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 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: {| class="wikitable" |----- ! a ! p (checked up to 5 × 1013) ! OEIS sequence |----- | 1 || 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... (All primes) | |----- | 2 || 1093, 3511 | |----- | 3 || 11, 1006003 | |----- | 4 || 1093, 3511 | |----- | 5 || 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 | |----- | 6 || 66161, 534851, 3152573 | |----- | 7 || 5, 491531 | |----- | 8 || 3, 1093, 3511 | |----- | 9 || 2, 11, 1006003 | |-----