A Pfaffian Proof and Generalization of a Conjecture of Sun Zhiwei

Let $p$ be an odd prime, let $n=(p-1)\mathrm{/}2$, and let $\chi =(\frac{\cdot }{p})$, with $\chi (0)=0$. For $a\in {\mathbb{F}}_p^{\times }$ define $$D_a(x)=\underset{1\le i,j\le n}{\det }(x+\chi (i^2-aj)),D_a^{(0)}(x)=\underset{0\le i,j\le n}{\det }(x+\chi (i^2-aj))\mathrm{.}$$ We prove $$D_a(0)=0⟺p\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)\text{and}\chi (an!)=1.$$ For $p\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$ we also give explicit Pfaffian-square factorizations of $D_a(x)$ and $D_a^{(0)}(x)$. Let $s_p=(-1{)}^{\lfloor (p+1)\mathrm{/}8\rfloor }$. If $\chi (an!)=1$, then $s_p{D}_a(x)\mathrm{/}x=s_p{D}_a^{(0)}(x)$ is a positive integer square. If $\chi (an!)=-1$, then there is a positive integer $\sigma$ such that $$s_p{D}_a(x)={\sigma }^2(nx-1),s_p{D}_a^{(0)}(x)=-{\sigma }^2(n+(2n+1)x)\mathrm{.}$$ The case $a=n!$ settles Sun’s Conjecture 4.1.