In this paper we study some products related to quadratic residues and quartic residues modulo primes. Let $p$ be an odd prime and let $A$ be any integer. We mainly determine completely the product $$f_p(A):=\prod_{ 1\le i,j\le(p-1)/2\atop p\nmid i^2-Aij-j^2}(i^2-Aij-j^2)$$ modulo $p$; for example, if $p\equiv1\pmod4$ then $$f_p(A)\equiv\begin{ cases}-(A^2+4)^{ (p-1)/4}\pmod p&\text{ if}\ (\frac{ A^2+4}p)=1, \\(-A^2-4)^{ (p-1)/4}\pmod p&\text{ if}\ (\frac{ A^2+4}p)=-1,\end{ cases}$$ where $(\frac{ \cdot}p)$ denotes the Legendre symbol. We also determine $$\prod^{ (p-1)/2}_{ i,j=1\atop p\nmid 2i^2+5ij+2j^2}\left(2i^2+5ij+2j^2\right) \ \text{ and}\ \prod^{ (p-1)/2}_{ i,j=1\atop p\nmid 2i^2-5ij+2j^2}\left(2i^2-5ij+2j^2\right)$$ modulo $p$.
Categories: math.NT