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Let $p$ be an odd prime. As in my paper, for $c,d\in\mathbb Z$ let us define $$[c,d]_p:=\det\left[\left(\frac{i^2+cij+dj^2}p\right)\right]_{0\le i,j\le p-1},$$ where $(\frac{\cdot}p)$ is the Legendre symbol. By Theorem 1.3 of the paper, if $(\frac dp)=1$ then $$[c,d]_p\equiv0\pmod{p-1}.$$ It is interesting to study $[c,d]_p$ in the case $(\frac dp)=-1$. Here I pose two new conjectures.

Conjecture 1. For any odd prime $p$ and $c,d\in\mathbb Z$ with $(\frac dp)=-1$, we have $(p-1)^2\mid[c,d]_p$.

Remark 1. I have verified this conjecture for all odd primes $p<750$. Moreover, I conjecture that for any positive odd integer $n$ and integer $d$ with $(\frac dn)=-1$, we have $$\det\left[\left(\frac{i^2+cij+dj^2}n\right)\right]_{0\le i,j\le n-1}\equiv0\pmod{\varphi(n)^2},$$ where $(\frac{\cdot}n)$ is the Jacobi symbol and $\varphi$ is Euler's totient function. This extended version has been verified for positive odd integers $n<200$.

Conjecture 2. (i) For any prime $p\equiv5\pmod 8$ with $p=x^2+y^2$ $ (x,y\in\mathbb Z\ \&\ 4\mid x-1)$, the number $[3,2]_p/((p-1)^2x)$ is a square.

(ii) For any prime $p\equiv3\pmod 8$ with $p=x^2+2y^2$ $(x,y\in\mathbb Z\ \&\ 4\mid x-1)$, there is a nonnegative integer $a_p$ such that $$[4,2]_p=-[8,8]_p=(-1)^{(p-3)/8}2^{p-3}(p-1)^2a_p^2\,x.$$

(iii) For any prime $p\equiv7\pmod{12}$ with $p=x^2+3y^2$ ($x,y\in\mathbb Z\ \&\ 3\mid x-1)$, the number $[3,3]_p/(2^{(p-7)/2}(p-1)^2x)$ is a square.

(iv) For any prime $p\equiv 11,15,23\pmod{28}$ with $p=x^2+7y^2$ ($x,y\in\mathbb Z\ \&\ (\frac x7)=1)$, the number $-[21,112]_p/((p-1)^2x)$ is a square.

(v) For any odd prime $p\equiv 1,2,4\pmod 7$ with $p=x^2+7y^2$ ($x,y\in\mathbb Z\ \&\ (\frac x7)=1)$, the number $(-1)^{(p-1)/2}[42,-7]_p/(8(p-1)x)$ is a square.

Remark 2. I have verified Conjecture 2 for odd primes $p<2000$.

For some related published conjectures and results, see Conjecture 4.8 in my paper and the preprint arXiv:1812.08080.

Conjecture 2 looks quite challenging, but Conjecture 1 might be relatively easy. Your comments are welcome!

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