Minimal least common multiple of a distinct covering system with given minimum modulus: the open values $L_{\min}(m)$ for $m\geq 8$

A covering system of the integers is a finite collection of residue classes $a_i \pmod{m_i}$ whose union is all of $\mathbb{Z}$; it is distinct when all moduli are different. For each integer $m\geq 2$, let $L_{\min}(m)$ be the smallest possible least common multiple of the moduli of a distinct covering system whose minimum modulus is exactly $m$. The exact values are known for $m=3,\dots,7$: $L_{\min}(3)=120$, $L_{\min}(4)=360$ (Dalton and Trifonov), $L_{\min}(5)=1440$, $L_{\min}(6)=5040$ (Klein), and $L_{\min}(7)=10080$ (Zhang and Zhang, 2026, disproving Klein's conjectured value 15120). The problem considered here is the continuation of this sequence at its natural generality: determine $L_{\min}(m)$ for $m\geq 8$ — the next case being $L_{\min}(8)$, for which the best published information is the upper bound $L_{\min}(8)\leq 50400$ from Krukenberg's construction, recorded in the introduction of Dalton and Trifonov, with no nontrivial lower bound published.

Background

A covering system is a finite set of arithmetic progressions $a \pmod{m}:=\{a+km:k\in\mathbb{Z}\}$, called residue classes, such that every integer belongs to at least one of them; the system is distinct if no two of its moduli $m_i$ are equal. Paul Erdős introduced covering systems in 1950 in connection with Romanoff's question on integers of the form $2^k+p$ with $p$ prime, and two quantitative threads have driven the field since. The first asks how large the minimum (i.e., smallest) modulus of a distinct covering system can be: Erdős conjectured it could be arbitrarily large, and Hough disproved this by showing the minimum modulus of any distinct covering system is bounded by an absolute constant (at most $10^{16}$ in the published version), while explicit constructions are known with minimum modulus up to 42 (as recounted in the introductions of the papers of Dalton–Trifonov, Klein, and Zhang–Zhang cited below). The second, more refined thread — the one this problem belongs to — fixes the minimum modulus $m$ and asks how small the least common multiple (LCM) of all moduli can be. Writing $L_{\min}(m)$ for the minimum of $\mathrm{lcm}(m_1,\dots,m_n)$ over all distinct covering systems with $\min_i m_i=m$, the known values are $L_{\min}(3)=120$ and $L_{\min}(4)=360$, proved by Dalton and Trifonov (2022), and $L_{\min}(5)=1440$ and $L_{\min}(6)=5040$, proved by Klein (2026). Two structural facts make these determinations tractable and set the conventions for any answer. First, if the moduli have LCM equal to $L$, the covering condition need only be checked on the finite set $\mathbb{Z}/L\mathbb{Z}$ of residues modulo $L$, and every modulus must divide $L$. Second, the divisor-completion lemma: any divisor $d\geq m$ of $L$ missing from the system may be added with an arbitrary residue without destroying the covering property or changing the LCM, so when ruling out a candidate $L$ one may assume all divisors of $L$ that are at least $m$ are used. Nonexistence proofs in the recent literature therefore proceed by enumerating candidate values $L$ and certifying infeasibility, via reciprocal-sum density obstructions ($\sum_i 1/m_i\geq 1$ is necessary since each class $a_i \pmod{m_i}$ covers density $1/m_i$) and by integer programming (IP) — finite feasibility models with one binary variable per candidate residue class, solved exactly by solvers such as Gurobi. The case $m=7$ illustrates both the state of the art and the difficulty: Klein constructed a system with minimum modulus 7 and LCM 15120 and conjectured this was minimal, while Churchhouse's earlier example had LCM 30240; in 2026 Zhang and Zhang gave a 66-congruence system with LCM 10080 and proved no smaller LCM is possible, using a chain of filters and complete Gurobi computations. Their proof highlights the main technical obstacle: dense near-covers, systems that cover 96–98% of residues modulo a candidate $L$ (they exhibit such partial covers at $L=5040$ and $L=7560$) without being genuine coverings, which force the computational certification to be tight. For $m=8$, the same introduction of Dalton–Trifonov records the historical constructions: Churchhouse's system with LCM $2^4\cdot 3^3\cdot 5^2\cdot 7=75600$ and Krukenberg's with LCM $2^5\cdot 3^2\cdot 5^2\cdot 7=50400$. In parallel, Harrington, Klein, Lowrance, and Trifonov (2026) classified which pairs (minimum modulus $m$, LCM $2^a3^b5^c$) with no prime divisors of the LCM beyond $2,3,5$ are realizable, completely for $m=2,\dots,6$ (with one exceptional family open), showed that such smooth LCMs force $m\leq 9$, and constructed a distinct covering system with minimum modulus 8 and smooth LCM $2^8\cdot 3^3\cdot 5^2=172800$.

Nothing is known to determine $L_{\min}(8)$: the best published upper bound is Krukenberg's $L_{\min}(8)\leq 50400$, no nontrivial lower bound has been published, and the smooth-LCM classification of Harrington–Klein–Lowrance–Trifonov does not apply to candidate LCMs divisible by 7 or other primes beyond $2,3,5$. Nor is any value $L_{\min}(m)$ for $m\geq 9$ known.

Problem Statement

Determine $L_{\min}(m)$, the smallest possible least common multiple of the moduli of a distinct covering system of the integers with minimum modulus exactly $m$, for every $m\geq 8$ for which such covering systems exist (by Hough's theorem this is a finite range of $m$) — the next open case being $m=8$, where the best published upper bound is $L_{\min}(8)\leq 50400$ (Krukenberg's construction, as recorded by Dalton and Trifonov) and no nontrivial lower bound has been published. No restriction is imposed on the number of congruences or on the moduli beyond distinctness and minimum modulus $m$, and any answer must respect Hough's boundedness of the minimum modulus, which limits the $m$ for which the quantity is defined; with explicit constructions known up to minimum modulus 42, the concrete open window is $8\leq m\leq 42$. The objective is the exact continuation of the known sequence $120, 360, 1440, 5040, 10080$ for $m=3,\dots,7$.

References

1. Jiheng Zhang and Shiliang Zhang, 'A Distinct Covering System with Minimum Modulus 7 and Minimal Least Common Multiple 10080', arXiv:2607.19029 (2026), https://arxiv.org/abs/2607.19029
2. Jonah Klein, 'On a Conjecture of Krukenberg and a Problem of Dalton and Trifonov', Integers: Electronic Journal of Combinatorial Number Theory 26 (2026), Article A38, DOI: 10.5281/zenodo.19402698, https://zenodo.org/records/19402698
3. Jack Dalton and Ognian Trifonov, 'Extreme Covering Systems of the Integers', arXiv:1905.07386 (2019; published as 'Extreme covering systems', Journal of Integer Sequences 25 (2022), Article 22.9.1), https://arxiv.org/abs/1905.07386, https://cs.uwaterloo.ca/journals/JIS/VOL25/Trifonov/trif3.html
4. 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
5. Bob Hough, 'Solution of the minimum modulus problem for covering systems', arXiv:1307.0874 (2013; published in Annals of Mathematics 181 (2015), no. 1, 361–382), https://arxiv.org/abs/1307.0874