Grant: Discovery Grants Program - Individual

Number theory is one of the most ancient fields within mathematics and yet, even today, continues to provide unexpected applications within and without the discipline. It is also somewhat notorious for having classical problems that have the feature that they are easy to state, yet, apparently, hard to solve. Our proposed research focusses on a number of results of this nature; we term our approach "explicit methods for Diophantine problems". The machinery we employ to prove our results is somewhat diverse. One of the basic fields we utilize is that of Diophantine approximation, which, classically, seeks to measure how well rational numbers approximate irrational ones. Where our proposal has a certain amount of novelty is in its combining these techniques with modifications of those famously used by Wiles to prove Fermat's Last Theorem (together with results coming from other areas of number theory, analytic and combinatorial).Our proposed work is centred upon two thematically-connected problems - theoretical and computational aspects of elliptic curves and explicit solution of classical problems from Diophantine equations. A common thread running through much of these two problems is their connection to solving a class of what are known as S-unit equations. Finding the solutions to such equations over cubic number fields enables us to tabulate all elliptic curves with rational coefficients and "small" conductor (known to be a finite problem since work of Shafarevich). Extending this to fields of higher degree allows one to carry this analysis to elliptic curves over number fields. Our proposed research will provide tables of such curves that greatly extend the current literature, as well as computational tools for solving S-unit equations that should find use in a wide variety of other settings.Our methods will also allow us to make progress on a number of other classical problems, including that of finding shifted powers in recurrence sequences, various polynomial-exponential equations, and the general n-term S-unit equation. To carry this out, we must first sharpen and generalize a number of recent results on ternary equations arising from the modularity of associated Galois representations, as well as the hypergeometric method of Thue-Siegel. In the course of carrying out this latter goal, we are led to a project in analytic and computational number theory, joint with Martin, O'Bryant and Rechnitzer, where we seek to obtain completely explicit bounds with error terms saving at least a logarithm for each standard function counting primes in arithmetic progression.