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Fermat quotient

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In number theory, the Fermat quotient of an integer a with respect to an odd prime p is defined as:

Contents

q p ( a ) = a p − 1 − 1 p .

or

δ p ( a ) = a − a p p .

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.

Properties

From the definition, it is obvious that

q p ( 1 ) ≡ 0 ( mod p ) q p ( − a ) ≡ q p ( a ) ( mod p ) , since p − 1 is even.

In 1850 Gotthold Eisenstein proved that if a and b are both coprime to p, then:

q p ( a b ) ≡ q p ( a ) + q p ( b ) ( mod p ) ; q p ( a r ) ≡ r q p ( a ) ( mod p ) ; q p ( p − a ) ≡ q p ( a ) + 1 a ( mod p ) ; q p ( p + a ) ≡ q p ( a ) − 1 a ( mod p ) ; q p ( p − 1 ) ≡ 1 ( mod p ) ; q p ( p + 1 ) ≡ − 1 ( mod p ) .

Eisenstein likened the first two of these congruences to properties of logarithms. These properties imply

q p ( 1 / a ) ≡ − q p ( a ) ( mod p ) ; q p ( a / b ) ≡ q p ( a ) − q p ( b ) ( mod p ) .

In 1895 Dmitry Mirimanoff pointed out that an iteration of Eisenstein's rules gives the corollary:

q p ( a + n p ) ≡ q p ( a ) − n ⋅ 1 a ( mod p ) .

From this, it follows that

q p ( a + n p 2 ) ≡ q p ( a ) ( mod p ) .

Special Values

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}:

− 2 q p ( 2 ) ≡ ∑ k = 1 p − 1 2 1 k ( mod p ) .

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:

− 3 q p ( 2 ) ≡ ∑ k = 1 ⌊ p 4 ⌋ 1 k ( mod p ) . 4 q p ( 2 ) ≡ ∑ k = ⌊ p 10 ⌋ + 1 ⌊ 2 p 10 ⌋ 1 k + ∑ k = ⌊ 3 p 10 ⌋ + 1 ⌊ 4 p 10 ⌋ 1 k ( mod p ) . 2 q p ( 2 ) ≡ ∑ k = ⌊ p 6 ⌋ + 1 ⌊ p 3 ⌋ 1 k ( mod p ) .

Eisenstein's series also has an increasingly complex connection to the Fermat quotients with other bases, the first few examples being:

− 3 q p ( 3 ) ≡ 2 ∑ k = 1 ⌊ p 3 ⌋ 1 k ( mod p ) . − 5 q p ( 5 ) ≡ 4 ∑ k = 1 ⌊ p 5 ⌋ 1 k + 2 ∑ k = ⌊ p 5 ⌋ + 1 ⌊ 2 p 5 ⌋ 1 k ( mod p ) .

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:

For more information, see, and.

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.

References

Fermat quotient Wikipedia
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