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
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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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