Vol. 5, No. 2, 2012

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Distribution of the exponents of primitive circulant matrices in the first four boxes of $\mathbb{Z}_n$

Maria Isabel Bueno, Kuan-Ying Fang, Samantha Fuller and Susana Furtado

Vol. 5 (2012), No. 2, 187–205

We consider the problem of describing the possible exponents of n-by-n boolean primitive circulant matrices. It is well known that this set is a subset of [1,n 1] and not all integers in [1,n 1] are attainable exponents. In the literature, some attention has been paid to the gaps in the set of exponents. The first three gaps have been proven, that is, the integers in the intervals [n 2 + 1,n 2], [n 3 + 2, n 2 2] and [n 4 + 3, n 3 2] are not attainable exponents. Here we study the distribution of exponents in between those gaps by giving the exact exponents attained there by primitive circulant matrices. We also study the distribution of exponents in between the third gap and our conjectured fourth gap. It is interesting to point out that the exponents attained in between the (i 1)-th and the i-th gap depend on the value of nmodi.

exponent, primitive circulant matrix, basis of a cyclic group, order, box
Mathematical Subject Classification 2010
Primary: 05C25, 05C50, 11P70
Received: 10 June 2011
Revised: 21 September 2011
Accepted: 22 September 2011
Published: 27 January 2013

Communicated by Joseph Gallian
Maria Isabel Bueno
Mathematics Department and College of Creative Studies
University of California, Santa Barbara
Santa Barbara, CA 93106
United States
Kuan-Ying Fang
Department of Mathematics
Northwestern University
Evanston, IL 60208
United States
Samantha Fuller
Department of Mathematics
Pennsylvania State University
University Park, PA 16802
United States
Susana Furtado
Faculdade de Economia do Porto
Universidade do Porto
Rua Doutor Roberto Frias
4200-464 Porto