# 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 $4(p-1)!\equiv-4-p\pmod{p^{2}+2p}$.

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 ClementsTheoremOnTwinPrimes 2013-03-22 17:58:32 2013-03-22 17:58:32 PrimeFan (13766) PrimeFan (13766) 4 PrimeFan (13766) Theorem msc 11N05