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Claim. If $r$ congruences $a_i\pmod{n_i}$, $1\le i\le r$, with moduli not necessarily distinct, cover $2^r$ consecutive integers, then they cover every integer; this is the statement of Problem 275. The bound is sharp: the $r$ classes $2^{i-1}\pmod{2^i}$ cover every integer that is not a multiple of $2^r$, so they cover $2^r-1$ consecutive integers without covering all of them. Erdős conjectured the statement, and the site records that Selfridge proved it independently of Crittenden and Vanden Eynden, citing no publication for Selfridge's proof, so it has no page of its own. The paper is not held in the library; later sources that consume or extend the theorem are: Klein, Koukoulopoulos and Lemieux's Claim 2.1, which uses it as its exact external input; Sun's finite-block criterion for real arithmetic sequences, which specializes to it; and Simpson's study of the authors' conjecture with a lower modulus cutoff, a conjecture whose cutoff-$1$ case is the theorem.

Depends on. Nothing in this wiki: the theorem is the paper's own.

Acceptance. Refereed: Proceedings of the American Mathematical Society 24 (1970), no. 3, 475–481, the DOI linked above, issued in March 1970; the page name carries the issue month, and the publication record gives no day. Reviewed: the site's curator, Thomas F. Bloom, records the problem as proved independently by Selfridge and by Crittenden and Vanden Eynden in the problem's commentary (page last edited 2026-01-23), and the theorem is a standard input in the later literature, as the sources above show. The site's label carries a Lean qualification: formal-conjectures marks the problem solved and points to a Lean proof in Boris Alexeev's lean-proofs repository, which follows the short proof of Balister, Bollobás, Morris, Sahasrabudhe and Tiba and is therefore recorded on their claim page; this corpus has audited no formal proof of the statement, so formalized is not listed here.

Not covered. The authors' conjecture for progressions with all moduli at least $k$, studied by Simpson, and the variants with distinct moduli are separate questions; the theorem has no finer content beyond the sharp threshold $2^r$.