Legendre Symbol of $\prod f(i,j)$ over $0<i<j<p/2,p\nmid f(i,j)$

Abstract

Let $p>3$ be a prime. We investigate Legendre symbol of $\prod _{\genfrac{}{}{0ex}{}{0<i<j<p/2}{p\nmid f(i,j)}}f(i,j)$, where $i,j\in \mathbb{Z},f(i,j)$ is a linear or quadratic form with integer coefficients. When $f=a{i}^2+bij+c{j}^2$ and $p\nmid c(a+b+c)$ , we prove that to evaluate the product is equivalent to determine $\sum _{y=1}^{p-1}\left(\frac{y(y+1)(y+k)}{p}\right)(\mathrm{mod}16)$ , where $4c(a+b+c)k\equiv (4ac-b^2)(\mathrm{mod}p).$ Parallel results are given for $\prod _{\genfrac{}{}{0ex}{}{i,j=1}{p\nmid f(i,j)}}^{(p-1)/2}\left(\frac{f(i,j)}{p}\right).$ Then we show that $\sum _{y=1}^{p-1}\left(\frac{y(y+1)(y+k)}{p}\right)(\mathrm{mod}16)$ can be evaluated explicitly when k=2,4,5,9,10 or k is a square. And for several classes of f(i,j) these two kinds of products can be evaluated explicitly. Finally when f is a linear form we give unified identities for these products. Thus we prove these kind of problems raised in Sun's paper.