Note On Elliptic Groups of Prime Orders

Abstract

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

Twenty One Pages. Keywords: Elliptic Prime; Group of Prime Order, Primitive Point; Koblitz Conjecture. arXiv admin note: text overlap with arXiv:1703.06806

References in corpus (1)

• A refinement of Koblitz's conjecture