On the Kummer system of congruences and the Fermat quotients (Q1342746)

In his study of the Fermat equation \(x^ p + y^ p = z^ p\) \((p\) an odd prime), Kummer introduced a famous system of congruences mod \(p\) depending on Bernoulli numbers. Later it was proved that the solutions of this system also satisfy certain congruences \[ (X-X^ m) (1-X)^{-1} q_ p(m) \equiv \sum^{m-1}_{k=1} H_ k^{(m)} (X) \pmod p, \] where \(m\) is any prime different from \(p\), \(q_ p(m)\) denotes the Fermat quotient \((m^{p-1} - 1)/p\), and the \(H_ k^{(m)} (X)\) are polynomials defined in a complicated way. The author provides a simpler proof for this result and studies properties of \(H_ k^{(m)} (X)\). He also discusses some applications.