Covering systems where the prime divisors of all moduli are only 2, 3, or 5
Pith reviewed 2026-05-20 08:02 UTC · model grok-4.3
The pith
If the least common multiple of the moduli is a power of 2, 3 or 5, then any distinct covering system has minimum modulus at most 9.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
If the least common multiple of the moduli in a distinct covering system is divisible only by the primes 2, 3 and 5, then the smallest modulus m satisfies m ≤ 9. Moreover, such a system exists when m=8, a=8, b=3, c=2. For all smaller values of m the paper supplies either an explicit covering or a proof that none exists (except the unresolved case m=6, b=c=1).
What carries the argument
A new upper bound on the density of the union of residue classes in a system of congruences, used together with an integer-programming encoding that decides existence of a covering with prescribed minimum modulus and prescribed 2-3-5-smooth period.
If this is right
- No distinct covering systems exist for any m > 9 when the least common multiple of the moduli is 2-3-5-smooth.
- All valid quadruples with m ≤ 6 are completely listed except the single open case m=6, b=c=1.
- An explicit covering exists for the quadruple (8,8,3,2).
- Nonexistence proofs for the remaining small-m cases rely either on exhaustive integer-program search or on the new density estimate.
Where Pith is reading between the lines
- The same density technique might be applied to moduli whose prime factors come from any fixed finite set of small primes.
- The explicit construction for m=8 suggests that the transition from possible to impossible coverings occurs sharply near m=9.
- Similar integer-programming formulations could be used to search for coverings with larger but still bounded m when the smoothness restriction is relaxed.
Load-bearing premise
The density estimate is tight enough to exclude every candidate quadruple with m greater than 9, and the integer program encodes the covering condition without missing feasible solutions.
What would settle it
Discovery of a distinct covering system whose smallest modulus is 10 or larger and whose moduli have least common multiple equal to 2^a 3^b 5^c for some a, b, c would falsify the claimed bound.
read the original abstract
We try to find all quadruples of positive integers $(m,a,b,c)$ with $a \geq b \geq c$ such that there exists a distinct covering system with minimum modulus $m$ and least common multiple of the moduli $2^a 3^b 5^c$. We obtain complete description of all such quadruples when $m=2,3,4,5$, or $6$, except when $m=6$ and $b=c=1$. We also show that if the LCM of the moduli has only $2$, $3$, or $5$ as prime divisors, then $m \leq 9$ and construct a distinct covering system with $m=8$, $a=8$, $b=3$, and $c=2$. When a covering system exists for a quadruple $(m,a,b,c)$ we provide an example. Nonexistence of covering systems is established via integer programming or by using a new estimate on the density of a set covered by a system of congruences.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper classifies all quadruples (m,a,b,c) with a≥b≥c such that a distinct covering system exists with minimum modulus m and LCM of the moduli equal to 2^a 3^b 5^c. It gives a complete description for m=2,3,4,5,6 (except the case m=6, b=c=1), proves that m≤9 whenever the LCM has only prime factors 2,3,5, constructs an explicit example for m=8,a=8,b=3,c=2, and proves non-existence in the remaining cases by integer programming or a new density estimate on the measure of the union of the congruences.
Significance. If the central claims hold, the work supplies a detailed classification and an absolute bound m≤9 for covering systems whose moduli are supported only on the primes 2,3,5. The explicit constructions and the computational verification for small m are concrete contributions; the new density estimate, if sharp, would be a useful technical tool for similar restricted-modulus problems.
major comments (2)
- [density estimate section] The section presenting the density estimate (used to rule out all m>9): the manuscript invokes a new upper bound on the density of the union to conclude that no covering exists for any quadruple with m>9. The precise statement of the estimate, the proof that it is strictly less than 1 for every admissible choice of moduli with minimum modulus >9, and verification that the bound remains valid when high powers of 2, 3 or 5 appear together should be supplied explicitly; any looseness in the estimate for particular exponent combinations would leave open the possibility of a covering with m≥10.
- [computational verification section] The integer-programming formulation for the small-m cases: the encoding of the existence question as an integer program must be shown to be complete (i.e., that every possible system of congruences with the given LCM is captured by the variables and constraints). If the solver returns “no solution” for certain quadruples, the manuscript should record the size of the search space or the timeout parameters used.
minor comments (2)
- Clarify the precise definition of “distinct covering system” at the first appearance; the current wording leaves open whether the residues must be distinct modulo each modulus or only that the moduli themselves are distinct.
- In the table or list of explicit constructions, add a short verification that each listed system is indeed a covering (i.e., every integer satisfies at least one congruence).
Simulated Author's Rebuttal
We thank the referee for the thorough reading and constructive feedback on our manuscript. We address each major comment below and will incorporate the requested clarifications and additions into the revised version.
read point-by-point responses
-
Referee: [density estimate section] The section presenting the density estimate (used to rule out all m>9): the manuscript invokes a new upper bound on the density of the union to conclude that no covering exists for any quadruple with m>9. The precise statement of the estimate, the proof that it is strictly less than 1 for every admissible choice of moduli with minimum modulus >9, and verification that the bound remains valid when high powers of 2, 3 or 5 appear together should be supplied explicitly; any looseness in the estimate for particular exponent combinations would leave open the possibility of a covering with m≥10.
Authors: We agree that additional explicit details are needed in the density estimate section. In the revised manuscript we will state the new upper bound on the measure of the union in full, include a complete self-contained proof that the bound is strictly less than 1 for every admissible quadruple with minimum modulus greater than 9, and verify that the estimate holds uniformly when high powers of 2, 3 or 5 occur simultaneously. These additions will close any potential gaps for m ≥ 10. revision: yes
-
Referee: [computational verification section] The integer-programming formulation for the small-m cases: the encoding of the existence question as an integer program must be shown to be complete (i.e., that every possible system of congruences with the given LCM is captured by the variables and constraints). If the solver returns “no solution” for certain quadruples, the manuscript should record the size of the search space or the timeout parameters used.
Authors: We accept that the integer-programming encoding requires an explicit completeness argument. In the revision we will add a subsection demonstrating that the chosen variables and constraints enumerate every possible distinct covering system whose moduli have the prescribed LCM. For all quadruples where the solver reports no solution we will also record the size of the search space explored and the timeout parameters that were employed. revision: yes
Circularity Check
No significant circularity detected; derivation relies on independent density bound and explicit computation
full rationale
The paper establishes nonexistence for m>9 via a new density estimate on unions of congruences and uses integer programming for small cases. Neither step reduces to a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The density bound is presented as a fresh analytic tool sufficient to exclude candidates, while the m=8 example is constructed explicitly. No equation or claim equates the target bound to its own inputs by construction, and external verification via computation keeps the argument self-contained against the stated assumptions.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Standard axioms of arithmetic and the correctness of integer linear programming over the integers
Reference graph
Works this paper leans on
-
[1]
P. Balister, B. Bollob´ as, R. Morris, J. Sahasrabudhe, and M. Tiba, On the Erd˝ os cov- ering problem: the density of the uncovered set, Invent. Math.228(2022), 377–414
work page 2022
-
[2]
J. Dalton and O. Trifonov, Extreme covering systems, J. of Int. Seq.,25, 22.9.1, 2022
work page 2022
-
[3]
M. Filaseta, K. Ford, S. Konyagin, C. Pomerance, and G. Yu, Sieving by large integers and covering systems of congruences, J. Amer. Math. Soc.,20(2007), 495–517
work page 2007
-
[4]
Klein, On a conjecture of Krukenberg and a problem of Dalton and Trifonov, arXiv, 2508.18062, 2025
J. Klein, On a conjecture of Krukenberg and a problem of Dalton and Trifonov, arXiv, 2508.18062, 2025
-
[5]
C. E. Krukenberg, Covering sets of the integers,Ph.D. dissertation, University of Illinois, Urbana-Champaign, 1971
work page 1971
-
[6]
N. McNew and J. Setty, On the densities of covering numbers and abundant numbers, arXiv, 2507.23041, 2025. Department of Mathematics, Cedar Crest College, Allentown, Pennsyl- vania 18104, USA Email address, Joshua HarringtonJoshua.Harrington@cedarcrest.edu 18 Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208, USA Emai...
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.