Large Zsigmondy Primes

Abstract

If $a>b$ and $n>1$ are positive integers and $a$ and $b$ are relatively prime integers, then a large Zsigmondy prime for $(a,b,n)$ is a prime $p$ such that $p|a^n-b^n$ but $p\nmid a^m-b^m$ for $1\le m<n$ and either $p^2|a^n-b^n$ or $p>n+1$. We classify all the triples of integers $(a,b,n)$ for which no large Zsigmondy prime exists.

10 pages