Eugène Catalan conjectured in 1842 that the only consecutive integers which are perfect powers are 8 and 9, i.e., the equation \(x^m-y^n=1\) has only one solution in the positive integers for \(m,n>2\). This conjecture was proven by Preda Mihailescu in 2003. After a short introduction about diophantine equations in general and a brief historical overview of the work on the problem of Catalan I will present some of the tools from algebraic number theory used in Mihailescu's proof. This will include some basic results from the theory of cyclotomic fields as well as Stickelberger's theorem and the group ring. These tools will be used to prove some partial results of Mihailescu. After this a short sketch of the remainder of the proof will be given.
The Catalan Conjecture
15.10.2018 13:15 - 14:45
Organiser:
H. Hauser
Location: