Genetic introduction in a proof of the Super Fermat Equation and beyond
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 structures in two blocks of 4 lecture hours and 2 exercise hours.
(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 structures in two blocks of 4 lecture hours and 2 exercise hours.
Lecturer
Date
17th ~ 18th August, 2023
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Thursday,Friday | 09:00 - 16:30 | Tsinghua-Ningzhai-W11 | ZOOM 07 | 559 700 6085 | BIMSA |
Prerequisite
Algebra, Algebraic Number Theory
Audience
Graduate
, Researcher
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
He 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.
During the last decade he specialized in abelian Iwasawa Theory and developed new methods for the investigation of open problems in the domain.
During the last decade he specialized in abelian Iwasawa Theory and developed new methods for the investigation of open problems in the domain.