Large Zsigmondy Primes

TL;DR

This paper classifies all integer triples (a, b, n) where no large Zsigmondy prime exists, extending understanding of prime divisors in exponential differences and their properties.

Contribution

It provides a complete classification of triples (a, b, n) lacking large Zsigmondy primes, a novel result in the study of prime divisors of exponential expressions.

Findings

Identifies all (a, b, n) with no large Zsigmondy prime

Characterizes conditions under which large Zsigmondy primes do not exist

Enhances understanding of prime divisors in exponential differences

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 $1 \leq m < n$ 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.