Problem 204

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claims/: The 1 claim page of Problem 204, one per claimant's result; the problem's standing derives from them.

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Statement. Are there $n$ such that there is a covering system with moduli the divisors of $n$ which is 'as disjoint as possible'?

That is, for all $d\mid n$ with $d>1$ there is an associated $a_d$ such that every integer is congruent to some $a_d\pmod{d}$, and if there is some integer $x$ with

$$x\equiv a_d\pmod{d}\textrm{ and }x\equiv a_{d'}\pmod{d'}$$

then $(d,d')=1$.

Status. DISPROVED (LEAN). The label is the site's (DISPROVED (LEAN), page last edited 28 December 2025). Adenwalla proved that no such $n$ exists, in a paper refereed and published in INTEGERS 26 (2026), #A52; the acceptance evidence and the Lean qualification are on his claim page.

Source. erdosproblems.com/204, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #204, https://www.erdosproblems.com/204.

References.

• [Ad25] S. Adenwalla, A Question of Erdős and Graham on Covering Systems. arXiv:2501.15170 (2025). Published as INTEGERS 26 (2026), #A52, doi:10.5281/zenodo.19949505.

Formalization. Statement in formal-conjectures.

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• adenwalla_2025_question_erdos_graham_covering_systems