The tangent function and power residues modulo primes,Czechoslovak Mathematical Journal

当前位置： X-MOL 学术 › Czechoslov. Math. J. › 论文详情

Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)

The tangent function and power residues modulo primes
Czechoslovak Mathematical Journal ( IF 0.5 ) Pub Date : 2023-04-13 , DOI: 10.21136/cmj.2023.0395-22
Zhi-Wei Sun

Let p be an odd prime, and let a be an integer not divisible by p. When m is a positive integer with p ≡ 1 (mod 2m) and 2 is an mth power residue modulo p, we determine the value of the product $\prod\limits_{k \in {R_m}(p)} {(1 + \tan (\pi ak/p))} $, where

$${R_m}(p) = \{0 < k < p:k \in \mathbb{Z}\,\,{\rm{is}}\,\,{\rm{an}}\,\,m{\rm{th}}\,\,{\rm{power}}\,\,{\rm{residue}}\,\,{\rm{modulo}}\,p\} .$$

In particular, if p = x² + 64y² with x, y ∈ $\mathbb{Z}$, then

$$\prod\limits_{k \in {R_4}(p)} {\left({1 + \tan \,\pi {{ak} \over p}} \right) = {{(- 1)}^y}{{(- 2)}^{(p - 1)/8}}.}$$

更新日期：2023-04-13

点击分享查看原文

点击收藏

公开下载

阅读更多本刊最新论文本刊介绍/投稿指南