Congruences for $q^{[p/8]}\ ({\rm mod}\ p)$

Zhi-Hong Sun¹

¹ School of Mathematical Sciences Huaiyin Normal University Huaian, Jiangsu 223001, P.R. China

Acta Arithmetica, Tome 159 (2013) no. 1, pp. 1-25

Résumé

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.

Détail

DOI : 10.4064/aa159-1-1

Mots-clés : mathbb set integers greatest common divisor integers equiv kern mod kern prime mathbb nmid y mathbb equiv kern mod kern suppose power paper using quartic reciprocity law determine kern mod kern terms where cdot greatest integer function hence partially solve conjectures posed previous papers