Fermat quotient


In number theory, the Fermat quotient of an integer a with respect to an odd prime p is defined as
or
This article is about the former; for the latter see p-derivation. The quotient is named after Pierre de Fermat.
If the base a is coprime to the exponent p then Fermat's little theorem says that qp will be an integer. If the base a is also a generator of the multiplicative group of integers modulo p, then qp will be a cyclic number, and p will be a full reptend prime.

Properties

From the definition, it is obvious that
In 1850, Gotthold Eisenstein proved that if a and b are both coprime to p, then:
Eisenstein likened the first two of these congruences to properties of logarithms. These properties imply
In 1895, Dmitry Mirimanoff pointed out that an iteration of Eisenstein's rules gives the corollary:
From this, it follows that:

Lerch's formula

M. Lerch proved in 1905 that
Here is the Wilson quotient.

Special values

Eisenstein discovered that the Fermat quotient with base 2 could be expressed in terms of the sum of the reciprocals modulo p of the numbers lying in the first half of the range :
Later writers showed that the number of terms required in such a representation could be reduced from 1/2 to 1/4, 1/5, or even 1/6:
Eisenstein's series also has an increasingly complex connection to the Fermat quotients with other bases, the first few examples being:

Generalized Wieferich primes

If qp ≡ 0 then ap−1 ≡ 1. Primes for which this is true for a = 2 are called Wieferich primes. In general they are called Wieferich primes base a. Known solutions of qp ≡ 0 for small values of a are:
For more information, see and.
The smallest solutions of qp ≡ 0 with a = n are:
A pair of prime numbers such that qp ≡ 0 and qr ≡ 0 is called a Wieferich pair.