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

Andrew Granville, Francesco Pappalardi

TL;DR

The paper studies when the values |a^m−b^n| can be prime as m,n vary, introducing a framework based on two-dimensional covering systems to classify obstructions to infinitely many primes. It establishes a precise correspondence between obstructions (a,b,Q) and covering systems, and shows how to construct (a,b,Q) from any given system via primes and the Chinese Remainder Theorem. Using Baker’s theory on linear forms in logarithms, it proves finiteness results for prime values under covers and develops a heuristic, refined by local corrections, to predict prime counts, supported by computational data. The work provides a cohesive method to analyze prime values in exponential families, connects combinatorial covering structures to number-theoretic obstructions, and offers conjectures on the density of exceptional pairs and asymptotic constants κ_{a,b} guiding prime counts.

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.

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

TL;DR

The paper studies when the values |a^m−b^n| can be prime as m,n vary, introducing a framework based on two-dimensional covering systems to classify obstructions to infinitely many primes. It establishes a precise correspondence between obstructions (a,b,Q) and covering systems, and shows how to construct (a,b,Q) from any given system via primes and the Chinese Remainder Theorem. Using Baker’s theory on linear forms in logarithms, it proves finiteness results for prime values under covers and develops a heuristic, refined by local corrections, to predict prime counts, supported by computational data. The work provides a cohesive method to analyze prime values in exponential families, connects combinatorial covering structures to number-theoretic obstructions, and offers conjectures on the density of exceptional pairs and asymptotic constants κ_{a,b} guiding prime counts.

Abstract

We exhibit a new application of two dimensional covering systems, examples of integer pairs for which has a prime divisor from some given finite set of primes, for every pair of integers . This leads us to conjecture what are the only possible obstructions to taking on infinitely many distinct prime values.
Paper Structure (16 sections, 4 theorems, 51 equations, 6 tables)

This paper contains 16 sections, 4 theorems, 51 equations, 6 tables.

Table of Contents

  1. Introduction
  2. A mixed case
  3. Two-dimensional covering systems
  4. Two-dimensional congruences
  5. Primes yielding a given reduced triple
  6. Two-dimensional covering systems
  7. Constructing $a,b$ and $Q$ for a given two-dimensional covering system
  8. Properties of non trivial reduced covering systems
  9. Linear forms in logarithms
  10. Heuristic for the number of primes $a^m-b^n$
  11. The number up to $x$
  12. The Cramér heuristic
  13. Linearity
  14. Local adjustments, and special form adjustment
  15. Convergence of the constant
  16. ...and 1 more sections

Key Result

Lemma 1

Every triple $(u,v,r)$ with $r\geqslant 1$ and $u$ and $v$ not both $0 \pmod r$, is equivalent to a reduced triple.

Theorems & Definitions (11)

  • Conjecture 1
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Corollary 1
  • Example 1
  • Example 2
  • ...and 1 more