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A Result In The Theory Of Twin Primes, Nelson Carella
A Result In The Theory Of Twin Primes, Nelson Carella
Publications and Research
This article determines a lower bound for the number of twin primes $p$ and $p+2$ up to a large number $x$.
Asymptotics For Primitive Roots Producing Polynomials And Primitive Points On Elliptic Curves, Nelson Carella
Asymptotics For Primitive Roots Producing Polynomials And Primitive Points On Elliptic Curves, Nelson Carella
Publications and Research
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 …