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.
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