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Abstract

Let $\mathbb {Z}$ be the set of integers, and let $(m,n)$ be the greatest common divisor of the integers $m$ and $n$. Let $p\equiv 1\kern 4pt ({\rm mod}\kern 4pt 4)$ be a prime, $q\in \mathbb {Z}$, $2\nmid q$ and $p=c^2+d^2=x^2+qy^2$ with $c,d,x,y\in \mathbb {Z}$ and $c\equiv 1\kern 4pt ( {\rm mod}\kern 4pt 4)$. Suppose that $(c,x+d)=1$ or $(d,x+c)$ is a power of $2$. In this paper, by using the quartic reciprocity law, we determine $q^{[p/8]}\kern 4pt ( {\rm mod}\kern 4pt p)$ in terms of $c,d,x$ and $y$, where $[\cdot ]$ is the greatest integer function. Hence we partially solve some conjectures posed in our previous two papers.