Fermat quotient

If a is an integer not divisible by a positive prime p, then Fermat’s little theorem (a.k.a. Fermat’s theorem) guarantees that the differencePlanetmathPlanetmath ap-1-1 is divisible by p.  The integer


is called the Fermat quotientMathworldPlanetmath of a modulo p.  Compare it with the Wilson quotientMathworldPlanetmath wp, which is similarly related to Wilson’s theorem.

Lerch’s formulaMathworldPlanetmathPlanetmath


for an odd prime p connects the Fermat quotients and the Wilson quotient.

If p is a positive prime but not a Wilson primeMathworldPlanetmath, and wp is its Wilson quotient, then the expression


is called the Fermat–Wilson quotient of p.  Sondow proves in [1] that the greatest common divisorMathworldPlanetmathPlanetmath of all Fermat–Wilson quotients is 24.


  • 1 Jonathan Sondow:  Lerch Quotients, Lerch Primes, Fermat–Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771. Available at http://arxiv.org/pdf/1110.3113v3.pdfarXiv.
Title Fermat quotient
Canonical name FermatQuotient
Date of creation 2013-03-22 19:34:22
Last modified on 2013-03-22 19:34:22
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 12
Author pahio (2872)
Entry type Definition
Classification msc 11A51
Classification msc 11A41
Defines Lerch’s formula
Defines Fermat–Wilson quotient