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Problem: Show that $2(n-1)! \equiv -1 \mod n+2 \iff n+2$ is a prime.

I know that Wilson's theorem states that $(n-1)! \equiv -1 \mod p $ for $p$ a prime, so that is the important thing to know with these type of problems.

I know that $-1 \equiv n+1 \mod n+2$ , and that if $n+2$ is prime, then $(n+1)! = -1 \mod n+2$. So this is all I have for now, any hints appreciated.

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  • $\begingroup$ Well, what is $(n+1)\times n\pmod {n+2}$? $\endgroup$ – lulu Apr 19 at 20:59
  • $\begingroup$ @lulu $(-n) \mod n+2 $ $\endgroup$ – IntegrateThis Apr 19 at 20:59
  • $\begingroup$ Yes, and what is $n\pmod {n+2}$? $\endgroup$ – lulu Apr 19 at 21:00
  • $\begingroup$ @lulu -2 I believe. $\endgroup$ – IntegrateThis Apr 19 at 21:00
  • $\begingroup$ Yes, and what is $-1\times -2$? $\endgroup$ – lulu Apr 19 at 21:00
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As per the comments since I know that $2 (n-1)! \equiv (-1)(-2)(n-1)! \equiv (n)(n+1)(n-1)! \equiv (n+1)! \equiv -1 \mod (n+2)$, then $n+2$ is prime.

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    $\begingroup$ Just to be clear: Wilson's Theorem plus the algebraic manipulation you provide just show one direction. Specifically: they show that $n+2$ prime $\implies 2(n-1)!\equiv -1 \pmod {n+2}$. You need to show the converse as well. However, that's pretty easy (Hint: if $n+2$ is composite then it has a non-trivial factor $<n-1$, at least for $n≥4$). $\endgroup$ – lulu Apr 19 at 21:27

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