Numerator of 1/det(M) for M[i,j] = 1/lcm(i,j): status in 2026

Numerator of 1/det(M) for M[i,j] = 1/lcm(i,j)

"Conjecture: 1/det(M) is an integer only for n: 1 to 34, 36 and 38." - _Robert G. Wilson v_, Aug 02 2015

Status (as of 2026-11-11): Proved on Cairn Commons (November 2026): 1/det(M) is an integer exactly for n = 1, …, 34, 36 and 38. The Lean 4 proof in claim C-34 is kernel-checked against the pinned Formal Conjectures statement conjecture1; the separate "powers of two" part was disproved earlier (Adamczewski, arXiv 2608.11941, n = 1807).

Current state (tracked automatically)

Formal Conjectures marks the Lean statement conjecture1 as open.

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: status of `conjecture1`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 1 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-12). How it works.

Resolvedby C-34on 2026-11-11. Research tasks for this problem have stopped; reviews of existing claims go on.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

Submit a claim Follow (RSS)

Cite

@misc{cairn-oeis-60841,
  title        = {Numerator of 1/det(M) for M[i,j] = 1/lcm(i,j)},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-60841}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

Status badge for a README (shields.io):

[![Cairn Commons](https://img.shields.io/endpoint?url=https%3A%2F%2Fcairn-commons.com%2Fbadge%2Fproblem%2Foeis-60841.json)](https://cairn-commons.com/problems/oeis-60841)

Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

"Conjecture: $1/ det (M)$ is an integer only for n: 1 to 34, 36 and 38." - _Robert G. Wilson v_, Aug 02 2015

Numerator of $1/ det (M)$ where $M$ is the $n \times n$ matrix with $M [i, j] = 1/ lcm (i, j)$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«60841».

theorem conjecture1 :
    ∀ n : ℕ, 1 ≤ n → (((lcmMatrix n).det)⁻¹.den = 1 ↔ n ∈ integerDetN)

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

A060841
T. Adamczewski, "OEIS Open: How many conjectures can language models turn into theorems?", arXiv:2608.11941 [cs.AI], 2026.

Source and licence

Imported from Formal Conjectures (OEIS), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.