Abstract
Smith showed in 1875 that if \({n \geqq 1}\) is an integer and \({G := {({\rm gcd}(i, j))}_{1 \leqq i, j \leqq n}}\) is the \({n \times n}\) matrix having gcd(i, j) as its i, j-entry for all integers i and j between 1 and n, then \({{\rm det}(G) = {\prod_{k=1}^{n}}\varphi(k)}\), where \({\varphi}\) is the Euler’s totient function. We show that if \({n \geqq 2}\) is an integer and \({H := {({\rm gcd}(i, j))}_{2 \leqq i, j \leqq n}}\) is the \({(n-1) \times (n-1)}\) matrix having gcd(i, j) as its i, j-entry for all integers i and j between 2 and n, then
We also calculate the determinants of the matrices \({{(f({\rm gcd}(x_{i}, x_{j})))}_{1 \leqq i, j \leqq n}}\) and \({{(f({\rm lcm}(x_{i},x_{j})))}_{1\leqq i, j\leqq n}}\) having f evaluated at \({{\rm gcd}(x_{i}, x_{j})}\) and \({{\rm lcm}(x_{i}, x_{j})}\) as their (i, j)-entries, respectively, where \({S = \{x_{1},\ldots, x_{n}\}}\) is a set of distinct positive integers such that \({x_{i} > 1}\) for all integers i with \({1 \leqq i \leqq n}\) and \({S \cup \{1\}}\) is factor closed (that is, \({S \cup \{1\}}\) contains every divisor of x for any \({x \in S \cup\{1\}}\)). Our result answers partially an open problem raised by Ligh [18].
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Hong, S., Hu, S. & Lin, Z. On a certain arithmetical determinant. Acta Math. Hungar. 150, 372–382 (2016). https://doi.org/10.1007/s10474-016-0664-4
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DOI: https://doi.org/10.1007/s10474-016-0664-4