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Adenwalla 2025 question erdos graham covering systems
Sarosh Adenwalla, A Question of Erdős and Graham on Covering Systems. arXiv preprint (2025). arXiv:2501.15170, doi:10.48550/arXiv.2501.15170. Published as INTEGERS 26 (2026), #A52, received 22 May 2025, accepted 31 March 2026, published 1 May 2026, doi:10.5281/zenodo.19949505. The arXiv record (https://arxiv.org/abs/2501.15170, read 2026-10-07) names the Creative Commons Attribution 4.0 license. The copy read for this card is arXiv:2501.15170v3 (3 December 2025).
arXiv 2501.15170 is a single-author paper by Sarosh Adenwalla. It answers a 1980 question of Erdos and Graham in the negative (Theorem 3.2): there is no integer n whose divisors greater than 1 form a distinct covering system with the property that whenever some integer satisfies both a mod d and a' mod d', the moduli d and d' are coprime. The paper also gives a necessary condition for the divisors of n greater than 1 to carry residue classes with that coprimality property, covering or not (Lemma 3.1: if p is the least prime factor of n, then n/p has fewer than p distinct prime factors). It proves the condition sufficient for n = p^k (Proposition 4.1) and for n = qp^k with p, q distinct primes, where such residue classes exist exactly when k = 1 or p > 2 (Proposition 4.2), and conjectures that it is sufficient in general (Conjecture 5.1), noting a claimed proof by Jia, Li and Liu (arXiv:2504.09579). The argument is elementary and combinatorial, working with the divisor lattice and density of the covered residues. Theorem 3.2 settles problem 204. For problem 203 the entry serves as a negative check: it settles a different question from the same Erdos-Graham 1980 source, not the r = 2 base-2-and-3 generalized Sierpinski question of problem 203.
Source: https://arxiv.org/abs/2501.15170.
Bears on. #204, which it settles; #203, as a negative check.