Determinants of trigonometric functions and class numbers (Q2676731)

Given an odd prime \(p\) and a positive integer \(n\), define the \((p-1)/2\)-square matrix \(A=(a_{jk})\) and the \(n\)-square matrix \(B=(b_{jk})\) by \[ a_{jk}=\cot\frac{jk\pi}{p},\quad b_{jk}=\tan\frac{jk\pi}{2n+1}. \] \textit{Z.-W. Sun} [``On some determinants involving the tangent function'', Preprint, \url{arXiv:1901.04837}] conjectured that \[ \Big(\frac{-2}{p}\Big)\frac{\det{A}}{2^\frac{p-3}{2}p^\frac{p-5}{4}} \] is a positive integer, divisible by the class number of \(\mathbb{Q}(\sqrt{-p})\) if \(p\equiv 3\pmod{4}\). The present author proves this and gives a formula for \(\det{A}\). Additionally, Sun conjectured that \[ \frac{\det{B}}{(2n+1)^\frac{n}{2}} \] is an integer. Assuming that \(2n+1\) is prime, the present author proves also this conjecture and gives a formula for \(\det{B}\).