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
Abstract

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 $\left[1,n-1\right]$ and not all integers in $\left[1,n-1\right]$ 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 $\left[\frac{n}{2}+1,n-2\right]$, $\left[\frac{n}{3}+2,\frac{n}{2}-2\right]$ and $\left[\frac{n}{4}+3,\frac{n}{3}-2\right]$ 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 $n\phantom{\rule{0.3em}{0ex}}mod\phantom{\rule{0.3em}{0ex}}i$.

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

Communicated by Joseph Gallian
Authors
 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 Portugal