On R. Chapman's ``evil determinant'': case \(p\equiv 1\pmod 4\) (Q2839287)

Let \(p\) be a prime number and let \(\left(\frac \cdot p\right)\) denote the Legrendre symbol. Among others, \textit{R. Chapman} [Acta Arith. 115, No. 3, 231--244 (2004; Zbl 1073.15006)] analyzed the determinant of the matrix \(C=\left[\left(\frac{j-i}p\right)\right]_{i,j=0,1,\ldots,(p-1)/2}\).NEWLINENEWLINE Recently, the author [Linear Algebra Appl. 436, No. 11, 4101--4106 (2012; Zbl 1317.11033)] proved that \(\det C=1\), for \(p\equiv 3\pmod 4\). Here, a similar approach is used to prove a formula for \(\det C\), conjectured also by \textit{R. Chapman} [``My evil determinant problem'', unpublished note (2009)], in terms of the fundamental unit and class number of \(\mathbb{Q}(\sqrt{p})\), for \(p\equiv 1\pmod 4\). Several expressions for parametric Cauchy-type determinants are evaluated and related to Dirichlet's class number formula for real quadratic fields.