The tangent function and power residues modulo primes

@article{Sun2023,
abstract = {Let $p$ be an odd prime, and let $a$ be an integer not divisible by $p$. When $m$ is a positive integer with $p\equiv 1\hspace\{4.44443pt\}(\@mod \; 2m)$ and $2$ is an $m$th power residue modulo $p$, we determine the value of the product $\prod _\{k\in R_m(p)\}(1+\tan (\pi ak/p))$, where \[ R\_m(p)=\lbrace 0<k<p\colon k\in \mathbb \{Z\}\ \text\{is an\}\ m\text\{th power residue modulo\}\ p\rbrace . \] In particular, if $p=x^2+64y^2$ with $x,y\in \mathbb \{Z\}$, then \[ \prod \_\{k\in R\_4(p)\} \Big (1+\tan \pi \frac\{ak\}\{p\}\Big )=(-1)^\{y\}(-2)^\{(p-1)/8\}. \]},
author = {Sun, Zhi-Wei},
journal = {Czechoslovak Mathematical Journal},
keywords = {power residues modulo prime; the tangent function; identity},
language = {eng},
number = {3},
pages = {971-978},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The tangent function and power residues modulo primes},
url = {http://eudml.org/doc/299110},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Sun, Zhi-Wei
TI - The tangent function and power residues modulo primes
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 3
SP - 971
EP - 978
AB - Let $p$ be an odd prime, and let $a$ be an integer not divisible by $p$. When $m$ is a positive integer with $p\equiv 1\hspace{4.44443pt}(\@mod \; 2m)$ and $2$ is an $m$th power residue modulo $p$, we determine the value of the product $\prod _{k\in R_m(p)}(1+\tan (\pi ak/p))$, where \[ R_m(p)=\lbrace 0<k<p\colon k\in \mathbb {Z}\ \text{is an}\ m\text{th power residue modulo}\ p\rbrace . \] In particular, if $p=x^2+64y^2$ with $x,y\in \mathbb {Z}$, then \[ \prod _{k\in R_4(p)} \Big (1+\tan \pi \frac{ak}{p}\Big )=(-1)^{y}(-2)^{(p-1)/8}. \]
LA - eng
KW - power residues modulo prime; the tangent function; identity
UR - http://eudml.org/doc/299110
ER -