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

Abstract

Let E/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(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 an elliptic curve analogue of classical lower bounds on the least primitive root of a prime.