Table of congruences

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In mathematics, a congruence is an equivalence relation on the integers. The following table lists important or interesting congruences.

2^{n-2} \equiv 1 \pmod{n}\,\! satisfied by infinitely many positive integers n[1][2][3][4]
2^{p-1} \equiv 1 \pmod{p}\,\! special case of Fermat's little theorem, satisfied by all odd prime numbers
2^{p-1} \equiv 1 \pmod{p^2}\,\! solutions are called Wieferich primes
(n-1)!\ \equiv\ -1 \pmod n by Wilson's theorem a natural number n is prime if and only if it satisfies this congruence
(p-1)!\ \equiv\ -1 \pmod{p^2} solutions are called Wilson primes
{2n-1 \choose n-1} \equiv 1 \pmod{n^3} by Wolstenholme's theorem satisfied by all prime numbers greater than 3
{2p-1 \choose p-1} \equiv 1 \pmod{p^4}, solutions are called Wolstenholme primes
{2p-1 \choose p-1} \equiv 1 \pmod{p^6}, must be satisfied by a counterexample to the converse of Wolstenholme's theorem[5]
F_{p - \left(\frac{{p}}{{5}}\right)} \equiv 0 \pmod{p} satisfied by all odd prime numbers
F_{p - \left(\frac{{p}}{{5}}\right)} \equiv 0 \pmod{p^2} solutions are called Wall–Sun–Sun primes

References[edit]