Some new evaluations of the Legendre symbol $\left(\frac{a+b\sqrt{q}}{p}\right)$

Abstract

We give some new evaluations of the Legendre symbol $\left(\frac{a+b\sqrt{q}}{p}\right)$ for certain integers $a$ and $b$ and certain primes $q$, where $p$ is an odd prime such that $\left(\frac{q}{p}\right)=1,$ and $\sqrt{q}$ denotes an integer whose square is $q(\mathrm{mod}p)$. For example it is shown that if $p$ is a prime $\equiv 1,19,25,31,37,55(\mathrm{mod}84)$ then $$\left(\frac{-5-2\sqrt{7}}{p}\right)=\left(\frac{x-2y}{7}\right)=\{\begin{array}{ll}+1& \text{if }x-2y\equiv 1,2,4(\mathrm{mod}7),\\ -1& \text{if }x-2y\equiv 3,5,6(\mathrm{mod}7),\end{array}$$ where $x$ and $y$ are the unique integers satisfying $p=x^2+xy+y^2$, $x\equiv 1(\mathrm{mod}4)$, $y\equiv 3(p-1)(\mathrm{mod}8)$ and $(1-(-1{)}^{(p-1)/2})x+y>0.$