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The generation of families of non-congruent numbers with arbitrarily many prime factors

University of British Columbia

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 examples.

Text

2019-10-03

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/71819

Mathematics

University of British Columbia

UBCO

http://creativecommons.org/licenses/by-nc-nd/4.0/