Theorem. A necessary and sufficient condition for the integers n>7 and n+2 to be twin primes is that 4(n−3)!+2+n is divisible by n but not by n+2.

http://arxiv.org/abs/math/0211034

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Theorem. A necessary and sufficient condition for the integers n>7 and n+2 to be twin primes is that 4(n−3)!+2+n is divisible by n but not by n+2.

http://arxiv.org/abs/math/0211034