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Another generalization of Smith’s determinant. (English) Zbl 0675.10002

The authors consider gcd-closed sets of positive integers, that is, sets \(S=\{x_ 1,x_ 2,...,x_ n\}\) of distinct positive integers such that \((x_ i,x_ j)=1\) for every \(i,j=1,...,n\). They calculate the determinant of the matrix \((s_{ij})\), where \(s_{ij}=(x_ i,x_ j)\), in terms of the Euler totient.
Reviewer: T.M.Apostol

MSC:

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

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[2] Niven, An Introduction to the Theory of Numbers (1980) · Zbl 0431.10001
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[4] DOI: 10.1080/03081088808817841
[5] DOI: 10.1016/0024-3795(89)90572-7 · Zbl 0672.15005
[6] McCarthy, Canad. Math. Bull. 29 pp 109– (1988) · Zbl 0588.10005
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