Some users are experiencing upload errors at the moment. If you receive a "UWSpace is down for maintenance" error, please email jordan.hale@uwaterloo.ca as soon as possible. We are very sorry for the inconvenience.

Show simple item record

dc.contributor.authorMurty, M. Ram
dc.contributor.authorSéguin, François
dc.contributor.authorStewart, Cameron L.
dc.date.accessioned2018-10-22 18:59:42 (GMT)
dc.date.available2018-10-22 18:59:42 (GMT)
dc.date.issued2019-01-01
dc.identifier.urihttps://dx.doi.org/10.1016/j.jnt.2018.06.017
dc.identifier.urihttp://hdl.handle.net/10012/14034
dc.descriptionThe final publication is available at Elsevier via https://dx.doi.org/10.1016/j.jnt.2018.06.017 © 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/en
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.language.isoenen
dc.publisherElsevieren
dc.rightsAttribution-NonCommercial-NoDerivatives 4.0 International*
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/4.0/*
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
dc.typeArticleen
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
uws.typeOfResourceTexten
uws.typeOfResourceTexten
uws.peerReviewStatusRevieweden
uws.scholarLevelFacultyen


Files in this item

Thumbnail
Thumbnail

This item appears in the following Collection(s)

Show simple item record

Attribution-NonCommercial-NoDerivatives 4.0 International
Except where otherwise noted, this item's license is described as Attribution-NonCommercial-NoDerivatives 4.0 International

UWSpace

University of Waterloo Library
200 University Avenue West
Waterloo, Ontario, Canada N2L 3G1
519 888 4883

All items in UWSpace are protected by copyright, with all rights reserved.

DSpace software

Service outages