# Does $(p-1)^2$ divide $\det[(\frac{i^2+cij+dj^2}p)]_{0\le i,j\le p-1}$ when $(\frac dp)=-1$?

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!