2025/04/13 by Jia, Zhengkun, Li, Huixi, Liu, Yushuo
#11B25(Primary)11B05 #11B30(Secondary ) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2504.09579
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.