Two Dimensional Covering Systems and Possible Prime Producing a^m - b^n

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conjecture_1: Predicts infinitely many prime values outside a fixed finite-prime obstruction when neither base is a perfect power.

conjecture_2: Extends the prime-value conjecture with exponent coprimality conditions for bases having maximal perfect-power exponents.

corollary_1: Characterizes a fixed finite-prime obstruction for all exponent pairs by a two-dimensional covering system.

proposition_3: Represents all nonnegative exponent pairs giving divisibility by a fixed prime as one uniquely reduced two-dimensional congruence lattice.

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Andrew Granville and Francesco Pappalardi, Two dimensional covering systems and possible prime producing $a^m-b^n$, arXiv:2601.10296v2, 11 April 2026, DOI 10.48550/arXiv.2601.10296.

The retained 13-page PDF is arXiv v2. The arXiv record says that v2 replaces the original submission of 15 January 2026 and corrects a mistake found by a referee. The record inspected on 5 September 2026 listed no journal reference. All result pages therefore cite v2; no claim is made that the v1 statements are interchangeable. The arXiv record (https://arxiv.org/abs/2601.10296, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

The paper begins with the finite-prime obstruction

$$\gcd(41^m-34^n,3\cdot5\cdot7)>1 \qquad(m,n\geq0).$$

For general $a,b$, condition (1.1) asks for an integer $Q$ such that

$$\gcd(a^m-b^n,Q)>1 \qquad\text{for all positive integers }m,n.$$

The paper's conjectures predict that this is the only obstruction, with an adjusted coprimality condition when $a$ or $b$ is a perfect power.

For its Section 2 classification, the source fixes positive integers $a,b,Q$ with $\gcd(a,b)=1$, takes $Q$ to be square-free, and assumes $\gcd(Q,ab)=1$. Corollary 1's standalone opening repeats only the last of these conditions. The global positivity and square-freeness conventions are retained on its result page because they are needed for the stated minimality equivalence.

For integers $u,v,r$ with $r\geq1$, put

$$S(u,v,r)=\{(m,n)\in\mathbb Z^2:mv\equiv nu\pmod r\}.$$

A finite set of triples is a two-dimensional covering system when the corresponding sets $S(u,v,r)$ cover $\mathbb Z^2$. Proposition 3 turns divisibility by a fixed prime into one uniquely reduced such lattice, and Corollary 1 classifies the obstruction for a fixed $Q$ by a two-dimensional covering. The print writes the lattices as a set; its minimality clause is read here with the lattices indexed by the primes dividing $Q$, so equal lattice sets arising from different primes remain separate entries.

Compiled scope

The introduction, Conjectures 1--2, the definitions in Section 2.1, Proposition 3, and Corollary 1 were read on PDF pp. 1--5. Their statements are transcribed below. Proofs, computations, and the later classification and construction sections were not compiled or independently checked.

Results and conjectures

• Conjecture 1
• Conjecture 2
• Proposition 3
• Corollary 1

No exact numbered Erdős-problem relationship is assigned in this source unit.