New results on power residues modulo primes

In this talk we introduce some new results on power residues modulo primes.

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 $m^\rm{th}$ power residue modulo $p$, the speaker determines the value of the product $\prod_{k\in R_m(p)}(1 + \tan \pi{ak\over p} )$, where $R_m(p) = \{0 \lt k \lt p : k \in {\Bbb Z}\ {\rm is\ an\ } m^{\rm th}\ {\rm power\ residue\ modulo\ } p\}$.

Let $p \gt 3$ be a prime. Let $b\in\Bbb Z$ and $\epsilon \in \{\pm1\}$. Joint with Q.-.H. Hou and H. Pan, we prove that the absolute value of $\left\{ N_p(a, b) : 1 \lt a \lt p\ {\rm and} \left( {a\over p} \right) = \epsilon \right\}$ is equal to ${3−\left({ −1\over p} \right)\over 2}$ , where $N_p(a, b)$ is the number of positive integers $x \lt p/2$ with $\{x^2 + b\}_p \gt \{ax^2 + b\}_p$, and $\{m\}_p$ with $m \in \Bbb Z$ is the least nonnegative residue of $m$ modulo $p$.

We will also mention some open conjectures.

Zhi-Wei Sun