Hong's multiplicative-function conjecture for odd GCD-closed sets

At least 11 years old · documented by 2014

Let $S=\{x_1,\ldots,x_n\}$ be an odd-GCD-closed set, and let $f$ be a completely multiplicative function, meaning $f(ab)=f(a)f(b)$ for positive integers $a,b$ . If $f$ is strictly monotone, then form the matrix $[f[x_i,x_j]]$ with $(i,j)$ -entry $f([x_i,x_j])$ . Hong's conjecture. The matrix $[f[x_i,x_j]]$ is nonsingular. The source says that this conjecture is false, using the function $N^\alpha$ and the paper's counterexamples.

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-sets lcm-matrices

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A 2014 paper gives explicit examples showing the conjecture is false, but the refutation has not been independently verified.

Hong’s conjecture asserts that the LCM matrix formed from a strictly monotone completely multiplicative function is nonsingular for every odd-GCD-closed set. The 2014 paper identifies the conjecture and claims it is false.

2014 counterexample

The paper gives the odd-GCD-closed set $S=\{1,3,5,7,195,291,1407,4025,1020180525\}$ , for which the ordinary LCM matrix is singular. Since $f(n)=n$ is strictly monotone and completely multiplicative, this is a claimed counterexample; the paper also constructs singular power LCM matrices for suitable $f(n)=n^\alpha$ . The claim has not been independently verified in the retrieved sources.

Current status (as of September 2026): The conjecture is claimed refuted by explicit counterexamples, but independent verification and any later dispute, withdrawal, or retraction are not recorded.

Sources

No solutions have been posted yet.