The Generalized Fermat Equation with exponents 2, 3, n

广义费马方程询问整数的r次方是否等于第p次方和第q次方之和。有充分的理由要求这三个整数不能有公素数。原始的费马方程(其中所有三个指数都相等)的解由费马大定理的陈述给出，它的证明花了300多年的时间，并推动了数学中几个领域的发展，这些领域的结果在数学中有许多应用，尽管原始问题似乎没有有趣的应用。广义Fermat方程(其完全解仍然是开放的)也以类似的方式作为现有方法的扩展和新的求解方法的发展的动力。这也是本项目试图达到的目标：以指数为2，3，n的广义Fermat方程的完全解为具体动机，我们计划开发和扩展方法，然后这些方法也将应用于许多其他问题。这与求解整数或有理数的两个变量的多项式方程有很大的关系。这是一个重要的悬而未决的问题，一般来说，这在算法上是可能的。这个项目的目的是让我们更接近(希望是积极的)答案。

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.