Two dimensional covering systems and possible prime producing $a^m-b^n$

TL;DR

This paper introduces a novel application of two-dimensional covering systems to identify integer pairs where $a^m-b^n$ always has a prime divisor from a finite set, and explores conjectures on prime values of such expressions.

Contribution

It presents a new application of covering systems to analyze prime divisors of exponential differences and proposes conjectures on their infinite prime values.

Findings

Identifies integer pairs with $a^m-b^n$ having prime divisors from finite sets

Proposes conjectures on the obstructions to $|a^m-b^n|$ taking infinitely many prime values

Introduces a new method linking covering systems to prime divisor analysis

Abstract

We exhibit a new application of two dimensional covering systems, examples of integer pairs $a,b$ for which $a^m-b^n$ has a prime divisor from some given finite set of primes, for every pair of integers $m,n\geq 0$. This leads us to conjecture what are the only possible obstructions to $|a^m-b^n|$ taking on infinitely many distinct prime values.