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Another generalisation of smith's determinant

Published online by Cambridge University Press:  17 April 2009

Scott Beslin  and
Steve Ligh
Scott Beslin
Affiliation:
Department of MathematicsNicholls State UniversityPO Box 2026Thibodaux LA70310United States of America
Steve Ligh
Affiliation:
Department of MathematicsUniversity of Southwestern LouisianaLafayette LA 70504United States of America
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Abstract

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Let S = {x1, x2, …, xn} be a set of distinct positive integers. The n × n matrix [S] = (Sij), where Sij, = (xi, xj), the greatest common divisor of xi, and xj, is called the greatest common divisor (GCD) matrix on S. H.J.S. Smith showed that the determinant of the matrix [E(n)], E(n) = { 1,2, …, n}, is ø(1)ø(2) … ø(n), where ø(x) is Euler's totient function. We extend Smith's result by considering sets S = {x1, x2, … xn} with the property that for all i and j, (xi, xj) is in S.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 40 , Issue 3 , December 1989 , pp. 413 - 415
Copyright
Copyright © Australian Mathematical Society 1989

References

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