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.''

Bibliography

1

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