A method of constructing specific polynomial representations $f(x)$ over the finite field $\mathbb{F}_p$ of the square roots function modulo a prime $p = 2^kn + 1$, $n$ odd, is presented. The formulas for the cases $k = 2$, $3$ and $4$ are given.
arxiv.orgLet $x\geq 1$ be a large number, and let $1 \leq a 0$ constant. This note proves that the counting function for the number of primes $p \in \{p=qn+a: n \geq1 \}$ with a fixed primitive root $u\ne \pm 1, v^2$ has the asymptotic formula $π_u(x,q,a)=δ(u,q,a)x/ \log x +O(x/\log^b x),$ where $δ(u,q,a)>0$ is the density, and $b=b(c)>1$ is a constant.
arxiv.org