Nonexistence of distinct covering systems with all moduli in $[n,12n]$ for $n \geq 4$ (the Dalton–Jones interval problem)

Dalton and Jones (arXiv:2506.11359, 2025) proved that for every integer $n \geq 3$ there is no distinct covering system whose moduli all lie in the interval $[n,10n]$, extending a line of results ([n,6n] in 2006, [n,8n] in 2022, [n,9n] in 2023). As the concrete next target of this program — a suggestion attributed to M. Filaseta — they pose the open question of whether the same holds with ratio 12: prove that for every integer $n \geq 4$ no distinct covering system has all moduli in $[n,12n]$, with the analogous statement for $[n,15n]$ and $n \geq 5$ posed as a more ambitious further extension. As of August 2026, the question is open. It can be settled in either direction: by a proof of nonexistence, or by an explicit distinct covering witness with all moduli in $[n,12n]$ for some $n \geq 4$.

Background

A covering system is a finite set of congruences $x \equiv a_i \pmod{n_i}$, $i = 1, \dots, k$, such that every integer satisfies at least one of them; it is called a distinct covering when the moduli are all different and larger than 1, written $1 < n_1 < n_2 < \dots < n_k$. Covering systems were introduced by Erdős in 1950 in work on integers of the form $2^k + p$. Two classical threads concern how restrictive distinctness is. First, the minimum modulus problem: Erdős conjectured that the smallest modulus of a distinct covering cannot be arbitrarily large; Hough (2015) proved this by showing the minimum modulus is at most $10^{16}$, a bound improved to 616000 by Balister, Bollobás, Morris, Sahasrabudhe and Tiba (2022). Second, interval problems: Erdős and Graham (1980) studied systems whose moduli all lie in $[n, cn]$ for a fixed constant $c$ and conjectured that for each $c$ the uncovered integers have positive density once $n$ is large; Filaseta, Ford, Konyagin, Pomerance and Yu (2007) proved that for every ratio $K > 1$ there is no distinct covering whose moduli all lie in $(N, KN]$ once $N$ is sufficiently large depending on $K$. These results leave entirely open what happens for small starting points $n$ at a fixed ratio $K$. A specific line of work addresses exactly this: in a 2006 research seminar, Filaseta, Trifonov and Yu showed there is no distinct covering with all moduli in $[n,6n]$ for $n \geq 3$ (unpublished; documented in Dalton–Trifonov 2022); Dalton and Trifonov (2022) improved the ratio to 8; Dalton's 2023 dissertation improved it to 9 (reported in [1]); and Dalton and Jones (2025) proved the ratio-10 case: for every $n \geq 3$ there is no distinct covering system with all moduli in $[n,10n]$. Their method combines two ingredients. One is a necessary density condition: since the congruence classes of a covering must cover all integers, the sum of reciprocals of the moduli, $\sum_i 1/n_i$, must be at least 1; for most $n$ the reciprocal sum over those moduli in $[n,10n]$ whose largest prime factor is below $\sqrt{9n+1}$ is below 1, which settles those $n$ immediately. The other is a set of 'reduction' lemmas in the tradition of Krukenberg's 1971 dissertation — rules showing that certain congruences can be discarded from a covering without destroying the covering property (for example: if a prime power $p^a$ divides the least common multiple of all moduli and fewer than $p$ congruences have moduli divisible by $p^a$, all of them can be discarded) — applied with computer-assisted case analysis to the exceptional $n$ where the density condition fails. The paper also flags the distortion method of Balister et al. — a reweighting of the density argument in which moduli carry unequal weights so the same counting obstruction yields more information — as a possible strengthening. For ratio 12 the density condition fails for many more values of $n$, and the authors state that their current method 'is undoubtedly not optimal' for the larger casework. In Section 4 they pose, attributing the target to private communication with Filaseta, that it 'would be nice to show' there is no distinct covering with all moduli in $[n,12n]$ for $n \geq 4$ — note the starting point shifts from $n \geq 3$ to $n \geq 4$ — and name $[n,15n]$ for $n \geq 5$ as the next interesting interval, left for further work.

Problem Statement

Determine whether there exists an integer $n \geq 4$ and a distinct covering system — a finite set of congruences $x \equiv a_i \pmod{n_i}$ with distinct moduli $1 < n_1 < \dots < n_k$, covering every integer — whose moduli all lie in the interval $[n, 12n]$.

As posed by Dalton and Jones (2025), following a suggestion of M. Filaseta, the expected answer is negative: prove that for every integer $n \geq 4$ there is no distinct covering system with all moduli in $[n, 12n]$. The problem can be settled in either direction: a rigorous proof of the nonexistence statement, or an explicit integer $n \geq 4$ together with an explicit finite list of congruences with distinct moduli in $[n, 12n]$ whose union is all of the integers.

The same source poses, as a strictly more ambitious and separate further goal that is not required for resolving the question above, the analogous statement with ratio 15: that there is no distinct covering system with all moduli in $[n, 15n]$ for any $n \geq 5$.

References

1. Jack Dalton and Nic Jones, On the intervals for the non-existence of covering systems with distinct moduli, arXiv:2506.11359 (2025), https://arxiv.org/abs/2506.11359
2. Jack Dalton and Ognian Trifonov, Extreme Covering Systems, Journal of Integer Sequences 25 (2022), Article 22.9.1, https://cs.uwaterloo.ca/journals/JIS/VOL25/Trifonov/trif3.html
3. Bob Hough, Solution of the minimum modulus problem for covering systems, Annals of Mathematics 181 (2015), DOI: 10.4007/annals.2015.181.1.6, arXiv:1307.0874, https://doi.org/10.4007/annals.2015.181.1.6
4. P. Balister, B. Bollobás, R. Morris, B. Sahasrabudhe, M. 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, https://doi.org/10.1007/s00222-021-01087-5
5. M. Filaseta, K. Ford, S. Konyagin, C. Pomerance, G. Yu, Sieving by large integers and covering systems of congruences, Journal of the American Mathematical Society 20 (2007), 495–517, DOI: 10.1090/s0894-0347-06-00549-2, https://doi.org/10.1090/s0894-0347-06-00549-2
6. Pál Erdős, On integers of the form 2^k+p and some related problems, Summa Brasil. Math. 2 (1950), 113–123, Zbl 0041.36808, https://zbmath.org/?q=an:0041.36808
7. Paul Erdős and Ronald L. Graham, Old and new problems and results in combinatorial number theory, Monographie No. 28 de L'Enseignement Mathématique, Genève (1980), Zbl 0434.10001, https://zbmath.org/?q=an:0434.10001
8. Michael Filaseta and Alexandros Kalogirou, Covering systems with the sum of the reciprocals of the moduli close to 1, Transactions of the American Mathematical Society 379 (2026), 4945–4972, DOI: 10.1090/tran/9670, arXiv:2407.15280, https://arxiv.org/abs/2407.15280
9. Jonah Klein, On a conjecture of Krukenberg and a problem of Dalton and Trifonov, arXiv:2508.18062 (2025), https://arxiv.org/abs/2508.18062
10. Joshua Harrington, Jonah Klein, Joshua Lowrence, 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
11. Ibrahim Mian and Shayaan Siddique, Kernel-Checked Exclusions for the Erdős-Selfridge Odd Covering Problem: Any Odd Covering of Z Has lcm Exceeding 10000, arXiv:2607.25628 (2026), https://arxiv.org/abs/2607.25628