An Inequality for the Determinant of the GCD Matrix and the GCUD Matrix
Received:July 15, 1997  
Key Words: Srith′s determinant   GCD matrix   unitary analogue   inequality.  
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Author NameAffiliation
PENTTI Haukkanen Dept. of Math. Sci
University of Tampers P.O.Box 607
FIN-33101 Tampere
Finland 
JUHA SIllanpaa Dept. of Math. Sci
University of Tampers P.O.Box 607
FIN-33101 Tampere
Finland 
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Abstract:
      Let S = {x1, x2,..., xn} be a set of distinct positive integers. The n x n matrix (S) whose i, j-entry is the greatest common divisor (xi, xj) of xi and xj is called the GCD matrix on S. A divisor d of x is said to be a unitary divisor of x if (d, x/d) = 1. The greatest common unitary divisor (GCUD) matrix (S**) is defined analogously. We show that if S is both GCD-closed and GCUD-closed, then det(S**) ≥ det(S), where the equality holds if and Only if (S**) = (S).
Citation:
DOI:10.3770/j.issn:1000-341X.2000.02.005
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