On some analogues of the Bourque–Ligh conjecture on LCM matrices

Pentti Haukkanen and Juha Sillanpää
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 3, 1997, Number 1, Pages 52–57
Full paper (PDF, 249 Kb)

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Authors and affiliations

Pentti Haukkanen
Department of Mathematical Sciences,
University of Tampere P.O.Box 607,
FIN-33101 Tampere, Finland

Juha Sillanpää
Department of Mathematical Sciences,
University of Tampere P.O.Box 607,
FIN-33101 Tampere, Finland

Abstract

Let S = {x_1, x_2, ..., x_n} be a set of distinct positive integers. The n \times n  matrix (S) whose i, j-entry is the greatest common divisor (x_i,x_j) of x_i and x_j is called the GCD matrix on S. The LCM matrix [S] on S is defined analogously. It is a direct consequence of a known determinant evaluation that the GCD matrix is always nonsingular on gcd-closed sets. Bourque and Ligh conjectured that the LCM matrix is always nonsingular on gcd-closed sets. It has been shown that this conjecture does not hold. In this paper we study certain analogues of this conjecture relating to GCD and LCM matrices on lcm-closed sets and some unitary analogues of GCD and LCM matrices.

AMS Classification

  • 11C20
  • 15A15
  • 11A25

References

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Cite this paper

Haukkanen, P. & Sillanpää, J. (1997). On some analogues of the Bourque–Ligh conjecture on LCM matrices. Notes on Number Theory and Discrete Mathematics, 3(1), 52-57.

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