Integer circulant determinants of order 15

Abstract

We consider the values taken by $n\times n$ circulant determinants with integer entries when $n$ is the product of two distinct odd primes $p,q$. These correspond to the integer group determinants for ${\mathbb{Z}}_{pq}$, the cyclic group of order $pq$. We show that $p^2$ and $q^2$ are not determinants (more generally we show that the classic necessary divisibility conditions are never sufficient when $n$ contains at least two odd primes). We obtain a complete description of the integer group determinants for ${\mathbb{Z}}_{15}$ (the smallest unresolved group) and partial results for general $n=3p.$