Let $p$ be an odd prime and $x$ be an indeterminate. Recently, Z.-W. Sun proposed the following conjecture: $$\det\left[x+\left(\frac{ j-i}{ p}\right)\right]_{ 0\le i,j\le \frac{ p-1}{ 2}}=\begin{ cases} (\frac{ 2}{ p})pb_px-a_p & \mbox{ if}\ p\equiv 1\pmod4, 1 & \mbox{ if}\ p\equiv 3\pmod4, \end{ cases}$$ where $a_p$ and $b_p$ are rational numbers related to the fundamental unit and class number of the real quadratic field $\mathbb{ Q}(\sqrt{ p})$. In this paper, we confirm the above conjecture of Sun based on Vsemirnov's decomposition of Chapman's "evil determinant".
Categories: math.NT