Prime power order circulant determinants

Abstract

Newman showed that for primes $p\ge 5$, an integral circulant determinant of prime power order $p^t$ cannot take the value $p^{t+1}$ once $t\ge 2$. We show that many other values are also excluded. In particular, we show that $p^{2t}$ is the smallest power of p attained for any $t\ge 3$, $p\ge 3$. We demonstrate the complexity involved by giving a complete description of the $25\times 25$ and $27\times 27$ integral circulant determinants. The former case involves a partition of the primes that are $1mod5$ into two sets, Tanner’s artiads and perissads, which were later characterized by E. Lehmer.