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.
If the base a is coprime to the exponent p then Fermat's little theorem says that qp(a) will be an integer. The quotient is named after Pierre de Fermat.
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
Eisenstein discovered that the Fermat quotient with base 2 could be expressed in terms of the sum of the reciprocals mod 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). 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:
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.
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