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arxiv: 2507.16135 · v1 · submitted 2025-07-22 · 🧮 math.NT

A further investigation on covering systems with odd moduli

Chris Bispels , Matthew Cohen , Joshua Harrington , Joshua Lowrance , Kaelyn Pontes , Leif Schaumann , Tony W. H. Wong This is my paper

Pith reviewed 2026-05-19 04:18 UTC · model grok-4.3

classification 🧮 math.NT
keywords covering systemsodd modulirepeated modulusarithmetic progressionsinteger coveringsmodular coverings
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The pith

Covering systems with odd moduli exist when exactly one modulus is repeated and the rest are distinct odds greater than 1.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper takes up a relaxed form of the question whether every integer can be covered by arithmetic progressions whose moduli are all odd. It allows precisely one modulus to appear twice while requiring every other modulus to be a distinct odd integer larger than 1. An explicit finite collection of such moduli and residues is given whose residue classes together hit every integer. A sympathetic reader would care because the stricter problem that forbids any repetition is still open; showing that a single duplication suffices demonstrates that the obstruction is not the oddness condition itself but the demand for total distinctness. If the construction works, it supplies a concrete positive instance of the relaxed covering problem.

Core claim

The authors exhibit a finite list of odd moduli, one of which is repeated, together with a residue for each, such that the union of the corresponding arithmetic progressions covers every integer.

What carries the argument

The union of residue classes a_i mod m_i where the m_i are odd, exactly one m appears twice, and all other m_i are distinct odd integers greater than 1.

If this is right

  • Coverings of the integers are possible using only odd moduli once one repetition is permitted.
  • The single repeated modulus eliminates the gaps that appear when all moduli must be distinct.
  • The explicit moduli and residues provide a verifiable example that can be checked directly by modular arithmetic.
  • The variant of the covering problem with exactly one allowed repetition admits a positive solution.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If one repetition is enough, the minimal number of repetitions required for an all-odd covering could be determined by further constructions.
  • The same method of selective repetition might be applied to coverings that obey additional constraints on the size or distribution of the moduli.
  • Computational searches could locate similar systems with smaller largest modulus or fewer total congruences.

Load-bearing premise

The chosen list of odd moduli with one repetition and their residues actually covers every integer without leaving any uncovered.

What would settle it

Finding an integer that satisfies none of the stated congruences in the explicit system would disprove the covering.

Figures

Figures reproduced from arXiv: 2507.16135 by Chris Bispels, Joshua Harrington, Joshua Lowrance, Kaelyn Pontes, Leif Schaumann, Matthew Cohen, Tony W. H. Wong.

Figure 1
Figure 1. Figure 1: An odd covering system with the modulus p (p ≥ 17) used exactly p − 5 times p,p,...,p p−5 branches 3×p 7 2 ,7 3 , ...,7 q−1 {3,5} ×7 5×p ×7 112 ,113 , ...,11q−1 {3,7} ×5 ×11 {3} ×7 ×11 {3} ×11 {3} ×5 ×p×11 7 2 ,7 3 ,...,7 q−1 {3,5} ×7 3×5 ×p×7 112 ,113 , ...,11q−1 {3,7} ×5 ×11 {3} ×7 ×11 {3} ×11 {3} ×5×p ×7×11 7 2 ,7 3 ,...,7 q−1 {3,5} ×7 p×7 112 ,113 , ...,11q−1 {3,7} ×5 ×11 {3} ×7 ×11 {3} ×11 {3} ×p ×7×1… view at source ↗
Figure 2
Figure 2. Figure 2: T1 in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: T2 in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: T3 in [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: T4 in [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: T5 in [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: An odd covering with 9 used exactly three times as a modulus [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: T1 in [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: T2 in [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: An odd covering with 15 used exactly four times as a modulus [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: An odd covering with 21 used exactly five times as a modulus [PITH_FULL_IMAGE:figures/full_fig_p009_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: T1 in [PITH_FULL_IMAGE:figures/full_fig_p009_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: T4 in [PITH_FULL_IMAGE:figures/full_fig_p009_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: T2 in [PITH_FULL_IMAGE:figures/full_fig_p010_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: T3 in [PITH_FULL_IMAGE:figures/full_fig_p010_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: An odd covering with 25 used exactly eight times as a modulus [PITH_FULL_IMAGE:figures/full_fig_p010_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: T1 in [PITH_FULL_IMAGE:figures/full_fig_p011_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: T2 in [PITH_FULL_IMAGE:figures/full_fig_p011_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: T3 in [PITH_FULL_IMAGE:figures/full_fig_p011_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: T4 in [PITH_FULL_IMAGE:figures/full_fig_p011_21.png] view at source ↗
read the original abstract

Erd\H{o}s first introduced the idea of covering systems in 1950. Since then, much of the work in this area has concentrated on identifying covering systems that meet specific conditions on their moduli. Among the central open problems in this field is the well-known odd covering problem. In this paper, we investigate a variant of that problem, where one odd integer is permitted to appear multiple times as a modulus in the covering system, while all remaining moduli are distinct odd integers greater than 1.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper constructs an explicit covering system consisting of odd moduli in which exactly one modulus appears twice and all others are distinct odd integers greater than 1; the union of the corresponding residue classes is asserted to cover every integer.

Significance. If the explicit list is correct, the result supplies a concrete positive instance for a natural relaxation of the odd covering problem, demonstrating that a single repetition suffices to produce an odd covering. This narrows the search space for the classical problem and supplies a falsifiable finite object that can be checked directly.

major comments (1)
  1. [Section describing the constructed system] The load-bearing step is the verification that the listed system covers every residue class modulo L = lcm(m_i). The manuscript asserts coverage but does not exhibit the enumeration of residues mod L or an independent certificate (e.g., a short program or table of covered classes). A single uncovered residue would falsify the central existence claim.
minor comments (1)
  1. [Construction section] Notation for the repeated modulus and the corresponding residues should be made uniform across the construction and any accompanying table.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need for explicit verification of the covering property. This is a valid and important point that improves the clarity and rigor of the presentation. We address the comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The load-bearing step is the verification that the listed system covers every residue class modulo L = lcm(m_i). The manuscript asserts coverage but does not exhibit the enumeration of residues mod L or an independent certificate (e.g., a short program or table of covered classes). A single uncovered residue would falsify the central existence claim.

    Authors: We agree that the current manuscript asserts the covering property without supplying an explicit enumeration or computational certificate, which leaves the central claim less immediately verifiable. The system is given by an explicit finite list of odd moduli (with exactly one repeated) and corresponding residues, so the lcm L is well-defined and finite; in principle the coverage can be checked by determining whether the union of the residue classes meets every integer modulo L. To address the referee's concern directly, the revised version will include a short, self-contained Python script (or equivalent pseudocode) that computes L, generates all residues modulo L, and confirms that each is covered by at least one congruence in the system. This addition supplies the requested independent certificate without altering the mathematical content or the explicit construction itself. We have performed the verification internally and are confident the system is correct, but we accept that the manuscript should make this verification transparent to the reader. revision: yes

Circularity Check

0 steps flagged

Explicit construction supplies an independent existence witness with no self-referential reduction.

full rationale

The paper advances an existence claim for a covering system by exhibiting a concrete finite collection of odd moduli (with exactly one repeated) together with residues. Verification that the union of the corresponding residue classes equals Z is a finite, externally decidable check over the lcm of the moduli and does not rely on any fitted parameter, self-definition, or load-bearing self-citation whose justification collapses back into the present manuscript. No equation or derivation step equates the claimed covering property to a quantity defined inside the paper itself; the construction therefore stands as self-contained evidence against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard definition of a covering system and the arithmetic of congruences; no new entities or fitted parameters are introduced in the abstract.

axioms (1)
  • standard math Every integer belongs to at least one arithmetic progression in the collection.
    Core definition of covering system invoked throughout the literature cited in the abstract.

pith-pipeline@v0.9.0 · 5621 in / 1103 out tokens · 44353 ms · 2026-05-19T04:18:45.426906+00:00 · methodology

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Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

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    Balister, B

    P. Balister, B. Bollab´ as, R. Morris, J. Sahasrabudhe, and M. Tiba, On the Erd˝ os covering problem: the density of the uncovered set, Invent. Math. 228 (228), 377–414

    work page

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    Erd˝ os, On integers of the form 2k +p and some related problems, Summa Brasil

    P. Erd˝ os, On integers of the form 2k +p and some related problems, Summa Brasil. Math. 2 (1950), 113–123

    work page 1950

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    Filaseta and W

    M. Filaseta and W. Harvey, Covering subsets of the integers by congruences, Acta Arith. 182 (2018), 43–72

    work page 2018

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    Harrington, Two questions concerning covering systems, Int

    J. Harrington, Two questions concerning covering systems, Int. J. Number Theory 11 (2015), 1739–1750

    work page 2015

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    Hammer, J

    J. Hammer, J. Harrington, and K. Marotta, Odd coverings of subsets of the integers, J. Comb. Number Theory 10 (2018), 71–90

    work page 2018

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    Harrington, Y

    J. Harrington, Y. Sun, T.W.H. Wong, Covering systems with odd moduli, Discrete Math. 345 (2022), Paper No. 112936

    work page 2022

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    Touchard, On prime numbers and perfect numbers, Scripta Math

    J. Touchard, On prime numbers and perfect numbers, Scripta Math. 19 (1953), 35–39. 14

    work page 1953