Diophantine Equations after Fermat's Last Theorem

丢番图问题是一个方程，其中一个感兴趣的是找到整数的解决方案。对丢番图问题的研究至少可以追溯到公元前世纪的丢番图时代。有史以来最著名的丢番图问题是“费马最后定理”。这个问题吸引了大量的专业和业余数学家的注意力超过350年，并最终解决了安德鲁怀尔斯在1994年。起初，似乎很少有丢番图问题可以用怀尔斯的技术解决。然而，几年前，首席研究员提出，要成功地解决其他有趣的丢番图问题，怀尔斯的想法必须与其他无关的方法相结合，这些方法被称为“丢番图分析”。首席研究员提出的策略由他自己、Bugeaud和Mittette组成的团队执行;这导致了巨大的成功，包括解决了几个著名的未解决问题。其中最著名的是找到斐波那契数列中的所有完美幂。这是一个50多年来未解决的问题，最终于2003年由上述团队解决。最近，首席研究员建议对费马大定理证明中使用的技术进行改进，他称之为“多重Frey”。这给了更多的信息，解决方案的丢番图问题比'单弗雷'，这是用于证明费马大定理。使用这种方法，上述团队成功地解决了几个丢番图方程，涉及5和6个未知数;一英尺被认为是没有平行。拟议研究的第一部分建立在最近的成功。人们希望在尽可能广泛的背景下调查这些想法，而不仅仅是在特定的情况下。“多弗雷”技术需要深入研究，它适用的丢番图方程必须分类。首席研究员工作的另一个方面是“曲线的算术”。丢番图方程是根据所谓的“维数”来分类的。那些具有一维的称为曲线。对于任何丢番图问题（甚至是曲线），一个显而易见的问题是：它有解吗？如果丢番图方程似乎没有解，另一个问题是：我们如何才能确定这一点？已经提出了许多方法来证明某些曲线没有解。首席研究员，在联合工作与马丁明亮提出了一个非常简单的方法，以显示一些曲线没有解决方案。这种方法似乎是最简单的方法，需要的信息和计算量最少。然而，我们还没有了解它是否在所有情况下都能提供与其他方法相同的信息，我们还没有从概念上理解所涉及的思想。丢番图方程的研究在很久以前就被分成了几个分支学科，它们之间很少或根本没有相互作用。这个项目的目的是联合收割机技术从几个这些子学科研究有趣的丢番图问题。它有望成为统一丢番图方程的一个步骤。

A Diophantine problem is an equation where one is interested in finding solutions that are whole numbers. The study of Diophantine problem goes back at least to the time of Diophantus in the third century BC. The most famous Diophantine problem of all time is 'Fermat's Last Theorem'. This problem attracted the attention of huge numbers of both professional and amatuer Mathematicians for over 350 years, and was finally solved by Andrew Wiles in 1994. At first, it seemed that very few Diophantine problems can be solved using Wiles' technique. However, a few years ago, the Principal Investigator proposed that to successfully solve other interesting Diophantine problems, Wiles' ideas must be combined with other, unrelated, methods from what is called 'Diophantine analysis'. The strategy proposed by the Principal Investigator was carried out by a team consisting of himself, and Bugeaud and Mignotte; this has lead to spectacular successes including the resolution of several famous unsolved problems. The best known of these is to find all the perfect powers in the Fibonacci sequence. This was an unsolved problem for over 50 years and was finally solved in 2003 by the above-mentioned team.Very recently, the Principal Investigator suggested a refinement of the technique used in the proof of Fermat's Last Theorem which he called 'Multi-Frey'. This gives far more information about the solutions of Diophantine problems than the 'Single-Frey' that is used in the proof of Fermat's Last Theorem. Using this approach the above-mentioned team successfully solved several Diophantine equations involving 5 and 6 unknowns; a feet believed to be without parallel.The first part of the proposed research builds on the recent successes. It is expected to investigate the ideas in the most general context possible instead of looking just at particular cases. The 'multi-Frey' technique needs to studied throughly and the Diophantine equations that it applies to have to be classified.Another aspect of the Principal Investigator's work is 'Arithmetic of Curves'. Diophantine equations are classified according to something called 'dimension'. Those having dimension one are called curves. An obvious question about to ask about any Diophantine problem (even curves) is: does it have solutions? If it seems that a Diophantine equation does not have solutions, another question is: how can we make sure of this? Many methods have been proposed for showing that certain curves do not have solutions. The Principal Investigator, in joint work with Martin Bright suggested a very simple method for showing that some curves do not have solutions. This method appears to be the simplest method yet and requires the least amount of information and computation. However, we have yet to understand if it gives the same information as the other methods in all cases, and we have yet to gain a conceptual understanding of the ideas involved. These are both directions of proposed research.The subject of Diophantine equations has long ago fragmented to several sub-disciplines with little or no interaction between them. This project aims to combine techniques from several of these sub-disciplines to study interesting Diophantine problems. It is expected to be a step forward toward re-unifying the subject of Diophantine equations.