Quartic, octic residues and binary quadratic forms

TL;DR

This paper uses quartic reciprocity to evaluate certain residues and powers modulo primes, deriving new congruences and criteria related to quadratic forms, Lucas sequences, and partially resolving previous conjectures.

Contribution

It introduces novel methods for computing residues and powers modulo primes using quartic reciprocity, and applies these to Lucas sequences and quadratic forms, advancing understanding in number theory.

Findings

Determines $q^{[p/8]} mod p$ in terms of quadratic form parameters.

Provides explicit formulas for certain binomial and quadratic expressions modulo $p$.

Establishes criteria for divisibility of Lucas sequence terms by primes.

Abstract

Let $\Bbb Z$ be the set of integers, and let $(m,n)$ be the greatest common divisor of integers $m$ and $n$. Let $p\equiv 1\mod 4$ be a prime, $q\in\Bbb Z$, $2\nmid q$ and $p=c^2+d^2=x^2+qy^2$ with $c,d,x,y\in\Bbb Z$ 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 $q^{[p/8]}\mod p$ in terms of $c,d,x$ and $y$, where $[\cdot]$ is the greatest integer function. We also determine $\big(\frac{b+\sqrt{b^2+4^{\alpha}}}2\big)^{\frac{p-1}4}\mod p$ for odd $b$ and $(2a+\sqrt{4a^2+1})^{\f{p-1}4}\mod p$ for $a\in\Bbb Z$. 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 $p\equiv 1\mod 8$), 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 $b\not\equiv 2\mod 4$. Hence we partially solve some conjectures posed by the author in two previous papers.