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# 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 $4(p-1)!\equiv-4-p\;\;(\mathop{{\rm mod}}p^{2}+2p)$.

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

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## Mathematics Subject Classification

11N05*no label found*

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