# Clement’s theorem on twin primes

Theorem. (P. Clement) Given a prime number^{} $p$, $p+2$ is also a prime (and $p$ and $p+2$ form a twin prime^{}) if and only if $4(p-1)!\equiv -4-p\phantom{\rule{veryverythickmathspace}{0ex}}(mod{p}^{2}+2p)$.

Richard Crandall and Carl Pomerance see this theorem as “a way to connect the notion of twin-prime pairs with the Wilson-Lagrange theorem.”

## References

- 1 Richard Crandall & Carl Pomerance, Prime Numbers: A Computational Perspective, 2nd Edition. New York: Springer (2005): 65, Exercise 1.57

Title | Clement’s theorem on twin primes |
---|---|

Canonical name | ClementsTheoremOnTwinPrimes |

Date of creation | 2013-03-22 17:58:32 |

Last modified on | 2013-03-22 17:58:32 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 4 |

Author | PrimeFan (13766) |

Entry type | Theorem |

Classification | msc 11N05 |