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# Table of congruences

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

\( 2^{n-1} \equiv 1 \pmod{n}\,\! \) | 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 |

\( F_{n - \left(\frac{{n}}{{5}}\right)} \equiv 0 \pmod{n} \) | 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 |

\( {2n-1 \choose n-1} \equiv 1 \pmod{n^3}\) | 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 |

\( (n-1)!\ \equiv\ -1 \pmod n\) | 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 |

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