Korselt's Criterion

$n$ Divides $a^n-a$ for all Integers $a$ Iff $n$ is Squarefree and $(p-1)\vert n/p-1$ for all Prime Divisors $p$ of $n$. Carmichael Numbers satisfy this Criterion.

References

Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgensohn, R. ``Giuga's Conjecture on Primality.'' Amer. Math. Monthly 103, 40-50, 1996.