# Note On Elliptic Groups of Prime Orders

## TL;DR

This paper investigates the distribution of primes p for which the elliptic curve group over finite fields has prime order, establishing a lower bound on their density among all primes.

## Contribution

It proves a lower bound on the number of primes p where the elliptic curve group order is prime, advancing understanding of prime distributions in elliptic curve groups.

## Key findings

- Establishes a lower bound for the count of such primes as proportional to x/log^2 x
- Supports the conjectured density of primes with prime group order in elliptic curves
- Provides theoretical evidence for the distribution of prime orders in elliptic groups

## Abstract

Let $E$ be an elliptic curve of rank $\text{rk}(E) \geq 1$, and let $E(\mathbb{F}_p)$ be the elliptic group of order $\#E(\mathbb{F}_p)=n$. The number of primes $p\leq x$ such that $n$ is prime is expected to be $\pi(x,E)=\delta(E)x/\log^2 x+o(x/\log^2 x)$, where $\delta(E)\geq 0$ is the density constant. This note proves a lower bound $\pi(x,E) \gg x/\log^2 x$.