×

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

Keywords:

LCM matrix
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Apostol, T.M., Arithmetical properties of generalized Ramanujan sums, Pacific J. math., 41, 281-293, (1972) · Zbl 0226.10045
[2] Beslin, S.; Ligh, S., Greatest common divisor matrices, Linear algebra appl., 118, 69-76, (1989) · Zbl 0672.15005
[3] Bourque, K.; Ligh, S., On GCD and LCM matrices, Linear algebra appl., 174, 65-74, (1992) · Zbl 0761.15013
[4] Bourque, K.; Ligh, S., Matrices associated with classes of arithmetical functions, J. number theory, 45, 367-376, (1993) · Zbl 0784.11002
[5] P. Haukkanen, J. Wang, and, J. Sillanpää, On Smith’s determinant, submitted for publication.
[6] Hong, S., LCM matrix on an r-fold gcd-closed set, J. sichuan university (natu. sci. ed.), 33, 650-657, (1996) · Zbl 0869.11021
[7] McCarthy, P.J., A generalization of Smith’s determinant, Canad. math. bull., 29, 109-113, (1988) · Zbl 0588.10005
[8] Smith, H.J.S., On the value of a certain arithmetical determinant, Proc. London math. soc., 7, 208-212, (1875 1876)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.