An Elliptic Curve Analogue of Pillai's Lower Bound on Primitive Roots

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve. For a prime $p$ of good reduction, let $r(E,p)$ be the smallest non-negative integer that gives the $x$-coordinate of a point of maximal order in the group $E({\mathbb{F}}_p)$. We prove unconditionally that $r(E,p)>0.72\log \log p$ for infinitely many $p$, and $r(E,p)>0.36\log p$ under the assumption of the Generalized Riemann Hypothesis. This can be viewed as elliptic curve analogues of classical lower bounds on the least primitive root of a prime.

14 pages, 7 tables

References in corpus (1)

• Primes in the Chebotarev density theorem for all number fields