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Families of congruent and non-congruent numbers Reinholz, Lindsey Kayla


A positive integer n is a congruent number if it is equal to the area of a right triangle with rational sides. Equivalently, the Mordell-Weil rank of the elliptic curve y²=x(x²-n²) is positive. Otherwise n is a non-congruent number. Although congruent numbers have been studied for centuries, their complete classification is one of the central unresolved problems in the field of pure mathematics. However, by using algorithms such as the method of 2-descent, various mathematicians have proven that numbers with prime factors of a specified form that satisfy a certain pattern of Legendre symbols are either always congruent or always non-congruent. In this thesis, we build upon these results and not only prove the existence of new families of congruent and non-congruent numbers, but also present a new method for generating families of non-congruent numbers. We begin by providing a technique for constructing congruent numbers with three prime factors of the form 8k+3, and then give a family of such numbers for which the rank of their associated elliptic curves equals two, the maximal rank for congruent number curves of this type. Following this, we offer an extension to work done by Iskra and present our new method for generating families of non-congruent numbers with arbitrarily many prime factors. This method employs Monsky's formula for the 2-Selmer rank. Unlike the method of 2-descent which involves a series of lengthy and complex calculations, Monsky's formula offers an elegant approach for determining whether a given positive integer is non-congruent. This theorem uses linear algebra, and through a series of steps, allows one to compute the 2-Selmer rank of a congruent number elliptic curve, which provides an upper bound for the curve's Mordell-Weil rank. By applying this method, we construct infinitely many distinct new families of non-congruent numbers with arbitrarily many prime factors of the form 8k+3. In addition, by utilizing the aforementioned method once again, we expand upon results by Lagrange to generate infinitely many new families of non-congruent numbers that are a product of a single prime of the form 8k+1 and at least one prime of the form 8k+3.

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