A lower bound for the two-variable Artin conjecture and prime divisors of recurrence sequences
| dc.contributor.author | Murty, M. Ram | |
| dc.contributor.author | Séguin, François | |
| dc.contributor.author | Stewart, Cameron L. | |
| dc.date.accessioned | 2018-10-22T18:59:42Z | |
| dc.date.available | 2018-10-22T18:59:42Z | |
| dc.date.issued | 2019-01-01 | |
| dc.description | The 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.abstract | In 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.sponsorship | Natural Sciences and Engineering Research Council of Canada | en |
| dc.description.sponsorship | Fonds de Recherche du Québec - Nature et Technologies | en |
| dc.description.sponsorship | Canada Research Chairs | en |
| dc.identifier.uri | https://dx.doi.org/10.1016/j.jnt.2018.06.017 | |
| dc.identifier.uri | http://hdl.handle.net/10012/14034 | |
| dc.language.iso | en | en |
| dc.publisher | Elsevier | en |
| dc.rights | Attribution-NonCommercial-NoDerivatives 4.0 International | * |
| dc.rights.uri | http://creativecommons.org/licenses/by-nc-nd/4.0/ | * |
| dc.subject | Artin's conjecture | en |
| dc.subject | Recurrence sequences | en |
| dc.subject | Thue equation | en |
| dc.title | A lower bound for the two-variable Artin conjecture and prime divisors of recurrence sequences | en |
| dc.type | Article | en |
| dcterms.bibliographicCitation | Murty, 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.2018.06.017 | en |
| uws.contributor.affiliation1 | Faculty of Mathematics | en |
| uws.contributor.affiliation2 | Pure Mathematics | en |
| uws.peerReviewStatus | Reviewed | en |
| uws.scholarLevel | Faculty | en |
| uws.typeOfResource | Text | en |
| uws.typeOfResource | Text | en |