Sharp values of the k-th smallest modulus bound B(k) for covering systems with distinct moduli, and Selfridge's problem on avoiding the modulus 2 in the squarefree case

A covering system is a finite collection of residue classes $a_j \pmod{m_j}$ whose union is all of the integers. Hough resolved Erdős's minimum modulus problem by proving that the least modulus of a covering system with distinct moduli is bounded, and Cummings, Filaseta and Trifonov proved that for every fixed $k$ the $k$-th smallest modulus — whenever it is genuinely needed for the covering — is bounded by an absolute constant $B(k)$, while in the squarefree-modulus case the minimum modulus is at most 118. The quantitative questions left open by these authors are: the smallest possible values of $B(k)$ in the general and the squarefree settings, effective explicit bounds, the related classification program for distinct covering systems, and the accompanying Selfridge problem, open since at least 1981, of whether a covering system with distinct squarefree moduli can avoid the modulus 2 entirely. The only general quantitative refinement is an explicit but far-from-sharp bound $\exp(c\,j^2/\log(j+1))$ due to Klein, Koukoulopoulos and Lemieux, no value of $B(k)$ or of its squarefree analogue is known sharply, and Selfridge's question remains open.

Background

A covering system (introduced by Erdős in 1950) is a finite set of congruences $x \equiv a_j \pmod{m_j}$, $1 \le j \le r$, with integer moduli $m_j > 1$, such that every integer satisfies at least one of them; for example $0 \pmod 2$, $0 \pmod 3$, $1 \pmod 4$, $3 \pmod 8$, $7 \pmod{12}$, $23 \pmod{24}$ covers the integers with distinct moduli. Erdős asked whether the minimum modulus of a covering system whose moduli are pairwise distinct can be arbitrarily large, believed it could, and offered $1000 for a resolution; the question is recorded in Guy's Unsolved Problems in Number Theory (section F13). Hough (Annals of Mathematics, 2015) proved the opposite: the minimum modulus of a distinct covering system is at most $10^{16}$. Balister, Bollobás, Morris, Sahasrabudhe and Tiba (BBMST) later re-proved this by a simpler and more versatile weighted second-moment argument (the distortion method) with the improved bound $615999$, alongside resolutions of several neighboring conjectures. In the opposite direction, the largest minimum modulus in a published construction of a distinct covering is $40$ (Nielsen, 2009), so the sharp constant is unknown to within four orders of magnitude.

A modulus is squarefree if it is not divisible by the square of any prime. Erdős gave a covering with $14$ distinct squarefree moduli $2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105$ whose minimum modulus is $2$, and Selfridge asked (a problem cited in 1981 in Guy's book, section F13) whether a covering with distinct squarefree moduli can have minimum modulus greater than $2$, i.e. avoid the modulus $2$ altogether. BBMST proved that a covering with distinct squarefree moduli must contain some even modulus, but even squarefree moduli such as $6, 10, 14$ are not excluded, so whether the modulus $2$ itself is forced remains open. Refining the BBMST exposition, Cummings, Filaseta and Trifonov (2022; published 2025) showed that every covering with distinct squarefree moduli has minimum modulus at most $118$ (their Theorem 1.1), and, as a consequence of Hough's work, that for every fixed $k \ge 0$ there is a constant $B(k+1)$ such that any covering with distinct moduli $m_1 < m_2 < \cdots < m_r$ with $r \ge k+1$, in which the congruences having the $k$ smallest moduli do not by themselves already cover the integers, must satisfy $m_{k+1} \le B(k+1)$ (their Theorem 1.2; $k=0$ recovers Hough's theorem, so $B(1)$ is exactly the sharp minimum modulus constant). They state that their proof of Theorem 1.2 can be made constructive but deliberately make no attempt at explicit bounds, and they explicitly ask: how small can $B(k)$ be made, what are the answers when the moduli are required squarefree, and can bounds of this shape support a complete classification of all distinct covering systems, given that the total number of moduli in a covering can be arbitrarily large. The only published general quantitative refinement so far is due to Klein, Koukoulopoulos and Lemieux (2024), who showed that the $j$-th smallest modulus of a minimal covering system (one in which no congruence can be removed) with distinct moduli is at most $\exp(c\,j^2/\log(j+1))$ for an absolute constant $c > 0$ — explicit, but very far from sharp, and saying nothing about the squarefree regime.

Problem Statement

For each integer $k \ge 1$, let $B(k)$ denote the least constant such that every covering system with distinct moduli $m_1 < m_2 < \cdots < m_r$, $r \ge k$, in which the congruences having the $k-1$ smallest moduli do not by themselves already cover the integers, satisfies $m_k \le B(k)$ (existence of such a finite $B(k)$ is Cummings–Filaseta–Trifonov's Theorem 1.2, a consequence of Hough's theorem; the hypothesis holds in particular for every level of a minimal covering, i.e. one in which no congruence can be removed). The common objective of the problem, as raised by that paper, is to determine how small these constants can be — quantitatively, how tightly the distinctness of the moduli forces the small moduli of a covering to be small:

1. Determine the exact value, or provably sharp asymptotics, of $B(k)$ for every $k \ge 1$ in the unrestricted distinct-moduli setting. Here 'provably sharp asymptotics' means matching upper and lower bounds for $B(k)$ that agree up to a factor $1+o(1)$ as $k \to \infty$. The $k=1$ case, $B(1)$ — the sharp minimum modulus constant of Hough's theorem — is itself open: the largest constructed minimum modulus is 40 (Nielsen) while the best proved upper bound is 615999 (BBMST).
2. Determine the corresponding optimal constants $B_{\mathrm{sf}}(k)$ in the setting where all moduli are required to be squarefree. The $k=1$ case is exactly Selfridge's problem, open since at least 1981: does there exist a covering system with distinct squarefree moduli whose minimum modulus exceeds 2, i.e. avoiding the modulus 2 entirely? (Equivalently: is $B_{\mathrm{sf}}(1) \ge 3$? A negative answer means $B_{\mathrm{sf}}(1) = 2$; the current proved upper bound is 118, and BBMST's result shows some even modulus is necessary without showing the modulus 2 is.) Effective explicit bounds for $B(k)$ and $B_{\mathrm{sf}}(k)$ that improve on the best currently proved bounds (for $k=1$, below $615999$ in the unrestricted case and below $118$ in the squarefree case) are also sought; the source proves Theorem 1.2 constructively in principle but computes no explicit values.
3. Using bounds of this shape, together with the fact that the total number of moduli in a covering can be arbitrarily large, decide whether a complete classification of all covering systems with distinct moduli (and of those with distinct squarefree moduli) is possible; this component is resolved either by carrying out such a classification, or by proving a precise obstruction showing that it cannot be completed.

Determining the exact value, or provably sharp asymptotics, of $B(k)$ (respectively $B_{\mathrm{sf}}(k)$) requires both a construction witnessing a large $k$-th smallest modulus and a matching upper-bound proof. Effective explicit bounds for $B(k)$ and $B_{\mathrm{sf}}(k)$ are a strictly weaker objective than the exact values or the provably sharp asymptotics.

References

1. Maria Cummings, Michael Filaseta, and Ognian Trifonov, 'An upper bound for the minimum modulus in a covering system with squarefree moduli', Acta Mathematica Hungarica 175 (2025). DOI: 10.1007/s10474-024-01496-x; arXiv:2211.08548; https://arxiv.org/abs/2211.08548
2. Bob Hough, 'Solution of the minimum modulus problem for covering systems', Annals of Mathematics (2) 181 (2015), 361–382. DOI: 10.4007/annals.2015.181.1.6; arXiv:1307.0874; https://arxiv.org/abs/1307.0874
3. Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe, and Marius Tiba, 'On the Erdős Covering Problem: the density of the uncovered set', Inventiones Mathematicae 228 (2022), 377–414. DOI: 10.1007/s00222-021-01087-5; arXiv:1811.03547; https://arxiv.org/abs/1811.03547
4. Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe, and Marius Tiba, 'The Erdős-Selfridge problem with square-free moduli', Algebra & Number Theory 15 (2021), 609–626. DOI: 10.2140/ant.2021.15.609; arXiv:1901.11465; https://arxiv.org/abs/1901.11465
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6. Jonah Klein, Dimitris Koukoulopoulos, and Simon Lemieux, 'On the $j$-th smallest modulus of a covering system with distinct moduli', International Journal of Number Theory 20 (2024). DOI: 10.1142/S1793042124500234; arXiv:2212.01299; https://arxiv.org/abs/2212.01299
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8. Joshua Harrington, Yewen Sun, and Wing Hong Tony Wong, 'Covering systems with odd moduli', arXiv:2104.00602 (2021); https://arxiv.org/abs/2104.00602
9. Chris Bispels, Matthew Cohen, Joshua Harrington, Joshua Lowrance, Kaelyn Pontes, Leif Schaumann, and Tony W. H. Wong, 'A further investigation on covering systems with odd moduli', arXiv:2507.16135 (2025); https://arxiv.org/abs/2507.16135
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11. Joshua Harrington, Jonah Klein, Joshua Lowrance, and Ognian Trifonov, 'Covering systems where the prime divisors of all moduli are only $2$, $3$, or $5$', arXiv:2605.18644 (2026); https://arxiv.org/abs/2605.18644
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