A lower bound for the two-variable Artin conjecture and prime divisors of recurrence sequences

TL;DR

This paper establishes an unconditional lower bound for the number of primes related to a two-variable version of Artin's conjecture, involving prime divisors of recurrence sequences, using three distinct mathematical approaches.

Contribution

It provides the first unconditional lower bound for the two-variable Artin conjecture variant and introduces three novel proofs of key estimates.

Findings

Proves an unconditional lower bound for primes dividing recurrence sequences.

Develops three independent proofs using transcendence, Thue equations, and height counting.

Establishes a disjunction theorem for primes satisfying either of two conditions.

Abstract

In 1927, Artin conjectured that any integer other than -1 or a perfect square generates the multiplicative group $\mathbb{Z}/p\mathbb{Z}^\times$ for infinitely many $p$. In \cite{MoSt}, Moree and Stevenhagen considered a two-variable version of this problem, and proved a positive density result conditionally to the generalized Riemann Hypothesis by adapting a proof by Hooley for the original conjecture (\cite{Ho}). In this article, we prove an unconditional lower bound for this two-variable problem. In particular, we prove an estimate for the number of distinct primes which divide one of the first $N$ terms of a non-degenerate binary recurrence sequence. We also prove a weaker version of the same theorem, and give three proofs that we consider to be of independent interest. The first proof uses a transcendence result of Stewart \cite{Stew}, the second uses a theorem of Bombieri and Schmidt on Thue equations \cite{BoSc} and the third uses Mumford's gap principle for counting points on curves by their height \cite{Mum}. We finally prove a disjunction theorem, where we consider the set of primes satisfying either our two-variable condition or the original condition of Artin's conjecture. We give an unconditional lower bound for the number of such primes.