The Generalized Fermat Equation with exponents 2, 3, n

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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.