Search

This note investigates the average density of prime numbers $p\in[x,2x]$ with respect to a random simultaneous primitive root $g\leq p^{1/2+\varepsilon}$ over the finite rings $\mathbb{Z}/p\mathbb{Z}$ and $\mathbb{Z}/p^2\mathbb{Z}$ as $x \to \infty$., Comment: Twenty Pages. Keywords: Prime number; Primitive root; Simultaneous primitive root; Hensel lemma; Artin Primitive Root Conjecture

Let $p \geq 2$ be a large prime, and let $k \ll \log p $ be a small integer. This note proves the existence of various configurations of $(k+1)$-tuples of consecutive and quasi consecutive primitive roots $n+a_0, n+a_1, n+a_2, \ldots, n+a_k$ in the finite field $\mathbb{F}_p$, where $a_0,a_1, \ldots, a_k$ is a fixed $(k+1)$-tuples of distinct integers., Comment: Forty Five Pages. Keywords: Primitive root; Consecutive primitive root; Consecutive Squarefree primitive root; Relative prime primitive root

Let $x \geq 1$ be a large number, let $f(x) \in \mathbb{Z}[x]$ be a prime polynomial of degree $\text{deg}(f)=m$, and let $u\ne \pm 1, v^2$ be a fixed integer. Assuming the Bateman-Horn conjecture, an asymptotic counting function for the number of primes $p=f(n) \leq x$ with a fixed primitve root $u$ is derived in this note., Comment: Keywords: Primitive root; Prime value of polynomial; Primitive root producing polynomial. arXiv admin note: substantial text overlap with arXiv:1504.00843