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(a) will be an integer. If the base a is also a generator of the multiplicative group of integers modulo p, then qp(a) 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 W_p 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 {1, ..., p − 1}: 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(a) ≡ 0 (mod p) then ap−1 ≡ 1 (mod p2). 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(a) ≡ 0 (mod p) for small values of a are: ! a ! p (checked up to 5 × 1013) ! OEIS sequence For more information, see and. The smallest solutions of qp(a) ≡ 0 (mod p) with a = n are: A pair (p, r) of prime numbers such that qp(r) ≡ 0 (mod p) and qr(p) ≡ 0 (mod r) is called a Wieferich pair.