bibkey: bloom2026erdos authors: T. F. Bloom year: 2026 title: “Erdős Problem #7” doi: null url: https://www.erdosproblems.com/7 claim: “Is there a distinct covering system all of whose moduli are odd? The unrestricted question remains open on the problem page.” strata_touched:

• D5/S3/Arith/Congruence/TwoOddPrimeUncoveredDensity license: citation-only triage: anchor

Odd covering systems

Source: https://www.erdosproblems.com/7, read 16 September 2026. The page attributes the question to Erdős and Selfridge, sometimes also with Schinzel, and marks it “Open, but could be proved with a finite example.” Discussion: https://www.erdosproblems.com/forum/discuss/7.

The squarefree restriction has a negative answer; arbitrary powers remain part of #7. The September source claims in schroeder2026nine and schroeder2026noncoverage concern, respectively, total prime support and prime factors of each individual modulus. Neither states the unrestricted conclusion. The forum also contains proposed proofs; their presence is not verification.

The supplied research search found no unrestricted proof in the repository, pinned mathlib, or linked formal-conjectures source. This is a bounded search finding, not a proof of absence from the literature. The existing frozen two-prime result supplies motivation only. The exact target and reusable counterexamples are in the canonical dossier.

The later paper Bispels–Cohen–Harrington–Lowrance–Pontes–Schaumann–Wong (2025/2026) likewise treats the unrestricted distinct odd problem as open and studies a repeated-modulus variant; its multiplicity constructions do not supply a distinct odd cover.