# Note On Elliptic Primitive Points

## TL;DR

This paper investigates the distribution of primes for which a point of infinite order on an elliptic curve generates the entire group over finite fields, establishing a lower bound on their density.

## Contribution

It proves a lower bound on the number of such primes, advancing understanding of elliptic primitive points and their distribution.

## Key findings

- Established a lower bound  x/  x on the count of primes with primitive points.
- Provides insight into the density (E,P) of primes with primitive points on elliptic curves.
- Contributes to the theory of elliptic curves and prime distributions in number theory.

## Abstract

Let $E$ be an elliptic curve of rank $\text{rk}(E) \geq 1$, and let $P \in E(\mathbb{Q})$ be a point of infinite order. The number of elliptic primes $p \leq x$ for which $\langle P\rangle=E(\mathbb{F}_p)$ is expected to be $\pi(x,E,P)=\delta(E,P)x/\log x+o(x/\log x)$, where $\delta(E,P)\geq 0$ is a constant. This note proves the lower bound $\pi(x,E,P) \gg x/\log x$.