My Favourite Math Joke

(I owe this one to Richard Pink.)

A famous number theorist dies and enters heaven. There he has the chance to talk to God, and he asks: “So, I’ve been trying to prove it for my whole life, but in vain… Is it true? Are all the zeroes of the Riemann zeta function on the critical line?”
And God answers: “Yes, the Riemann hypothesis is true!”
The number theorist is excited: “Great! I knew it! But tell me, how does one prove it?”
God chuckles and answers: “Prove it? I don’t know, I just see that all the zeroes ARE on the critical line…”

A Proof for the Fundamental Theorem of Algebra using nothing but Real Analysis

Inspired by Felix’ post I want to show you how to proof the fundamental theorem of algebra using only real analysis. Some steps are quite similar to Felix’ proof, but alas, we must not use the power of holomorphic functions. I’ve seen this proof more than four years ago in my undergraduate Analysis I course at ETH, taught by Dietmar Salamon.

Theorem: Every non-constant polynomial has a complex zero.

Proof: Without loss of generality, let be a normalized complex polynomial and define and and set

such that holds. Then we get for

and hence

This shows that, in general, zeroes of polynomials cannot be too large in terms of their coefficients. But what happens inside the circle? The set is bounded and closed, hence compact by Heine-Borel. Polynomials are continuous, hence  has a minimum at some point . This minimum is even a global minimum, since for all .

If there is nothing to do. Hence assume that the global minimum is not zero. Then we can define the polynomial

which has the property that it has a global minimum of at . Hence there exist a non-zero complex number and a positive integer such that

Let be any complex -th root of . Then Hence there exists a polynomal such that

Since polynomials are continuous, has to be bounded on , say by . Using this and restricting to the unit interval we get

It is now not hard to see that one can choose small enough to make positive, in turn getting , in contradiction to the global mininmum of value , QED.

On the 4-rank of Class Groups of Quadratic Number Fields

In their paper, Etienne Fouvry and Jürgen Klüners prove that the 4-rank of class groups of quadratic number fields behave as predicted by the Cohen-Lenstra heuristics und their extension by Gerth. My current task is it to understand the paper thoroughly. I’m doing quite well so far, and I want to describe the key steps in their proof in the easiest case.

Summary on Mihailescu’s Proof of Catalan’s Conjecture

I finally submitted my semester project on Catalan’s Conjecture. In this post, I want to give an idea of the concept behind the proof.

Announcement [Update!]: A Talk on “Catalan’s Conjecture: A Purely Algebraic Proof”

Update: The talk starts at 14.00 sharp, not at 14.15 as announced previously.

Tomorrow at 14.15 14.00 in room HG G43 at ETH Zürich, I will present the results of my semester project on Catalan’s Conjecture that I wrote under the supervision of Prof. E. Kowalski. I know that this announcement comes on a very short notice, but unfortunately we could not get a confirmation earlier.

Catalan’s Conjecture states that the only consecutive powers of integers are 8 and 9. The question remained open for over 150 years and was answered positively only very recently. My talk will cover the main ideas of the complete proof, given by Mihailescu in 2004. Unlike the initial proof, given in 2002 by Mihailescu, that was still dependent on a computer calculation and the theory of linear forms in logarithms, this version of the proof only requires the arithmetic of cyclotomic number fields.

I expect that the non-expert will not be able to follow through every part of my talk. I will do my best to present the ideas behind the propositions and theorems. The final written version of my project should be on-line by the end of February and will contain proofs for nearly all statements.