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

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 $nmodi$.