Closed-form nonexistence certificates for odd covering systems

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Ege Öztürk, Closed-form nonexistence certificates for odd covering systems, preprint, version 2 (related work updated), August 2026, DOI 10.13140/RG.2.2.13216.55049.

The edition read is the eight-page version 2 PDF, obtained from the preprint's ResearchGate record on 2026-10-16. It is not peer reviewed.

For odd $M$, let $u(M)$ be the least number of residues modulo $M$ left uncovered when one residue class is chosen for every divisor $d > 1$ of $M$; a covering system with distinct odd moduli dividing $M$ exists exactly when $u(M) = 0$. The preprint builds closed-form certificates that $u(M) > 0$ (a peel threshold, a first-order grid volume certificate from the Chinese remainder structure of the survivors of the prime-power classes, and a second-order refinement through a matching of coprime-support box pairs), each evaluable in exact rational arithmetic without search. Its main claimed application is that no covering system with distinct odd moduli greater than one has least common multiple below $11{,}486{,}475$. It also claims infinite families of covering-free moduli, the exact value $u(qr) = qr - q - r$ for distinct odd primes $q$ and $r$, and an account of where certificates of this type must stop short of the full question.

Bears on. An author claim of partial progress on Problem 7, the existence of a covering system with distinct odd moduli. Read status: unread; only the abstract and the first page of the introduction were read on 2026-10-16, no statement is checked, and no proof is verified on this card.

No file of this source is held: no license on record permits its redistribution. The PDF prints no license or copyright notice in its text, read 2026-10-16, and the DataCite record for the DOI, read 2026-10-16, lists no rights; the ResearchGate page was not read, so the term is unstated.