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Greatest common divisor matrices. (English) Zbl 0672.15005

Authors’ summary: Let \(S=\{x_ 1,x_ 2,...,x_ n\}\) be a set of distinct positive integers. The \(n\times n\) matrix \([S]=(s_{ij}),\) where \(s_{ij}=(x_ i,x_ j),\) the greatest common divisor of \(x_ i\) and \(x_ j\), is called the greatest common divisor (GCD) matrix of S. The paper studies the GCD matrices in the direction of their structure, determinant, and arithmetic in \(Z_ n\). Several open problems are posed.
Reviewer: R.Covaci

MSC:

15B36 Matrices of integers
15A15 Determinants, permanents, traces, other special matrix functions
11C20 Matrices, determinants in number theory
11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors
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