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I assume that is nigh-impossible to prove when the conditions on the integers are very general. However, my algebra professor told me that the following is true:

If $n$ is a composite positive integer, $(n - 1)! + 1$ is not a power of $n$.

I assume this is an easy number theory problem, but I don't know how to approach it. The form of $(n-1)!+1$ makes me think of somehow splitting the term into its primes and using Wilson's, however improbable it is to do so. And for a proof by contradiction, finding parameters on $x$ such that $n^x=(n-1)!+1$ gets me nowhere, since it would just translate to a discrete log problem.

I appreciate any hints or input!

2 Answers 2

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HINT Since $n$ is composite, consider a prime dividing $n$. Will it divide $(n-1)! + 1$? (Remember that the prime dividing $n$ will occur in $(n-1)!$ and hence will divide $(n-1)!$).

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    That's kind of a heavy-handed hint :P2012-06-30
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The number $(n-1)! + 1$ cannot even be divisible by $n$, nevermind actually being $n$ or a power of $n$.

See this new question for proof of this fact (at least for $n\geq 5$):

link

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    I just copied and pasted a url, the text appeared by itself.2012-07-09