On Hong’s conjecture for power LCM matrices

Abstract

A set S={x ₁,...,x _n } of n distinct positive integers is said to be gcd-closed if (x _i , x _j ) ∈ S for all 1 ⩽ i, j ⩽ n. Shaofang Hong conjectured in 2002 that for a given positive integer t there is a positive integer k(t) depending only on t, such that if n ⩽ k(t), then the power LCM matrix ([x _i , x _j ]^t) defined on any gcd-closed set S={x ₁,...,x _n } is nonsingular, but for n ⩾ k(t) + 1, there exists a gcd-closed set S={x ₁,...,x _n } such that the power LCM matrix ([x _i , x _j ]^t) on S is singular. In 1996, Hong proved k(1) = 7 and noted k(t) ⩾ 7 for all t ⩾ 2. This paper develops Hong’s method and provides a new idea to calculate the determinant of the LCM matrix on a gcd-closed set and proves that k(t) ⩾ 8 for all t ⩾ 2. We further prove that k(t) ⩾ 9 iff a special Diophantine equation, which we call the LCM equation, has no t-th power solution and conjecture that k(t) = 8 for all t ⩾ 2, namely, the LCM equation has t-th power solution for all t ⩾ 2.