Abstract

The purpose of the lecture series is to provide a comprehensive development of the basic mathematical tools used in the proof of the Super Fermat equation having no integer solutions. The equation is

(1) (x^p+y^p)/(x+y) = p^e z^p; e = 0 if p does not divide z, and 1 otherwise; p >3 and (x, y, z) = 1. This generalizes and implies Fermat's Last Theorem. Along the line, we also review some classical methods used for attempting to prove FLT. The topics we shall develop during the first lectures include:

(a) logarithmic derivatives and Kummerian results on FLT

(b) Gauss and Jacobi sums and the Stickelberger ideal

(c) Semilocal products of local fields and their Galois theory, together with some convergence results. We consider particular formal binomial series, which first appeared in the lecturer's proof of Catalan's Conjecture, 20 years ago, and discuss the connection to the present result

The last lectures will be dedicated entirely to the completion of the proof of

(1) not having solutions, as well as giving indications for the proof of stronger results on the generalization called strong Fermat-Catalan equation:

(2) (x^p+y^p)/(x+y) = p^e z^q; e = 0 if p does not divide z, and 1 otherwise; p >3 and (x, y, z) = 1. here q is a prime different from $q$. We show how to obtain upper bounds for possible solutions and discuss consequences.

The lecture series will be structured in two blocks of 4 lecture hours and 2 exercise hours.

Audience

Graduate, Researcher

Prerequisite

Algebra, Algebraic Number Theory

Lecturer Intro

Preda Mihăilescu studied mathematics and computer science in Zürich, receiving a PhD from ETH Zürich in 1997. His PhD thesis, titled Cyclotomy of rings and primality testing, was written under the direction of Erwin Engeler and Hendrik Lenstra. After his first studies, during close to 20 years, he worked in Zürich in the industry, first as a numerical analyst, then as a developer and consultant in IT-Security: cryptogrphy and fingerprint-identification. After 2000, during five years, he did research at the University of Paderborn, Germany, where he proved the long standing 'Conjecture of Catalan', in 2002. In 2005 he received a Wolkswagen Foundation professorship at the University of Göttingen, where he has been a professor ever since.