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 q_p(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]

The smallest solutions of q_p(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

External links

Gottfried Helms. Fermat-/Euler-quotients (a^p-1 – 1)/p^k with arbitrary k.