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Claim. If a finite covering system has pairwise distinct squarefree moduli greater than one, then its smallest modulus is at most $118$. This is Theorem 1.1 of M. Cummings, M. Filaseta and O. Trifonov, An upper bound for the minimum modulus in a covering system with squarefree moduli, Acta Math. Hungar. 175 (2025), 1--25. The paper builds on the distortion method of Balister, Bollobás, Morris, Sahasrabudhe and Tiba, whose general bound $616000$ is on their claim page; the squarefree hypothesis lets the sieve run over primes alone and gives the much smaller bound. The paper's second result, that the $k$-th smallest modulus of a covering system with distinct moduli is bounded by an absolute constant when that modulus is needed for the covering, is outside the question. Sun's lecture of 6 May 2026, cited on the problem page, distinguishes this squarefree bound from the general threshold.

Covers. The case of Problem 2 in which every modulus is squarefree: no covering system with distinct squarefree moduli greater than one has smallest modulus above $118$. The unrestricted question is settled on Hough's page and the page of Balister, Bollobás, Morris, Sahasrabudhe and Tiba linked above; this page sharpens the bound for the squarefree class only, and the largest attainable squarefree minimum modulus is not identified.

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

Acceptance. Refereed: Acta Mathematica Hungarica 175 (2025), 1--25, doi:10.1007/s10474-024-01496-x; the arXiv v1 of 2022-11-15 names this page. Not reviewed: the site's page for the problem does not cite the paper. Not formalized: no Lean proof of the theorem is recorded. The proof is not checked in this corpus, and the library has no card for the paper.