Prove that $n$ is prime number if $n > 1$ and $n$ divides $(n-1)!+1$

Asked Jul 10 '21 at 01:32

Active Oct 19 '23 at 16:43

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Let $n\in \Bbb Z$ such that $n > 1$ and $n$ divides $(n-1)!+1$. Prove that $n$ is prime number.

One of the ways I can think of is to do it by contradiction:

Using the prime number definition that says that a number $n$ is prime if and only if $n>1$ and its only positive divisors are 1 and $n$.

Then I’d have to assume that $n = ab$ such that $1<a,b<n$ and would have to come to that $a = 1$ o $b = 1$.

How should I do the contradiction? Is there any easier way to do that?

edited Jul 10 '21 at 13:57

Angelo

asked Jul 10 '21 at 01:32

Roy M

7

If $n$ isn’t prime, then there’s some $a$ in $[2,n-1]$ that divides $n$ and hence divides $(n-1)!+1$. But this is impossible, because every such $a$ goes into $(n-1)!+1$ with a remainder of $1$. – mjqxxxx Jul 10 '21 at 01:37
2

@mjqxxxx This should be an answer. – Deepak Jul 10 '21 at 02:07
6

FYI Wilson's theorem, the proof is in the article. – rtybase Jul 10 '21 at 07:57

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