Wilson's theorem

For an integer $j\ge 2$:

Equivalently, $j\mid (j-1)!+1$ precisely when $j$ is prime (and trivially when $j=1$).

Consequence: This gives a primality criterion — in principle it detects primes, though computing $(j-1)!$ makes it impractical for large $j$.but

Related: prime counting function, prime number theorem.