Linear Forms in Logarithms and Fibonacci Numbers

The main work included in these pages is from a paper co-written by myself and my brother, Simon Earp-Lynch, under the supervision of Omar Kihel, pertaining to Diophantine triples of Fibonacci numbers. To go along with this will be introductory material not included in said paper which establishes the mathematical concepts therein and offers some historical perspective and motivation. The initial aim of the paper was to explore the possibility of a generalization of the main result in [2] on D(4)-Diophantine triples of Fibonacci numbers. The paper managed to extend the ideas in [2] to results for D(9)-Diophantine triples and D(64)-Diophantine triples. A generalization of Lemma 1 of [1] was also found, a lemma on Diophantine triples and Pellian equations which is key in establishing the main result in [2]. This paper includes this result and its proof, which involves a correction of the proof of Lemma 1 of [1]. This result may prove useful in the extension of the results in the paper, and potentially others as well. I will begin by introducing Diophantine equations, leading to Diophantine triples, followed by a section on the necessary preliminaries on Fibonacci num- bers, which concludes with the statements of our main results. Following this, I establish the primary machinery used in the proof of the main result, linear forms in logarithms. I then move to the generalization of the aforementioned Lemma 1 of [1], before finally commencing the proof of the main results.

Name thesis:degree_name

M.Sc. Mathematics and Statistics

Level thesis:degree_level

Masters

Discipline thesis:degree_discipline

Faculty of Mathematics and Science

Department dc:contributor.department

Department of Mathematics

Grantor

Brock University

Year dc:date.issued

2019

Author dc:creator

• Earp-Lynch, Benjamin

• Linear forms and logarithms; Fibonacci numbers; Diophantine triples; Pellian equations.

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/10464/14406

OAI identifier oai:identifier

oai:brocku.scholaris.ca:10464/14406