Asymptotics For Primitive Roots Producing Polynomials And Primitive Points On Elliptic Curves

Let $x \geq 1$ be a large number, let $f(n) \in \mathbb{Z}[x]$ be a prime producing polynomial of degree $\deg(f)=m$, and let $u\neq \pm 1,v^2$ 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 primitive root $u$ is derived in this note. This asymptotic result has the form \pi_f(x)=\# \{ p=f(n)\leq x:\ord_p(u)=p-1 \}=\left (c(u,f)+ O\left (1/\log x )\right ) \right )x^{1/m}/\log x, where $c(u,f)$ is a constant depending on the polynomial and the fixed integer. Furthermore, new results for the asymptotic order of elliptic primes with respect to fixed elliptic curves $E:f(X,Y)=0$ and its groups of $\mathbb{F}_p$-rational points $E(\mathbb{F}_p)$, and primitive points are proved in the last chapters

Results for nonWieferich Primes

Let $v\geq 2$ be a fixed integer, and let $x \geq 1$ be a large number. The exact asymptotic counting function for the number of nonWieferich primes $p\leq x$ such that $v^{p-1}-1 \equiv 0 \bmod p^2$ in the interval $[1,x]$ is proposed in this note. The current results in the literature provide lower bounds, which are conditional on the $abc$ conjecture or the Erdos binary additive conjecture

Squarefree Totients p-1 And Primitive Roots

This note determines an effective asymptotic formula for the number of squarefree totients $p-1$ with a fixed primitive root t u 6= ±1, v

Sign Patterns of the Liouville Function and Mobius Function over the Integers

Let $x\geq1$ be a large number, and let $0\leq a_

A Result in the Theory of Twin Primes

This article determines a lower bound for the number of twin primes $p$ and $p+2$ up to a large number $x$

Sum of Mobius Functions over the Shifted Primes

This article provides an asymptotic result for the summatory Mobius function ∑ p≤x μ(p + a) = O (x log−c x) over the shifted primes, where a ̸ = 0 is a fixed parameter, and c \u3e 1 is a constant

Topics In Analytic Number Theory And Consecutive Primitive Roots

This monograph proves the existence of various configurations of $(k+1)$-tuples of consecutive and quasi consecutive primitive roots $n+a_0, n+a_1, n+a_2, ,,,, n+a_k$ in the finite field \F_p, where $a_0,a_1, \ldots, a_k$ is a fixed $(k+1)$-tuples of distinct integers

Densities For The Repeating Decimals Problems

Let $p\geq 2$ be a prime, and let $1/p=0.\overline{x_{w-1} \ldots x_1x_0}$, $x_i \in \{0,1, 2, \ldots , 9\}$. The period $w\geq 1$ of the repeating decimal $1/p$ is a divisor of $p-1$. This note shows that the counting function for the number of primes with maximal period $w=p-1$ has an effective lower bound \pi_{10}(x)=\# \{ p\leq x:\ord_p(10)=p-1 \}\gg x/ \log x. This is a lower bound for the number of primes $p\leq x$ with a fixed primitive root $10 \bmod p$ for all large numbers $x\geq 1$. An extension to repeating decimal $1/p$ with near maximal period $w=(p-1)/r$, where $r \geq 1$ is a small integer, is also provided