On the quartic character of quadratic units
Volume 159 / 2013
Abstract
Let $\mathbb Z$ be the set of integers, and let $(m,n)$ be the greatest common divisor of integers $m$ and $n$. Let $p$ be a prime of the form $4k+1$ and $p=c^2+d^2$ with $c,d\in\mathbb Z$, $d=2^rd_0$ and $c\equiv d_0\equiv 1\pmod 4$. In the paper we determine $\def\sls#1#2{\bigl(\frac{#1}{#2}\bigr)}\sls{b+\sqrt{b^2+4^{\alpha}}}2^{\frac{p-1}4}\pmod p$ for $p=x^2+(b^2+4^{\alpha})y^2$ $(b,x,y\in\mathbb Z,\ 2\nmid b)$, and $(2a+\sqrt{4a^2+1})^{\frac{p-1}4}\pmod p$ for $p=x^2+(4a^2+1)y^2$ $(a,x,y\in\mathbb Z)$ on the condition that $(c,x+d)=1$ or $(d_0,x+c)=1$. As applications we obtain the congruence for $U_{(p-1)/4}\pmod p$ and the criterion for $p\,|\, U_{(p-1)/8}$ (if $p\equiv 1\pmod 8$), where $\{U_n\}$ is the Lucas sequence given by $U_0=0,\ U_1=1$ and $U_{n+1}=bU_n+U_{n-1}\ (n\ge 1)$, and $b\not\equiv 2\pmod 4$. Hence we partially solve some conjectures that we posed in 2009.
