Diophantine Equations after Fermat's Last Theorem

丢芬图问题是指人们对求整数解感兴趣的方程。对丢番图问题的研究至少可以追溯到公元前三世纪的丢番图时代。最著名的丢芬图问题是“费马大定理”。这个问题吸引了大量的专业和业余数学家的关注超过350年，最终由Andrew Wiles在1994年解决。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，为了成功解决其他有趣的丢芬图问题，怀尔斯的想法必须与其他不相关的“丢芬图分析”方法相结合。首席调查员提出的策略由他本人、比戈和米诺特组成的团队执行；这导致了惊人的成功，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的所有完美幂。这是一个50多年来没有解决的问题，最终在2003年由上述团队解决。最近，首席研究员提出了对费马大定理证明中使用的技术的改进，他称之为“多重弗雷”。这比证明费马大定理中使用的“单-弗雷”给出了更多关于丢芬图问题解的信息。利用这种方法，上述团队成功地解决了几个涉及5和6个未知数的丢番图方程；被认为没有平行的脚。拟议研究的第一部分以最近的成功为基础。我们希望在最普遍的背景下研究这些想法，而不是只关注特定的情况。需要对“多弗雷”技术进行深入研究，并且必须对其应用的丢番图方程进行分类。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“维数”来分类的。维数为1的称为曲线。对于丢芬图问题（甚至曲线问题），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：我们如何确定这一点？人们提出了许多方法来证明某些曲线没有解。首席研究员，在与马丁·布莱特的联合工作中，提出了一个非常简单的方法来证明一些曲线没有解。这种方法似乎是迄今为止最简单的方法，所需的信息和计算量最少。然而，我们尚未了解它是否在所有情况下提供与其他方法相同的信息，并且我们尚未对所涉及的思想获得概念性的理解。这两个都是建议研究的方向。丢番图方程的主题很久以前就分裂成几个子学科，它们之间很少或根本没有相互作用。这个项目旨在结合这些分支学科的技术来研究有趣的丢番图问题。这被认为是朝重新统一丢番图方程这一主题迈出的一步。

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.