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 $\displaystyle \prod_{0<i<j<p/2 \atop p\nmid f(i,j) } f(i,j) \ $, where $i,j\in \Bbb Z, f(i,j)$ is a linear or quadratic form with integer coefficients. When $f=ai^2+bij+cj^2$ and $p\nmid c(a+b+c)$ , we prove that to evaluate the product is equivalent to determine $ \displaystyle \sum_{y=1}^{p-1} \bigg(\frac{y(y+1)(y+k)}{p}\bigg) \pmod{16}$ , where $4c(a+b+c)k \equiv (4ac-b^2)\pmod{p}.$ Parallel results are given for $\displaystyle \prod_{i,j=1 \atop p\nmid f(i,j) }^{(p-1)/2} \bigg(\frac{ f(i,j) }{p}\bigg).$ Then we show that $ \displaystyle \sum_{y=1}^{p-1} \bigg(\frac{y(y+1)(y+k)}{p}\bigg) \pmod{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.