Erdős Problem #7 — Covering systems with all moduli odd

Erdős

Let a covering system be a finite collection of residue classes $C=\{a_m\bmod{m}\}_{m\in S}$ with distinct moduli $S\subset\mathbb{N}_{>1}$ whose union is $\mathbb{Z}$ . Erdős's Odd Covering conjecture. No covering system consists only of residue classes whose moduli $m>1$ are odd.

This is Erdős's negatively stated Odd Covering Problem: it asserts that every covering system contains a modulus divisible by $2$ . The source describes a weaker result proved in the paper, namely that every covering system has a modulus divisible by a prime $p\leq 19$ , while stronger work of Hough and Nielsen proves that every covering system contains a modulus divisible by either $2$ or $3$ ; the conjecture itself is not resolved in the supplied text.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Covering systems with all moduli odd
Many further unsolved problems can be asked about covering systems. Selfridge and I asked: Is there a covering system all whose moduli are odd?
source: P. Erdős, Some of my favourite problems in number theory, combinatorics, and geometry, Resenhas IME-USP 2 (1995), 165-186.

Sources & referencesView supporting material

Primary source

Jackson Hopper, “On covering systems of integers”, arXiv:1705.04372 (2017).

3 weeks ago

Progress summary

Refreshed 1 week ago

Partially solved

The question remains open: a new computer-checked result only shows that any example would need a very large common period.

Erdős and Selfridge asked whether the integers can be covered by finitely many residue classes with distinct, odd moduli greater than $1$ . The unrestricted question remains unanswered.

Known results

Hough and Nielsen proved that at least one modulus is divisible by $2$ or $3$ .
Balister, Bollobás, Morris, Sahasrabudhe, and Tiba ruled out the case of odd squarefree moduli.
The same authors showed that any odd covering would have least common multiple divisible by $9$ or $15$ .
McNew and Setty classified covering numbers up to $10^6$ .

2026 formalized exclusion

A Lean $4$ formalization proves that any such covering would have least common multiple exceeding $10^4$ , via density and abundancy arguments plus checked exclusions below $10^4$ . It verifies known partial mathematics and does not prove nonexistence. A separate repository claims a resolution, but its stated theorem only establishes the same lower bound and is contradicted by the official formalization and paper’s open status.

Current status (as of July 2026): The problem is open; the strongest supplied recent progress is a formally verified lower bound exceeding $10^4$ for the least common multiple of any hypothetical covering.

Sources

No solutions have been posted yet.