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

q p ( a ) = a p − 1 − 1 p , {\displaystyle q_{p}(a)={\frac {a^{p-1}-1}{p}},}

or

δ p ( a ) = a − a p p {\displaystyle \delta _{p}(a)={\frac {a-a^{p}}{p}}}.

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

q p ( 1 ) ≡ 0 ( mod p ) q p ( − a ) ≡ q p ( a ) ( mod p ) ( since 2 ∣ p − 1 ) {\displaystyle {\begin{aligned}q_{p}(1)&\equiv 0&&{\pmod {p}}\\q_{p}(-a)&\equiv q_{p}(a)&&{\pmod {p}}\quad ({\text{since }}2\mid p-1)\end{aligned}}}

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 ∓ 1 ) ≡ ± 1 ( mod p ) {\displaystyle {\begin{aligned}q_{p}(ab)&\equiv q_{p}(a)+q_{p}(b)&&{\pmod {p}}\\q_{p}(a^{r})&\equiv rq_{p}(a)&&{\pmod {p}}\\q_{p}(p\mp a)&\equiv q_{p}(a)\pm {\tfrac {1}{a}}&&{\pmod {p}}\\q_{p}(p\mp 1)&\equiv \pm 1&&{\pmod {p}}\end{aligned}}}

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 ) {\displaystyle {\begin{aligned}q_{p}\!\left({\tfrac {1}{a}}\right)&\equiv -q_{p}(a)&&{\pmod {p}}\\q_{p}\!\left({\tfrac {a}{b}}\right)&\equiv q_{p}(a)-q_{p}(b)&&{\pmod {p}}\end{aligned}}}

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 ) . {\displaystyle q_{p}(a+np)\equiv q_{p}(a)-n\cdot {\tfrac {1}{a}}{\pmod {p}}.}

From this, it follows that:

q p ( a + n p 2 ) ≡ q p ( a ) ( mod p ) . {\displaystyle q_{p}(a+np^{2})\equiv q_{p}(a){\pmod {p}}.}

Lerch's formula

M. Lerch proved in 1905 that

∑ j = 1 p − 1 q p ( j ) ≡ W p ( mod p ) . {\displaystyle \sum _{j=1}^{p-1}q_{p}(j)\equiv W_{p}{\pmod {p}}.}

Here W p {\displaystyle W_{p}} 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}:

− 2 q p ( 2 ) ≡ ∑ k = 1 p − 1 2 1 k ( mod p ) . {\displaystyle -2q_{p}(2)\equiv \sum _{k=1}^{\frac {p-1}{2}}{\frac {1}{k}}{\pmod {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 ) . {\displaystyle -3q_{p}(2)\equiv \sum _{k=1}^{\lfloor {\frac {p}{4}}\rfloor }{\frac {1}{k}}{\pmod {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 ) . {\displaystyle 4q_{p}(2)\equiv \sum _{k=\lfloor {\frac {p}{10}}\rfloor +1}^{\lfloor {\frac {2p}{10}}\rfloor }{\frac {1}{k}}+\sum _{k=\lfloor {\frac {3p}{10}}\rfloor +1}^{\lfloor {\frac {4p}{10}}\rfloor }{\frac {1}{k}}{\pmod {p}}.}

2 q p ( 2 ) ≡ ∑ k = ⌊ p 6 ⌋ + 1 ⌊ p 3 ⌋ 1 k ( mod p ) . {\displaystyle 2q_{p}(2)\equiv \sum _{k=\lfloor {\frac {p}{6}}\rfloor +1}^{\lfloor {\frac {p}{3}}\rfloor }{\frac {1}{k}}{\pmod {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 ) . {\displaystyle -3q_{p}(3)\equiv 2\sum _{k=1}^{\lfloor {\frac {p}{3}}\rfloor }{\frac {1}{k}}{\pmod {p}}.}

− 5 q p ( 5 ) ≡ 4 ∑ k = 1 ⌊ p 5 ⌋ 1 k + 2 ∑ k = ⌊ p 5 ⌋ + 1 ⌊ 2 p 5 ⌋ 1 k ( mod p ) . {\displaystyle -5q_{p}(5)\equiv 4\sum _{k=1}^{\lfloor {\frac {p}{5}}\rfloor }{\frac {1}{k}}+2\sum _{k=\lfloor {\frac {p}{5}}\rfloor +1}^{\lfloor {\frac {2p}{5}}\rfloor }{\frac {1}{k}}{\pmod {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:

a p (checked up to 5 × 1013) OEIS sequence 1 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... (All primes) A000040 2 1093, 3511 A001220 3 11, 1006003 A014127 4 1093, 3511 5 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 A123692 6 66161, 534851, 3152573 A212583 7 5, 491531 A123693 8 3, 1093, 3511 9 2, 11, 1006003 10 3, 487, 56598313 A045616 11 71 12 2693, 123653 A111027 13 2, 863, 1747591 A128667 14 29, 353, 7596952219 A234810 15 29131, 119327070011 A242741 16 1093, 3511 17 2, 3, 46021, 48947, 478225523351 A128668 18 5, 7, 37, 331, 33923, 1284043 A244260 19 3, 7, 13, 43, 137, 63061489 A090968 20 281, 46457, 9377747, 122959073 A242982 21 2 22 13, 673, 1595813, 492366587, 9809862296159 A298951 23 13, 2481757, 13703077, 15546404183, 2549536629329 A128669 24 5, 25633 25 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 26 3, 5, 71, 486999673, 6695256707 27 11, 1006003 28 3, 19, 23 29 2 30 7, 160541, 94727075783

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.

External links

  • Gottfried Helms. .
  • Richard Fischer. .