Czechoslovak Mathematical Journal, Vol. 73, No. 3, pp. 971-978, 2023
The tangent function and power residues modulo primes
Zhi-Wei Sun
Received September 12, 2022. Published online April 13, 2023.
Abstract: Let $p$ be an odd prime, and let $a$ be an integer not divisible by $p$. When $m$ is a positive integer with $p\equiv1\pmod{2m}$ 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 $R_m(p)=\{0<k<p k\in\mathbb Z \text{is an} m\text{th power residue modulo} p\}.$ In particular, if $p=x^2+64y^2$ with $x,y\in\mathbb Z$, then $\prod_{k\in R_4(p)} \Big(1+\tan\pi\frac{ak}p\Big)=(-1)^y(-2)^{(p-1)/8}$.
Keywords: power residues modulo prime; the tangent function; identity
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Affiliations: Zhi-Wei Sun, Department of Mathematics, Nanjing University, West Building, Gulou Campus, No. 22 Hankou Road, Nanjing 210093, P. R. China, e-mail: zwsun@nju.edu.cn
