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Keywords:
power residues modulo prime; the tangent function; identity
power residues modulo prime; the tangent function; identity
Summary:
Let $p$ be an odd prime, and let $a$ be an integer not divisible by $p$. When $m$ is a positive integer with $p\equiv 1\pmod {2m}$ 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 $$ R_m(p)=\{0<k<p\colon k\in \mathbb Z\ \text {is an}\ m\text {th power residue modulo}\ p\}. $$ In particular, if $p=x^2+64y^2$ with $x,y\in \mathbb Z$, then $$ \prod _{k\in R_4(p)} \Big (1+\tan \pi \frac {ak}p\Big )=(-1)^{y}(-2)^{(p-1)/8}. $$
References:
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