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On the basis of my computation, I have the following conjecture involving the secant function. Conjecture. Let $p$ be an odd prime and define $$S_p:=\det\left[\sec2\pi\frac{jk}p\right]_{0\le j,k\le (...

$\newcommand\Legendre{\genfrac(){}{}}$Let $p$ be an odd prime, and let $\Legendre\cdot p$ be the Legendre symbol. In 2003, Robin Chapman evaluated the determinants $$\det\left[\Legendre{i+j}p\right]_{...