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arXiv · 2205.12439

Prime power order circulant determinants

Michael J. Mossinghoff·Christopher Pinner
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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.

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Michael J. Mossinghoff, Christopher Pinner. 2022-05-25. Prime power order circulant determinants. https://arxiv.org/abs/2205.12439

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