I have found this interesting theorem but with no proof:
"$p$ and $p+2$ are twin primes iff $4((p-1)!+1)\equiv -p\mod{(p(p+2)})$"
I have tried to prove it with classical N.T. theorems but I just get nowhere. Any advise?
Thanks in advance.
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2Chinese remainder theorem: prove separately modulo $p$ and $p+2$. – Angina Seng Aug 31 '17 at 11:19
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do you mean "iff" instead of "iif"? if yes, which of the two directions have you been able to prove? – supinf Aug 31 '17 at 11:32
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wilson's theorem may also help. – Aug 31 '17 at 11:43
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To apply Wilson's Theorem for $p+2$ try multiplying through by $p$ (these are coprime if $p$ is odd). – Mark Bennet Aug 31 '17 at 11:51
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2https://arxiv.org/pdf/math/0408018.pdf – user328442 Aug 31 '17 at 12:03
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@supinf Sorry, my mistake! – Cragfelt Aug 31 '17 at 15:41
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@user328442 Also here https://link.springer.com/article/10.1007%2FBF02836740 – Cragfelt Aug 31 '17 at 15:42
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Thank you @LordSharktheUnknown, Mark Bennet and Roddy Macphee – Cragfelt Aug 31 '17 at 15:44