The tangent function and power residues modulo primes

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 ≡ 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

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