Resolving Adenwalla's conjecture related to a question of Erdős and Graham about covering systems

TL;DR

This work resolves Adenwalla's conjecture by proving that the necessary condition for a nice integer—namely that if $n>1$ has smallest prime divisor $p$ and $rac{n}{p}$ has fewer than $p$ distinct prime factors—indeed suffices to guarantee niceness. The authors construct a novel hierarchical residue-assignment framework and leverage a non-uniform application of the Chinese Remainder Theorem to produce good sets of congruences for all such $n$, thereby completing the characterization of nice integers. They also present an algorithmic implementation that generates explicit residue sets and provide concrete examples illustrating both the construction and a non-nice case. The results advance the understanding of covering systems and resolve a notable open problem in combinatorial number theory with potential implications for related density and factorization questions.

Abstract

Erdős and Graham posed the question of whether there exists an integer $n$ such that the divisors of $n$ greater than $1$ form a distinct covering system with pairwise coprime moduli for overlapping congruences. Adenwalla recently proved no such $n$ exists, introducing the concept of nice integers, those where such a system exists without necessarily covering all integers. Moreover, Adenwalla established a necessary condition for nice integers: if $n$ is nice and $p$ is its smallest prime divisor, then $n/p$ must have fewer than $p$ distinct prime factors. Adenwalla conjectured this condition is also sufficient. In this paper, we resolve this conjecture affirmatively by developing a novel constructive framework for residue assignments. Utilizing a hierarchical application of the Chinese Remainder Theorem, we demonstrate that every integer satisfying the condition indeed admits a good set of congruences. Our result completes the characterization of nice integers, resolving an interesting open problem in combinatorial number theory.

Theorems & Definitions (20)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 1: Adenwalla, Adenwalla2025
• Theorem 2
• Lemma 1
• proof
• Lemma 2
• proof