The two-variable Artin conjecture and elliptic analogues

Abstract

In 1927, Emil Artin conjectured that for any integer a other than -1 or squares, the set of primes p for which a is a primitive root modulo p has positive density in the set of all primes. This was proven subject to the generalized Riemann hypothesis (GRH) by Hooley in 1967. In 2002, P. Moree and P. Stevenhagen formulated an analogous two-variable conjecture, and used a result of Stephens on binary recurrence sequences to prove the conjecture conditionally on GRH. In this thesis, we show unconditional lower bounds for this two-variable conjecture. In particular, we obtain a result about general binary recurrence sequences that can be applied to this problem. We also formulate an analogue of the two-variable conjecture in the context of elliptic curves, and prove an unconditional lower bound for elliptic curves of rank 1. Finally, we obtain some results about the largest prime factor of the nth cyclotomic polynomial evaluated at a fixed integer, and where we let n vary.