Quartic, octic residues and binary quadratic forms

Abstract

Let $\mathbb{Z}$ be the set of integers, and let $(m,n)$ be the greatest common divisor of integers $m$ and $n$. Let $p\equiv 1\mathrm{mod}4$ be a prime, $q\in \mathbb{Z}$, $2\nmid q$ and $p=c^2+d^2=x^2+q{y}^2$ with $c,d,x,y\in \mathbb{Z}$ and $c\e 1\mod 4$. Suppose that $(c,x+d)=1$ or $(d,x+c)$ is a power of 2. In the paper, by using the quartic reciprocity law we determine $q^{[p/8]}\mathrm{mod}p$ in terms of $c,d,x$ and $y$, where $[\cdot ]$ is the greatest integer function. We also determine $(\frac{b+\sqrt{b^2+4^{\alpha }}}{2}{)}^{\frac{p-1}{4}}\mathrm{mod}p$ for odd $b$ and $(2a+\sqrt{4a^2+1})^{\f{p-1}4}\mod p$ for $a\in \mathbb{Z}$. As applications we obtain the congruence for $U_{\f{p-1}4}\mod p$ and the criterion for $p\mid U_{\frac{p-1}{8}}$ (if $p\equiv 1\mathrm{mod}8$), where $\{U_n\}$ is the Lucas sequence given by $U_0=0,U_1=1$ and $U_{n+1}=b{U}_n+U_{n-1}(n\ge 1)$, and $b\not\equiv 2\mathrm{mod}4$. Hence we partially solve some conjectures posed by the author in two previous papers.

45 pages

Cited by in corpus (1)

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