Zsigmondy's Theorem

Theorem
Let $a, b > 0$ be coprime positive integers.

Let $n \ge 1$ be a (strictly) positive integer.

Then there is a prime number $p$ such that
 * $p$ divides $a^n - b^n$
 * $p$ does not divide $a^k - b^k$ for all $k < n$

with the following exceptions:


 * $n = 1$ and $a - b = 1$
 * $n = 2$ and $a + b$ is a power of $2$
 * $n = 6$, $a = 2$, $b = 1$