Diophantine problems

数论是数学中最古老的领域之一，但即使在今天，仍然在学科内外提供意想不到的应用。它也有一些臭名昭著的经典问题，这些问题的特点是，它们很容易陈述，但显然很难解决。我们建议的研究集中在一些结果，这种性质，我们称我们的方法“明确的方法丢番图问题”。我们用来证明我们的结果的机器有些不同。我们使用的基本领域之一是丢番图近似，它在经典上试图衡量有理数近似无理数的程度。我们的建议有一定的新奇是在其结合这些技术与修改的那些著名的怀尔斯证明费马大定理（连同结果来自其他领域的数论，分析和组合）。我们提出的工作是集中在两个主题连接的问题-理论和计算方面的椭圆曲线和显式解决经典问题的丢番图方程。贯穿这两个问题的一个共同线索是它们与求解一类被称为S-单位方程的问题的联系。在立方数域上找到这些方程的解使我们能够将所有具有有理系数和“小”导体的椭圆曲线制表（已知这是Shafarevich工作以来的有限问题）。扩展到更高程度的领域，允许进行这种分析，椭圆曲线在数域。我们提出的研究将提供这样的曲线表，大大扩展了目前的文献，以及计算工具，解决S-单位方程，应该找到使用在各种各样的其他settings.Our方法也将使我们能够取得进展的其他一些经典问题，包括寻找转移的权力，在递归序列，各种多项式指数方程，和一般的n项S单位方程。为了实现这一点，我们必须首先锐化和推广一些最近的结果所产生的三元方程的模块化相关联的伽罗瓦表示，以及超几何方法的Thue-Siegel。在执行这后一个目标的过程中，我们导致了一个项目在分析和计算数论，联合马丁，奥布莱恩特和Rechnitzer，在那里我们寻求获得完全明确的界限与错误项保存至少一个对数为每个标准函数计数素数在算术级数。

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.