The generation of families of non-congruent numbers with arbitrarily many prime factors
department or school: Arts and Sciences, Irving K. Barber School of (Okanagan)
department or school: Computer Science, Mathematics, Physics and Statistics, Department of (Okanagan)
data provider: DSpace
degree campus: UBCO
creator: Reinholz, Lindsey Kayla
issued: 2019-10-03T18:26:40Z
issued: 2019
related degree: Doctor of Philosophy - PhD
degree grantor: University of British Columbia
description: A congruent number n is a positive integer that is equal to the area of a right triangle with rational side lengths. Positive integers for which such a representation does not exist are called non-congruent numbers. Equivalently, n is non-congruent if and only if the arithmetic rank of the cubic curve En:y²=x³-n²x, known as a congruent number elliptic curve, is zero. Determining whether or not a given positive integer is congruent in a finite number of steps is a problem of significant interest in the field of pure mathematics. Although a complete solution to this classical problem has yet to be discovered, progress has been made in describing particular families of congruent and non-congruent numbers. The classification of numbers into such families is often done by imposing conditions on the prime divisors of the numbers and on the Legendre symbols relating the primes. This thesis focuses on the generation of both odd and even non-congruent numbers. We present a new family of even non-congruent numbers that are a product of arbitrarily many distinct primes; these non-congruent numbers have at least one prime factor in each odd congruence class modulo eight. Our main contribution is the development of a general approach for constructing families of non-congruent numbers. We show that existing families of non-congruent numbers can be extended by working over the finite field with two elements and using a formula by Monsky for computing the 2-Selmer rank of congruent number elliptic curves. The new non-congruent numbers are produced by multiplying known non-congruent numbers, corresponding to congruent number elliptic curves with 2-Selmer rank of zero, by arbitrarily many suitable primes. This novel technique allows an infinite collection of non-congruent numbers to be generated, including both odd and even non-congruent numbers with arbitrarily many distinct prime divisors in each odd congruence class modulo eight. Our results are illustrated by numerous numerical e…
aggregated CHO: https://circle.library.ubc.ca/rest/handle/2429/71819
note: The Generation of Families ofNon-congruent Numbers withArbitrarily Many Prime FactorsbyLindsey Kayla ReinholzB.Sc., The University of British Columbia, 2011M.Sc., The University of British Columbia, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe College of Graduate Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)October 2019c© Lindsey Kayla Reinholz, 2019The following individuals certify that they have read, and recommendto the College of Graduate Studies for acceptance, a thesis/dissertation en-titled:The Generation of Families of Non-congruent Numbers withArbitrarily Many Prime Factorssubmitted by Lindsey Kayla Reinholz in partial fulfilment of the re-quirements of the degree of Doctor of PhilosophyDr. Qiduan Yang, Department of Computer Science, Mathematics, Physics, and StatisticsCo-supervisorDr. Sylvie Desjardins, Department of Computer Science, Mathematics, Physics, and StatisticsSupervisory Committee MemberDr. Shawn Wang, Department of Computer Science, Mathematics, Physics, and StatisticsSupervisory Committee MemberDr. Karen Perry, Department of ChemistryUniversity ExaminerDr. Yang Zhang, Department of Mathematics, University of ManitobaExternal ExamineriiAbstractA congruent number n is a positive integer that is equal to the area ofa right triangle with rational side lengths. Positive integers for which sucha representation does not exist are called non-congruent numbers. Equiv-alently, n is non-congruent if and only if the arithmetic rank of the cubiccurveEn : y2 = x3 − n2x,known as a congruent number elliptic curve, is zero. Determining whether ornot a given positive integer is congruent in a finite number of steps is a prob-lem of significant interest in the field of pure mathematics. Although a com-plete solution to this classical problem has yet to be discovered, progress hasbeen made in describing particular families of congruent and non-congruentnumbers. The classification o…
has type: Thesis/Dissertation
date issued: 2019-11
is shown at: 10.14288/1.0383245
language: eng
degree discipline: Mathematics
provider: Vancouver : University of British Columbia Library
publisher: University of British Columbia
rights: Attribution-NonCommercial-NoDerivatives 4.0 International
rights URI: http://creativecommons.org/licenses/by-nc-nd/4.0/
scholar level: Graduate
type: Text
identifier URI: http://hdl.handle.net/2429/71819