-1}-1)B_m/m\), where \(e\geq 1\) and \(p\) is a prime \(\geq 5\) with \(p-1\) not dividing \(m\). This expression contains a similar sum and moreover an additional term depending on the Fermat-Euler quotient \((r^{\varphi(p^e)}-1)/p^e\), where \(r\) is a primitive root mod \(p^e\). As applications the author proves several congruences involving Bernoulli numbers and, occasionally, the Fermat quotient \((r^{p-1}-1)/p\). Some of these are known, but the proofs are different. A particular theme is the connection of the Bernoulli numbers \(B_{(p+1)/2}\) and \(B_{(p-1)/2}\) to the class numbers of quadratic fields.