On Fermat and Wilson quotients (Q1917642)

This is a survey of arithmetic properties of the Fermat quotient \(q_p(a) = (a^{p-1} - 1)/p\) and the Wilson quotient \(W_p=((p-1)!+1)/p\) \((p\) an odd prime). Special attention is paid to relationships between these quotients and Bernoulli numbers \(B_m\) as well as between \(q_p(a)\) and Mirimanoff polynomials. The usual power sums required to establish these kinds of relationships are dealt with by a method relying on properties of symmetric polynomials. As a by-product there is an apparently new expression for \({B_m\over m} + {B_{m+p-1} \over m+p-1} \bmod p\). The article ends with three open problems related to symmetric polynomials.