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This is a homework question.

I know there exists infinitely many primes. Let $n = p-1$ and so by Wilson's theorem we know there exists atleast one prime $p$ that divides $n! + 1$. I used wolframalpha and checked for a couple of $n = p-1$ values and all show me that there are in fact two distinct primes and one of them is in fact $p$.

How can I use this to conclude there exists a second prime $q$, $q \neq p$ such that $q$ dividies $n!+1$

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1 Answer

Equivalently you can show that $(p-1)!+1=p^n$ has no solution for $p \geq 7$ and $n \in \mathbb{N}.$

Hint: $p^{n-1}+...+p+1=(p-2)!$ and $p-1|(p-2)!$ (since $p-1$ is composite), so what can you say about $n$ then? can you derive a contradiction?

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why is showing the statement $(p-1)! + 1 = p^n$ has no solutions equivalent? And anything special about $p =7$ and $n=2$? –  Tyler Hilton Feb 14 '13 at 5:13
    
You have set $n=p-1$ so if you could rule out the existence of primes where $(p-1)!+1$ is only divisible by one prime you'd be done. That bounds come from the an inequality you'll get after determining a condition on $n.$ –  Ehsan M. Kermani Feb 14 '13 at 5:20
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What is $p^{n-1}+...+p+1$ mod $p-1?$ –  Ehsan M. Kermani Feb 14 '13 at 5:44
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No, you have $n$ $1$'s, right? –  Ehsan M. Kermani Feb 14 '13 at 6:23
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Second step: Assume the contrary that there is a $p>5$ s.t. for some $n$ we have $(p-1)!+1=p^n$ or indeed $(p-1)!=p^n-1=(p-1)(p^{n-1}+...+p+1)$ which is $(p-2)!=p^{n-1}+...+p+1.$ As you said, $p^{n-1}+...+p+1$ mod $p-1$ is $n$ and since $p-1|(p-2)!$ then $p-1|n$ and indeed $p-1 \leq n.$ Thus, it is enough to show that for $p>5,$ we have $p^{p-1}>(p-1)!+1.$ –  Ehsan M. Kermani Feb 14 '13 at 7:31
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