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Hong's odd-set LCM nonsingularity conjecture

About 12 years old · traced to

Let SS be a GCD-closed set of odd positive integers, meaning that every element of SS is odd and the greatest common divisor of any two elements of SS belongs to SS. Let [S]=[[xi,xj]][S]=[[x_i,x_j]] be the LCM matrix on SS. Hong's conjecture. The matrix [S][S] is nonsingular. The source states that this conjecture is disproved by a counterexample.

References

Primary source

Mika Mattila, Pentti Haukkanen and Jori Mäntysalo, “Studying the singularity of LCM-type matrices via semilattice structures and their Möbius functions”, arXiv:1403.5389 (2014).

gcd-closed-setslcm-matrices
Posted

Progress summary

Refreshed
Claimed solved

A 2014 preprint gives an explicit nine-number example that allegedly breaks the conjecture, but no independent verification was found.

Hong’s conjecture asserts that the least-common-multiple matrix of every odd GCD-closed set is nonsingular; it was raised by Hong in 2002. A 2014 preprint claims the conjecture fails for a set with 99 elements.

Known results

  • Odd GCD-closed sets with at most 88 elements reportedly have nonsingular LCM matrices (2014 preprint).

2014 claimed counterexample

Theorem 4.24.2 gives S={1,3,5,7,195,291,1407,4025,1020180525}S=\{1,3,5,7,195,291,1407,4025,1020180525\} and computes ΨS,1/N(1020180525)=0\Psi_{S,1/N}(1020180525)=0, claiming that the associated LCM matrix is singular. This is a claimed disproof, not an independently verified result in the retrieved evidence.

Current status (as of September 2026): The conjecture has a reported explicit 99-element counterexample, but its claimed refutation remains unverified here.

Sources

Solutions 0

No solutions have been posted yet.