Note On Elliptic Primitive Points

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 $\pi(x,E,P) \gg x/\log x$.