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dc.contributor.authorMurty, M. Ram
dc.contributor.authorSéguin, François
dc.contributor.authorStewart, Cameron L. 18:59:42 (GMT) 18:59:42 (GMT)
dc.descriptionThe final publication is available at Elsevier via © 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license
dc.description.abstractIn 1927, Artin conjectured that any integer other than −1 or a perfect square generates the multiplicative group (Z/pZ)× for infinitely many p. In 2000, 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. 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, the second uses a theorem of Bombieri and Schmidt on Thue equations and the third uses Mumford's gap principle for counting points on curves by their height. 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.en
dc.description.sponsorshipNatural Sciences and Engineering Research Council of Canadaen
dc.description.sponsorshipFonds de Recherche du Québec - Nature et Technologiesen
dc.description.sponsorshipCanada Research Chairsen
dc.rightsAttribution-NonCommercial-NoDerivatives 4.0 International*
dc.subjectArtin's conjectureen
dc.subjectRecurrence sequencesen
dc.subjectThue equationen
dc.titleA lower bound for the two-variable Artin conjecture and prime divisors of recurrence sequencesen
dcterms.bibliographicCitationMurty, M. R., Séguin, F., & Stewart, C. L. (2019). A lower bound for the two-variable Artin conjecture and prime divisors of recurrence sequences. Journal of Number Theory, 194, 8–29. doi:10.1016/j.jnt.2018.06.017en
uws.contributor.affiliation1Faculty of Mathematicsen
uws.contributor.affiliation2Pure Mathematicsen

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