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Fermat quotient

In number theory, the Fermat quotient of an integer a ≥ 2 with respect to a prime base p is defined as:[1][2][3]

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

If a is coprime to p then Fermat's little theorem says that qp(a) will be an integer. The quotient is named after Pierre de Fermat.

Properties

In 1850 Ferdinand Eisenstein proved that if a and b are both coprime to p, then:[2]

$q_p(ab)\equiv q_p(a)+q_p(b) \pmod{p}$
$q_p(p-1)\equiv 1 \pmod{p}$
$q_p(p+1)\equiv -1 \pmod{p}$
$-2q_p(2) \equiv \sum_{k=1}^{\frac{p-1}{2}} \frac{1}{k} \pmod{p}.$

Generalized Wieferich primes

If $q_p(a) ≡ 0 (mod p)$ then $a ^{ p-1 }≡ 1 (mod p^2).$ 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 $q_p(a) ≡ 0 (mod p)$ for small prime values of a are:[2]

a p OEIS sequence
2 1093, 3511 A001220
3 11, 1006003 A014127
5 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 A123692
7 5, 491531 A123693
11 71
13 2, 863, 1747591 A128667
17 2, 3, 46021, 48947, 478225523351 A128668
19 3, 7, 13, 43, 137, 63061489 A090968
23 13, 2481757, 13703077, 15546404183, 2549536629329 A128669

The smallest solutions of qp(a) ≡ 0 (mod p) with a = the nth prime are"

1093, 11, 2, 5, 71, 2, 2, 3, 13, 2, 7, 2, 2, 5, … (sequence A174422 in 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

^ Weisstein, Eric W., "Fermat Quotient" from MathWorld.
^ a b c Fermat Quotient at The Prime Glossary
^ Paulo Ribenboim, My Numbers, My Friends, p216