Prime power order circulant determinants

TL;DR

This paper investigates the possible values of integral circulant determinants of prime power order, establishing new exclusions and providing detailed classifications for specific cases like 25x25 and 27x27 matrices.

Contribution

It extends Newman’s results by identifying additional excluded values and offers a complete description of certain large prime power cases, including prime classifications.

Findings

$p^{2t}$ is the smallest power of $p$ attained for $t extgreater 2$, $p extgreater 2$

Complete classification of 25x25 and 27x27 integral circulant determinants

Partition of primes $1 mod 5$ into Tanner's perissads and artiads

Abstract

Newman showed that for primes $p\geq 5$ an integral circulant determinant of prime power order $p^t$ cannot take the value $p^{t+1}$ once $t\geq 2.$ 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 $t\geq 3$, $p\geq 3.$ 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 $1\bmod5$ into two sets, Tanner's \textit{perissads} and \textit{artiads}, which were later characterized by E. Lehmer.