Exercise 1134Number theory › PrimesOlympiadAdded 3 Nov 2026

Primes and factorials

#factorial#primes#Wilson

Statement

Show that $n!+1$ is not divisible by any prime $\le n$ .

Hint

For prime $p\le n$ : $p\mid n!$ , and if $p\mid n!+1$ , then $p\mid1$ .

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Setup. Let $p$ be prime with $p\le n$ . Then $p$ appears in the product $n!=1\cdot2\cdots n$ , so $p\mid n!$ .

Suppose $p\mid n!+1$ . Then $p\mid(n!+1)-n!=1$ , contradiction.

Conclusion. All prime divisors of $n!+1$ are $>n$ .

Application. This gives Euclid-style proofs that infinitely many primes exist: $n!+1$ has a prime divisor $>n$ .