Quartic, octic residues and binary quadratic forms

Details Quartic, octic residues and binary quadratic forms Quartic, octic residues and binary quadratic forms

2011-08-15

arxiv.org/abs/1108.3027v1

Mathematics

Number Theory

31 pages

Scientific paper

Let $p\equiv 1\mod 4$ be a prime, $q$ be an odd number and $p=c^2+d^2=x^2+qy^2$ for some integers $c,d,x$ and $y$. Suppose that $c\equiv 1\mod 4$ and $c$ is coprime to $x+d$. 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. When $q=b^2+4^{\alpha}$, we also determine $\big(\frac{b+\sqrt{b^2+4^{\alpha}}}2\big)^{\frac{p-1}4}\mod p$. As an application we obtain the congruence for $U_{\frac{p-1}4}\mod p$ 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 $U_{n+1}=bU_n+4^{\alpha-1}U_{n-1}\ (n\ge 1)$. Hence we partially solve some conjectures posed by the author in previous two papers.

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