Diophantine Equations after Fermat's Last Theorem

丢番图问题是一个方程，其中一个人感兴趣的是寻找整数解。丢番图问题的研究至少可以追溯到公元前3世纪丢番图时代。有史以来最著名的丢番图问题是“费马大定理”。350多年来，这个问题吸引了大量专业和业余数学家的注意，并最终由安德鲁·怀尔斯在1994年解决。起初，似乎很少有丢番图问题可以用怀尔斯的方法解决。然而，几年前，首席研究员提出，为了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，这就是所谓的丢番图分析。首席调查员提出的战略是由他本人、布格奥和米格诺特组成的一个小组实施的；这导致了巨大的成功，包括解决了几个著名的悬而未决的问题。其中最著名的是在斐波纳契数列中找到所有完美的幂。这是一个50多年来一直没有解决的问题，最终在2003年被上述团队解决。就在最近，首席调查员建议改进费马大定理证明中所使用的技术，他称之为‘多重弗雷’。与费马大定理证明中使用的“单弗雷”相比，这提供了更多关于丢番图问题解的信息。使用这种方法，上述团队成功地解决了几个涉及5个和6个未知数的丢番图方程；一个被认为是不平行的脚。拟议研究的第一部分建立在最近的成功基础上。预计它将尽可能在最一般的背景下调查这些想法，而不是只关注特定的案例。“多重弗雷”技术需要深入研究，它所适用的丢番图方程必须进行分类。首席研究人员的另一个方面是“曲线的算法”。丢番图方程式是根据所谓的“维”来分类的。维度为一的称为曲线。对于任何丢番图问题(即使是曲线)，一个明显的问题是：它有解决方案吗？如果一个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？已经提出了许多方法来证明某些曲线没有解。首席调查员与马丁·布莱特合作，提出了一种非常简单的方法来证明某些曲线没有解。这种方法似乎是迄今为止最简单的方法，并且需要的信息量和计算量最少。然而，我们尚未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，我们也尚未对所涉及的想法有一个概念性的理解。这是建议研究的两个方向。丢番图方程的主题很久以前就被分割成几个子学科，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子学科中的几个技术来研究有趣的丢番图问题。预计这将是朝着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.