Note On Elliptic Primitive Points

Let $E$ be an elliptic curve of rank $\text{rk}(E)\ge 1$, and let $P \in E($\mathbb{Q}$)$beapointofinfiniteorder\mathrm{.}Thenumberofellipticprimes$p \leq x$forwhich$\langle P\rangle=E($\mathbb{F}$_p)$isexpectedtobe$\pi(x,E,P)=\delta(E,P)x/\log x+o(x/\log x)$,where$\delta(E,P)\geq 0$ is a constant. This note proves the lower bound $\pi (x,E,P)\gg x\mathrm{/}\log x$.