Zsigmondy Theorem

If $1\leq b<a$ and $(a,b)=1$ (i.e., $a$ and $b$ are Relatively Prime), then $a^n-b^n$ has a Primitive Prime Factor with the following two possible exceptions:

1. $2^6-1^6$.

2. $n=2$ and $a+b$ is a Power of 2.

Similarly, if $a>b\geq 1$, then $a^n+b^n$ has a Primitive Prime Factor with the exception $2^3+1^3=9$.

References

Ribenboim, P. The Little Book of Big Primes. New York: Springer-Verlag, p. 27, 1991.