Lower bounds for the minimal least common multiple of a distinct covering system with minimum modulus m ≥ 8
Abstract
A covering system is distinct when its moduli are pairwise different, and Lₘᵢₙ(m) denotes the least value of the least common multiple of the moduli over all distinct covering systems of ℤ whose smallest modulus is m. The values Lₘᵢₙ(3)=120, Lₘᵢₙ(4)=360, Lₘᵢₙ(5)=1440 and Lₘᵢₙ(6)=5040 are known, and Lₘᵢₙ(7)=10080 has recently been claimed; above m=7 the literature records only upper bounds, and the case m=8 has been posed as a problem for which integer programming is "currently not computationally feasible". We prove Lₘᵢₙ(8) ≥ 7200, Lₘᵢₙ(9) ≥ 7920, Lₘᵢₙ(10) ≥ 10080, Lₘᵢₙ(11) ≥ 27720, Lₘᵢₙ(12) ≥ 25200, which together with Krukenberg's construction sandwiches Lₘᵢₙ(8) between 7200 and 50400. Every published value that a bound consumes is carried as an explicit hypothesis. The engine is an exclusion certificate in exact integer arithmetic: a recursive sharpening of the reciprocal-sum bound along coprime splittings of the period, with no solver anywhere. The same certificate excludes twelve of the eighteen candidates for Lₘᵢₙ(7) without an integer program, and shows that every covering of ℤ by distinct odd moduli greater than 1 has least common multiple at least 45045, closing three cells left open by the previously certified bound of 10⁴. All statements are machine-checked in Lean 4.
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Archived files
- Version 1 · current (opens in a new tab)
Source snapshot 2026-08-30 15:34 UTC
File fingerprint
e6ca15ea3a1a5bced95919ffc5af0cad94d2799e41807b9bca1f09611baea4d4
Claim ledger
Stated results
cl-01known2026-08-28
Covering systems as a Finset of moduli, the reduction to one period, and the density (reciprocal-sum) bound in integer form
cl-02routine2026-08-28
The recursive Chinese-remainder split bound, and the soundness of the resulting decidable certificate
cl-03routine2026-08-28
Divisor descent (adjoining 0 mod k transfers a lower bound from k to m), and the sweep that assembles a lower bound from per-cell certificates
cl-04known data2026-08-28
The 66-class system of arXiv:2607.19029 Section 6 covers Z/10080Z, so Lₘin(7) <= 10080 and Klein's Conjecture 2 is false
cl-05known data2026-08-28
The source's own filter output recomputed from its definitions: the 18-element list L₇, the 9-element list below 5040, the 58 divisors M₉240, and the exact rational 877/4620
cl-06known2026-08-28
Twelve of the eighteen sum-filter candidates for Lₘin(7) are excluded by exact integer arithmetic with no solver: 5544, 5880, 6048, 6552, 6930, 7056, 7392, 8064, 8568, 8820, 9072, 9576
cl-07routine2026-08-28
The reach of the certificate on the Lₘin(7) problem: over all 720 multiples of 7 in [5040, 10080) it leaves exactly six survivors, and below 5040 exactly two
cl-08known2026-08-28
Lₘin(7) >= 5040 from Klein's Lₘin(6) = 5040, and exactly which six cells remain for Lₘin(7) >= 10080
cl-09known2026-08-28
Any covering of Z by distinct odd moduli > 1 has lcm at least 45045; in particular the three cells 10395, 12285, 17325 that arXiv:2607.25628 records as beyond its certificate
cl-10routine2026-08-28
The odd frontier: below 200000 the certificate leaves exactly five values, 45045, 135135, 155925, 176715, 197505
cl-11routine2026-08-28
The classical distinct covering system with lcm 12, and the certificate's soundness and limit controls
cl-12candidate2026-08-28
First lower bounds for Lₘin(m), m >= 8, resting on Klein's peer-reviewed theorem alone: Lₘin(8) >= 5040, Lₘin(9) >= 5040, Lₘin(10) >= 10080
cl-13candidate2026-08-28
Lower bounds for Lₘin(m), m >= 8, using the full published chain including Lₘin(7) = 10080: Lₘin(8) >= 7200, Lₘin(9) >= 7920, Lₘin(11) >= 27720, Lₘin(12) >= 25200
cl-14routine2026-08-28
Negative controls
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- A covering system is a finite set of congruences x ≡ aᵢ (mod mᵢ), mᵢ ≥ 2, satisfied by every integer; it is distinct when the moduli are pairwise different. Two families of questions about the least common multiple L of the moduli meet in this family, because one certificate settles cells of both.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7