Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Refining the conditions on the Fermat quotient

Andrew J. Granvillea1

a1 Trinity College, Cambridge University

The aim of this paper is to establish a result on a family of congruences arising from the Fermat quotient: this result has an interesting application to the Fermat problem. For over 150 years the Fermat problem has been divided into two cases; the First Case being the assertion that for each odd prime p,


has no solution in non-zero integers x, y, z. Kummer's work on ideal numbers led, in 1847, to the complete solution of the Fermat problem for regular primes.

(Received July 10 1984)

(Revised December 06 1984)