Let \(\mathcal{S}(\mathbb{Z}_n)=\{D(a_0,\ldots,a_{n-1}):a_i\in\mathbb{Z}\}\) be the set of all the determinants of integral \(n\times n\) circulant matrices, i.e., \[ D(a_0,\ldots,a_{n-1})=\det \begin{pmatrix} a_0&a_{n-1}&\cdots &a_1\\ a_1&a_0&\ldots&a_2\\ \vdots & \vdots &\ddots &\vdots\\ a_{n-1}& a_{n-2}&\ldots &a_0 \end{pmatrix}. \] The set \(\mathcal{S}(\mathbb{Z}_n)\) is closed under multiplication and can be defined more broadly for a finite group \(G\). Here, \(G=\mathbb{Z}_n\) is a cyclic group of order \(n\). The complete description of \(\mathcal{S}(G)\) is known only for groups of order less than \(16\). By the results of H. T. Laquer [in: The Fibonacci sequence, Collect. Manuscr., 18th anniv. Vol., The Fibonacci Assoc., 212–217 (1980; Zbl 0524.15007)] and M. Newman [Ill. J. Math. 24, 156–158 (1980; Zbl 0414.15007)], we have \(n^2\mathbb{Z}\) and \(\{m\in \mathbb{Z}:\gcd(m,n)=1\}\) are both contained in \( \mathcal{S}(\mathbb{Z}_n)\). Furthermore, certain divisibility conditions for \(m\in\mathcal{S}(\mathbb{Z}_n)\) are known.
The current paper describes the integers contained in \(\mathcal{S}(\mathbb{Z}_{p^t}),\) where \(p\ge 5\) is a prime number and \(t\) is an integer greater or equal to two, or \(p=3\) and \(t\ge 3.\) The complete description of \(\mathcal{S}(\mathbb{Z}_{25})\) is given in Theorem 1.2, with representation including the use of special prime numbers congruent to 1 modulo 5, so-called perissads. Similarly, the description of \(\mathcal{S}(\mathbb{Z}_{27})\) is given (in Theorem 1.3), invoking a certain family of primes congruent to 1 modulo 3 and 9. Finally, in Proposition 1.4, it is shown that \(\mathcal{S}(\mathbb{Z}_{p^t})\subseteq \mathcal{S}(\mathbb{Z}_{p^{t-1}})\) for \(t\ge 2.\)