Quartic, octic residues and binary quadratic forms

Let $\mathbb{Z}$ be the set of integers, and let $(m,n)$ be the greatest common divisor of integers $m$ and $n$. Let $p\equiv 1\mathrm{m}\mathrm{o}\mathrm{d}4$ 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$c\e 1\mod 4$\mathrm{.}Supposethat$(c,x+d)=1$or$(d,x+c)$ is a power of 2. In the paper, by using the quartic reciprocity law we determine $q^{[p\mathrm{/}8]}\mathrm{m}\mathrm{o}\mathrm{d}p$ in terms of $c,d,x$ and $y$, where $[\cdot ]$ is the greatest integer function. We also determine $(\frac{b+\sqrt{b^2+4^{\alpha }}}{2}{)}^{\frac{p-1}{4}}\mathrm{m}\mathrm{o}\mathrm{d}p$ for odd $b$ and $(2a+\sqrt{4a^2+1}{)}^{\text{f}p-14}\mathrm{m}\mathrm{o}\mathrm{d}p$ for $a\in \mathbb{Z}$. As applications we obtain the congruence for $U_{\text{f}p-14}\mathrm{m}\mathrm{o}\mathrm{d}p$ and the criterion for $p\mid U_{\frac{p-1}{8}}$ (if $p\equiv 1\mathrm{m}\mathrm{o}\mathrm{d}8$), 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.