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Sep 10, 2019 at 12:23 history closed Gerry Myerson nt.number-theory Duplicate of Stronger versions of Wilson's Theorem
Sep 10, 2019 at 11:26 history edited apple CC BY-SA 4.0
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Sep 10, 2019 at 7:53 comment added Greg Martin The usual heuristic is that the "probability" that a prime $p$ satisfies $p^3 \mid \big( (p-1)!+1\big)$ is $1/p^2$. Since there are no such primes up to $2\times10^{13}$ (as per the Wikipedia article on Wilson primes), our heuristic is that there is a (much) less than $1$ in $2\times10^{13}$ chance that any such primes exist. But yes, a proof seems hopeless at present.
Sep 10, 2019 at 7:26 comment added Gerry Myerson I suspect the answer to all three of your questions is, nobody knows.
Sep 10, 2019 at 6:08 history asked apple CC BY-SA 4.0