DISPROVED (LEAN)
This has been solved in the negative and the proof verified in Lean.
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 $(d,d')=1$.
The density of such $n$ is zero. Erdős and Graham believed that no such $n$ exist.
Adenwalla
[Ad25] has proved there are no such $n$.
In general, for any $n$ one can try to choose such $a_d$ to maximise the density of integers so covered, and ask what this density is. This was also investigated by Adenwalla
[Ad25].
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This page was last edited 28 December 2025. View history
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Additional thanks to: Sarosh Adenwalla
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #204, https://www.erdosproblems.com/204, accessed 2026-04-11
