Fermat quotients for composite moduli (Q1366667)

Let \(a\) and \(m \geq 2\) be relatively prime integers. The authors call the integer \[ q(a,m)=(a^{\varphi (m)}-1)/m \] the Euler quotient of \(m\) with base \(a\). If \(m\) is a prime, this reduces to the widely studied Fermat quotient. Some basic properties of \(q(a,m)\) were observed by \textit{M. Lerch} [C. R. Acad. Sci., Paris 142, 35-38 (1906; JFM 37.0225.02)]. The present paper constitutes a systematic study of this number. As in the case of the Fermat quotient, there are many connections to Bernoulli polynomials and numbers. Of special interest are the Wieferich numbers \(m\) with base \(a\), that is, the numbers \(m\) satisfying \(a^{\varphi (m)} \equiv 1 \pmod {m^2}\). The authors completely characterize these in terms of the primes \(p\) with \(a^{p-1} \equiv 1 \pmod {p^2}\) (the Wieferich primes). Also included are numerical examples and tables of Wieferich numbers.