On prime factors of determinants of circulant matrices (Q1044588)

Let \(n\) and \(p\) be distinct primes. Consider a random \(n\times n\) circulant whose entries are integers and row sums are not divisible by~\(p\). The author proves that its determinant is divisible by \(p\) with probability \[ \mathrm{Pr}_{n,p}=1-\left(1-\frac{1}{p^t}\right)^\frac{n-1}{t}, \] where \(t\) is the smallest positive integer satisfying \(p^t\equiv 1\,(\mathrm{mod}\,n)\). He also studies \(\lim_{i\to\infty}\mathrm{Pr}_{n_i,p_i}\), where \((n_i)\) and \((p_i)\) are sequences of primes satisfying certain conditions. For example, if \(\lim_{i\to\infty}n_i=\infty\) and if there is a positive (necessarily even) integer \(k\) such that \(p_i=kn_i+1\) for all \(i=1,2,\dots\), then this limit is \(1-e^{-\frac{1}{k}}\). The author also discusses shortly this matter when \(n\) is composite. The case \(p=2\) traces back to his previous paper [Linear Algebra Appl.~428, 1949--1955 (2008; Zbl 1143.15004)].