In number theory, the Fermat quotient of an integer a with respect to an odd prime p is defined as[1][2][3][4] or This article is about the former; for the latter see p-derivation.
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.
These properties imply In 1895, Dmitry Mirimanoff pointed out that an iteration of Eisenstein's rules gives the corollary:[6] From this, it follows that:[7] M. Lerch proved in 1905 that[8][9][10] Here
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: If qp(a) ≡ 0 (mod p) then ap−1 ≡ 1 (mod p2).