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

Andrew Granville; Francesco Pappalardi

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

Number Theory 2026-04-14 v2

Authors: Andrew Granville , Francesco Pappalardi

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.

Keywords

prime numbers perfect numbers prime number theory

Cite

@article{arxiv.2601.10296,
  title  = {Two dimensional covering systems and possible prime producing $a^m-b^n$},
  author = {Andrew Granville and Francesco Pappalardi},
  journal= {arXiv preprint arXiv:2601.10296},
  year   = {2026}
}

Comments

Replaces original submission. Corrects mistake found by referee

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

Abstract

Keywords

Cite

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