The two-variable Artin conjecture and elliptic analogues

dc.contributor.authorSeguin, Francoisen
dc.contributor.departmentMathematics and Statisticsen
dc.contributor.supervisorMurty, M. Ramen's University at Kingstonen
dc.description.abstractIn 1927, Emil Artin conjectured that for any integer a other than -1 or squares, the set of primes p for which a is a primitive root modulo p has positive density in the set of all primes. This was proven subject to the generalized Riemann hypothesis (GRH) by Hooley in 1967. In 2002, P. Moree and P. Stevenhagen formulated an analogous two-variable conjecture, and used a result of Stephens on binary recurrence sequences to prove the conjecture conditionally on GRH. In this thesis, we show unconditional lower bounds for this two-variable conjecture. In particular, we obtain a result about general binary recurrence sequences that can be applied to this problem. We also formulate an analogue of the two-variable conjecture in the context of elliptic curves, and prove an unconditional lower bound for elliptic curves of rank 1. Finally, we obtain some results about the largest prime factor of the nth cyclotomic polynomial evaluated at a fixed integer, and where we let n vary.en
dc.relation.ispartofseriesCanadian thesesen
dc.rightsQueen's University's Thesis/Dissertation Non-Exclusive License for Deposit to QSpace and Library and Archives Canadaen
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dc.rightsThis publication is made available by the authority of the copyright owner solely for the purpose of private study and research and may not be copied or reproduced except as permitted by the copyright laws without written authority from the copyright owner.en
dc.subjectArtin's conjectureen
dc.subjectBinary recurrence sequencesen
dc.subjectPrimitive rootsen
dc.subjectWieferich primesen
dc.titleThe two-variable Artin conjecture and elliptic analoguesen
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