Fermat quotient

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


 * $$q_p(a) = \frac{a^{p-1}-1}{p},$$

or


 * $$\delta_p(a) = \frac{a - a^p }{p}$$.

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


 * $$\begin{align}

q_p(1) &\equiv 0 && \pmod{p} \\ q_p(-a)&\equiv q_p(a) && \pmod{p}\quad (\text{since } 2 \mid p-1) \end{align}$$

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


 * $$\begin{align}

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{align}$$

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


 * $$\begin{align}

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{align}$$

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


 * $$q_p(a+np)\equiv q_p(a)-n\cdot\tfrac{1}{a} \pmod{p}.$$

From this, it follows that:


 * $$q_p(a+np^2)\equiv q_p(a) \pmod{p}.$$

Lerch's formula
M. Lerch proved in 1905 that


 * $$\sum_{j=1}^{p-1}q_p(j)\equiv W_p\pmod{p}.$$

Here $$W_p$$ 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&thinsp;−&thinsp;1}:


 * $$-2q_p(2) \equiv \sum_{k=1}^{\frac{p-1}{2}} \frac{1}{k} \pmod{p}.$$

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:


 * $$-3q_p(2) \equiv \sum_{k=1}^{\lfloor\frac{p}{4}\rfloor} \frac{1}{k} \pmod{p}.$$


 * $$4q_p(2) \equiv \sum_{k=\lfloor\frac{p}{10}\rfloor + 1}^{\lfloor\frac{2p}{10}\rfloor} \frac{1}{k} + \sum_{k=\lfloor\frac{3p}{10}\rfloor + 1}^{\lfloor\frac{4p}{10}\rfloor} \frac{1}{k} \pmod{p}.$$


 * $$2q_p(2) \equiv \sum_{k=\lfloor\frac{p}{6}\rfloor+1}^{\lfloor\frac{p}{3}\rfloor} \frac{1}{k} \pmod{p}.$$

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


 * $$-3q_p(3) \equiv 2\sum_{k=1}^{\lfloor\frac{p}{3}\rfloor} \frac{1}{k} \pmod{p}.$$


 * $$-5q_p(5) \equiv 4\sum_{k=1}^{\lfloor\frac{p}{5}\rfloor} \frac{1}{k} + 2\sum_{k=\lfloor\frac{p}{5}\rfloor+1}^{\lfloor\frac{2p}{5}\rfloor} \frac{1}{k} \pmod{p}.$$

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:


 * {| class="wikitable"

! a ! p (checked up to 5 × 1013) ! OEIS sequence
 * 1 || 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... (All primes)
 * 2 || 1093, 3511
 * 3 || 11, 1006003
 * 4 || 1093, 3511
 * 5 || 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
 * 6 || 66161, 534851, 3152573
 * 7 || 5, 491531
 * 8 || 3, 1093, 3511
 * 9 || 2, 11, 1006003
 * 10 || 3, 487, 56598313
 * 11 || 71
 * 12 || 2693, 123653
 * 13 || 2, 863, 1747591
 * 14 || 29, 353, 7596952219
 * 15 || 29131, 119327070011
 * 16 || 1093, 3511
 * 17 || 2, 3, 46021, 48947, 478225523351
 * 18 || 5, 7, 37, 331, 33923, 1284043
 * 19 || 3, 7, 13, 43, 137, 63061489
 * 20 || 281, 46457, 9377747, 122959073
 * 21 || 2
 * 22 || 13, 673, 1595813, 492366587, 9809862296159
 * 23 || 13, 2481757, 13703077, 15546404183, 2549536629329
 * 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
 * }
 * 11 || 71
 * 12 || 2693, 123653
 * 13 || 2, 863, 1747591
 * 14 || 29, 353, 7596952219
 * 15 || 29131, 119327070011
 * 16 || 1093, 3511
 * 17 || 2, 3, 46021, 48947, 478225523351
 * 18 || 5, 7, 37, 331, 33923, 1284043
 * 19 || 3, 7, 13, 43, 137, 63061489
 * 20 || 281, 46457, 9377747, 122959073
 * 21 || 2
 * 22 || 13, 673, 1595813, 492366587, 9809862296159
 * 23 || 13, 2481757, 13703077, 15546404183, 2549536629329
 * 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
 * }
 * 18 || 5, 7, 37, 331, 33923, 1284043
 * 19 || 3, 7, 13, 43, 137, 63061489
 * 20 || 281, 46457, 9377747, 122959073
 * 21 || 2
 * 22 || 13, 673, 1595813, 492366587, 9809862296159
 * 23 || 13, 2481757, 13703077, 15546404183, 2549536629329
 * 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
 * }
 * 23 || 13, 2481757, 13703077, 15546404183, 2549536629329
 * 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
 * }
 * 26 || 3, 5, 71, 486999673, 6695256707
 * 27 || 11, 1006003
 * 28 || 3, 19, 23
 * 29 || 2
 * 30 || 7, 160541, 94727075783
 * }
 * 28 || 3, 19, 23
 * 29 || 2
 * 30 || 7, 160541, 94727075783
 * }
 * 29 || 2
 * 30 || 7, 160541, 94727075783
 * }
 * 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, ...

A pair (p,&thinsp;r) of prime numbers such that qp(r) ≡ 0 (mod p) and qr(p) ≡ 0 (mod r) is called a Wieferich pair.