# Research Projects in Number Theory

Research projects in Probability Theory will be directed by Pamela Harris, Assistant Professor of Mathematics at Williams College.
### Project #1: Searching for Elliptic Curves with Rank 9

There is only one abelian group of order 8 which is noncyclic yet contains a cyclic subgroup of order 4; in 1973, Andrew Ogg showed there exist infinitely many elliptic curves defined over the rationals with this group as its torsion subgroup. It is natural to ask what are the properties of the elliptic curves in this family. In 2005, Ansaldi et al. generalized ideas of Nick Rogers from 2000 to find curves with rank 0 through 6. That same year, Noam Elkies did better: he found a curve in this family with rank 8. This project seeks to extend ideas in the paper by Ansaldi et al. to find an elliptic curve in this family with rank 9. We explain how to sort through such curves by focusing on a subfamily with aforementioned torsion subgroup and positive rank, and discuss the implementation of our ideas on a high performance computing cluster.
### Project #2: $ABC$-Triples in Families

Given three positive, relative prime integers such that the first two sum to the third, it is rare to have the product of the primes dividing them to be smaller than each of the three. In 1985, David Masser and Joseph Osterlé made this precise by defining a "quality" for such a triple of integers; their celebrated "ABC Conjecture" asserts that it is rare for this quality to be greater than 1 -- even through there are infinitely many examples where this happens. In 1987, Gerhard Frey offered an approach to understanding this conjecture by introducing elliptic curves. In this project, we introduce families of triples so that the Frey curve has nontrivial torsion subgroup, and explain how certain triples with large quality appear in these families. We also discuss how these families contain infinitely many examples where the quality is greater than 1.
### Project #3: Rational Distance Sets in Conic Sections

Leonhard Euler noted that there exists an infinite set of rational points on the unit circle such that the pairwise distance of any two is also rational; the same statement is nearly always true for lines and other circles. In 2004, Garikai Campbell considered the question of a rational distance set consisting of four points on a parabola. We introduce new ideas to discuss a rational distance set of four points on a hyperbola. We will also discuss the issues with generalizing to a rational distance set of five points on an arbitrary conic section.
### Project #4: Squares and Cubes in Arithmetic Progressions

In 1640, Pierre de Fermat sent a letter to Bernard Frénicle de Bessy claiming that that there are no four or more rational squares in a nontrivial arithmetic progression; this statement was shown in a posthumous work by Leonhard Euler in 1780. In 1823, Adrien-Marie Legendre showed that there are no three or more rational cubes in a nontrivial arithmetic progression. A modern proof of either claim reduces to showing that certain elliptic curves have no rational points other than torsion. A 2009 paper by Enrique González-Jiménez and Jörn Steuding, extended by a 2010 paper by Alexander Diaz, Zachary Flores, and Markus Vasquez discussed a generalization by looking at four squares in an arithmetic progression over quadratic extensions of the rational numbers. Similarly, a 2010 paper by Enrique González-Jiménez discussed a generalization by looking at three cubes in an arithmetic progression over quadratic extensions of the rational numbers. In this project, we give explicit examples of four squares and three cubes in arithmetic progressions, and recast many ideas by performing a complete 2-descent of quadratic twists of certain elliptic curves.