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Fermat quotient
From Wikipedia, the free encyclopedia
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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. 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.
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Properties
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From the definition, it is obvious that
In 1850, Gotthold Eisenstein proved that if a and b are both coprime to p, then:[5]
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:[6]
From this, it follows that:[7]
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Lerch's formula
M. Lerch proved in 1905 that[8][9][10]
Here is the Wilson quotient.
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Special values
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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:
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Generalized Wieferich primes
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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:[2]
For more information, see [17][18][19] and.[20]
The smallest solutions of qp(a) ≡ 0 (mod p) with a = n are:
- 2, 1093, 11, 1093, 2, 66161, 5, 3, 2, 3, 71, 2693, 2, 29, 29131, 1093, 2, 5, 3, 281, 2, 13, 13, 5, 2, 3, 11, 3, 2, 7, 7, 5, 2, 46145917691, 3, 66161, 2, 17, 8039, 11, 2, 23, 5, 3, 2, 3, ... (sequence A039951 in the OEIS)
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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