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.


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


LCM matrix
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