The Generalized Fermat Equation with exponents 2, 3, n

指数为 2, 3, n 的广义费马方程

• 批准号: 239402565
• 负责人: Professor Dr. Michael Stoll
• 金额: --
• 依托单位: Lehrstuhl Mathematik II - Computeralgebra, Algorithmische Arithmetische Geometrie
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2013
• 资助国家: 德国
• 起止时间: 2012-12-31 至 2016-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/239402565?language=en
• 关键词: Generalized; Fermat; Equation; exponents

The Generalized Fermat Equation asks whether the r th power of an integer can be equal to the sum of a pth power and a qth power. There are good reasons to require the three integers to be without common prime divisor. The solution of the original Fermat Equation (where all three exponents are equal), is given by the statement of ‘Fermat’s Last Theorem’, whose proof took over 300 years and has driven the development of several areas within mathematics whose results have many applications within mathematics, even though the original question does not seem to have interesting applications. The Generalized Fermat Equation (whose complete solution is still open) has served in a similar way as a motivation for the extension of existent and the development of new solution methods. This is what also this project tries to achieve: with the goal of a complete solution of the Generalized Fermat Equation with exponents 2, 3, n as concrete motivation, we plan to develop and extend methods that then will also have applications to many other problems. This is related quite generally to the solution of polynomial equations in two variables in integers or rational numbers. It is an important open question, wether this is algorithmically possible in general. The project is meant to bring us closer to a (hopefully positive) answer.

广义费马方程的问题是，一个整数的r次幂是否可以等于p次幂和q次幂之和。有充分的理由要求这三个整数没有公素因子。原始费马方程的解（其中所有三个指数都相等）由“费马最后定理”的陈述给出，其证明花了300多年的时间，并推动了数学中几个领域的发展，其结果在数学中有许多应用，即使原始问题似乎没有有趣的应用。广义费马方程（其完整解仍然是开放的）以类似的方式成为扩展现有求解方法和发展新求解方法的动力。这也是这个项目试图实现的目标：以指数为2，3，n的广义费马方程的完整解为目标，我们计划开发和扩展方法，然后也将应用于许多其他问题。这是相当普遍的解决方案的多项式方程在两个变量的整数或有理数。这是一个重要的悬而未决的问题，这在算法上是否可能。该项目旨在让我们更接近一个（希望是积极的）答案。