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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 $\#E(\mathbb{F}_p)=n$. Thenumber 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 thedensity constant. This note proves a lower bound $\pi(x,E) \gg x/\log^2 x$.

Note On Elliptic Groups of Prime Orders