## On the Bourque-Ligh conjecture of least common multiple matrices.(English)Zbl 1015.11007

From the text: Let $$S= \{x_1,\dots, x_n\}$$ be a set of $$n$$ distinct positive integers. The matrix $$[S]_n$$ having the least common multiple $$[x_i,x_j]$$ of $$x_i$$ and $$x_j$$ as its $$i,j$$-entry is called the least common multiple (LCM) matrix on $$S$$. A set $$S$$ is gcd-closed if $$(x_i,x_j)\in S$$ for $$1\leq i,j\leq n$$. K. Bourque and S. Ligh [J. Number Theory 45, 367-376 (1993; Zbl 0784.11002)] conjectured that the LCM matrix $$[S]_n$$, defined on a gcd-closed set $$S$$, is nonsingular. P. Haukkanen, J. Wang and J. Sillanpää [Linear Algebra Appl. 258, 251-269 (1997; Zbl 0883.15002)] gave a counterexample for $$n=9$$. In [J. Sichuan Univ., Nat. Sci. Ed. 35, No. 2, 155-157 (1998; Zbl 1015.11062)] the author showed that the conjecture is true for $$n\leq 5$$. In this paper he proves that the conjecture is true for $$n\leq 7$$ and is not true for $$n\geq 8$$. So the conjecture is solved completely.

### MSC:

 11C20 Matrices, determinants in number theory 15B36 Matrices of integers 15A15 Determinants, permanents, traces, other special matrix functions

LCM matrix

### Citations:

Zbl 0784.11002; Zbl 0883.15002; Zbl 1015.11062
Full Text:

### References:

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