2017/11/17 by M. Ram Murty, Murty, M. Ram, François Séguin +3
Mathematics · #11B37 #11D59 #11N69 #FOS: Mathematics #Geometric and Algebraic Topology #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1711.06410
openalex publication_date 2017/11/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 1927, Artin conjectured that any integer other than -1 or a perfect square\ngenerates the multiplicative group \ℤ/p\ℤ^\× for\ninfinitely many p. In citeMoSt, Moree and Stevenhagen considered a\ntwo-variable version of this problem, and proved a positive density result\nconditionally to the generalized Riemann Hypothesis by adapting a proof by\nHooley for the original conjecture ( citeHo). In this article, we prove an\nunconditional lower bound for this two-variable problem. In particular, we\nprove an estimate for the number of distinct primes which divide one of the\nfirst N terms of a non-degenerate binary recurrence sequence. We also prove a\nweaker version of the same theorem, and give three proofs that we consider to\nbe of independent interest. The first proof uses a transcendence result of\nStewart citeStew, the second uses a theorem of Bombieri and Schmidt on Thue\nequations citeBoSc and the third uses Mumford's gap principle for counting\npoints on curves by their height citeMum. We finally prove a disjunction\ntheorem, where we consider the set of primes satisfying either our two-variable\ncondition or the original condition of Artin's conjecture. We give an\nunconditional lower bound for the number of such primes.\n