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 $q \pmod p$. For example it is shown that if $p$ is a prime $\equiv 1, 19, 25, 31, 37, 55 \pmod{84}$ then $$ \left(\frac{-5-2\sqrt{7}}{p} \right) = \left(\frac{x-2y}{7}\right) = \begin{cases} +1 & \text{if } x-2y \equiv 1,2,4 \pmod 7, \cr -1 & \text{if } x-2y \equiv 3,5,6 \pmod 7, \end{cases} $$ where $x$ and $y$ are the unique integers satisfying $p=x^2+xy+y^2$, $x \equiv 1 \pmod 4$, $y \equiv 3(p-1) \pmod 8$ and $\left(1-(-1)^{(p-1)/2}\right)x+y>0.$

Publication
In Acta Arithmetica, Volume 170 (2015), 361-380
Date
Links