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Large Zsigmondy Primes

Ömer Avcı

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.

Large Zsigmondy Primes

Abstract

If and are positive integers and and are relatively prime integers, then a large Zsigmondy prime for is a prime such that but for and either or . We classify all the triples of integers for which no large Zsigmondy prime exists.

Paper Structure

This paper contains 3 sections, 15 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Results on Zsigmondy Primes

Key Result

Theorem 1.1

If $a>b$ are relatively prime positive integers and $n$ is an integer greater than 1, then there exists a large Zsigmondy prime for $(a,b,n)$ except the following cases.

Theorems & Definitions (21)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2: Lifting the Exponent Lemma - LTE
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Corollary 2.6
  • Lemma 2.7
  • Corollary 2.8
  • Lemma 2.9
  • ...and 11 more