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
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