New Diophantine directions


Diophantine equations are polynomial equations whose coefficients and unknowns must be integers. There are many famous problems in mathematics involving Diophantine equations. They play a central role in the modern theory of arithmetic geometry and the subject is more alive than ever before. This is to a large extend due to recent exciting developments after the celebrated proof of Fermat's Last Theorem. Soon after this famous problem was solved, generalizations of Fermat's equations where the exponents (p,p,p) are replaced by three possibly different exponents (p,q,r) were investigated. The complete solution of these generalized Fermat equations is now considered the new holy grail of number theory.

First of all, two important cases of the generalized Fermat equation are considered in this proposal. We propose novel ways of combining both Hilbert and Bianchi modular approaches with finding rational points on curves, to resolve many new generalized Fermat equations with exponents (p,p,r) where p and r are distinct odd primes greater than 3. We also develop an anabelian descent method, which should, for the first time, resolve equations with all three exponents distinct primes and not given by 2,3,n (modulo permutation).

Second, we look at generalizations of the superelliptic equation and aim to greatly extend our knowledge there. This should have applications to other types of Diophantine problems, like finding all perfect powers in certain recurrence sequences.

The development of the underlying methods should also lead to novel results and insights into other branches of number theory.





Dr. S.R. Dahmen

Verbonden aan

Vrije Universiteit Amsterdam, Faculteit der B├Ętawetenschappen, Afdeling Wiskunde


Dr. S.R. Dahmen, J.M. van Langen MSc, P.H.C. Putz MSc