A Pfaffian Proof and Generalization of a Conjecture of Sun Zhiwei

Let $p$ be an odd prime, let $n=(p-1)/2$, and let $\chi=(\frac{\cdot}{p})$, with $\chi(0)=0$. For $a\in\mathbb F_p^\times$ define \[
D_a(x)=\det_{1\le i,j\le n}(x+\chi(i^2-aj)),
\qquad
D_a^{(0)}(x)=\det_{0\le i,j\le n}(x+\chi(i^2-aj)). \] We prove \[
D_a(0)=0
\quad\Longleftrightarrow\quad
p\equiv 3 \pmod 4
\quad\text{and}\quad
\chi(a n!)=1. \] For $p\equiv3\pmod4$ we also give explicit Pfaffian-square factorizations of $D_a(x)$ and $D_a^{(0)}(x)$. Let $s_p=(-1)^{\lfloor(p+1)/8\rfloor}$. If $\chi(a n!)=1$, then $s_pD_a(x)/x=s_pD_a^{(0)}(x)$ is a positive integer square. If $\chi(a n!)=-1$, then there is a positive integer $\sigma$ such that \[
s_pD_a(x)=\sigma^2(nx-1),\qquad
s_pD_a^{(0)}(x)=-\sigma^2\bigl(n+(2n+1)x\bigr). \] The case $a=n!$ settles Sun's Conjecture 4.1.