Wieferich's Criterion

Theorem
Suppose Fermat's equation:
 * $x^p + y^p = z^p$

has a solution in which $p$ is an odd prime that does not divide any of $x$, $y$ or $z$.

Then $2^{p - 1} - 1$ is divisible by $p^2$.

Also known as
Some sources give this as Wieferich's theorem, but this is also used for his result concerning the Hilbert-Waring Theorem for cubes.

Also see

 * Definition:Wieferich Prime