Monthly Archives: October 2012

Fermat’s Lost Theorem

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Pierre Fermat, known for his Last Theorem and for rarely proving any of his claims.


Last week, I posted a series of increasingly difficult challenges in the post One line proof. Today, I would like to spend some time with the second challenge:

Fermat’s Lost Theorem: Show that (x+y)^n = x^m+y^m has only one solution for integers x > y > 0 and m,n > 1.

The truth is, I don’t know how to solve this problem myself. But, I think that we can figure it out together. Below, I will give the solution to the simpler case of y=1 and x > 1. I expect that many of you know the following variant of the problem:

Fermat’s Last Theorem: Show that z^m = x^m + y^m has no solutions for integers z > x > y > 0 and m > 2.

The above theorem was one of the most important unsolved problems in mathematics, until Andrew Wiles presented his proof to the public at a conference in Cambridge in 1993. Then someone pointed out a serious flaw in his proof and the extreme high Wiles was feeling turned into a dark abyss of despair. But being awesome implies that you pick yourself up and run full force towards the wall as if you didn’t get floored the last time you tried breaking through. And so Andy went back to his office and, with a little help from his friend (and former student) Richard Taylor, he fixed the flaw and published the massive proof in the most prestigious journal of mathematics, the Annals of Mathematics, in 1995.
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One line proof

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It is not often that we come across a problem whose solution can fit in the margins of a notebook. In fact, many of the problems I have worked on in the field of quantum many-body physics require proofs that often exceed 30 pages. And that is without taking into account the several references included as sources for results used as “elementary” tools in the proof (referees love these papers…) So it is natural to think that a proof is a proof, so long as it is correct, and once confirmed by the academic community it is time to move on to something new.

But… sometimes things get interesting. A 30-page proof collapses to a 3-page proof when a different point of view is adopted (see the famous Prime Number Theorem). Below, you will find two problems that may, or may not have a “one-line” proof. The challenge is for you to find the shortest, most elegant proof for each problem:

A unit triangle: A triangle with sides a\le b \le c has area 1. Show that b^2 \ge 2.

Fermat’s Lost Theorem: Show that (x+y)^n = x^m+y^m has only one solution with positive integers x > y > 0 and m,n > 1.

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Individual quantum systems

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When I went to school in the 20th century, “quantum measurements” in the laboratory were typically performed on ensembles of similarly prepared systems. In the 21st century, it is becoming increasingly routine to perform quantum measurements on single atoms, photons, electrons, or phonons. The 2012 Nobel Prize in Physics recognizes two of the heros who led these revolutionary advances, Serge Haroche and Dave Wineland. Good summaries of their outstanding achievements can be found at the Nobel Prize site, and at Physics Today.

Serge Haroche developed cavity quantum electrodynamics in the microwave regime. Among other impressive accomplishments, his group has performed “nondemolition” measurements of the number of photons stored in a cavity (that is, the photons can be counted without any of the photons being absorbed). The measurement is done by preparing a Rubidium atom in a superposition of two quantum states. As the Rb atom traverses the cavity, the energy splitting of these two states is slightly perturbed by the cavity’s quantized electromagnetic field, resulting in a detectable phase shift that depends on the number of photons present. (Caltech’s Jeff Kimble, the Director of IQIM, has pioneered the development of analogous capabilities for optical photons.)
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No calculators allowed

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During Friday evenings at Caltech, the Institute for Quantum Information and Matter runs a seminar series focusing on research talks geared towards a diverse audience whose common background is quantum mechanics (just that, nothing else…) But, the fun doesn’t end after the 45 minute talks are over. The seminar is followed by a mingling session on the brand new patio of East Bridge, the building where Richard Feynman delivered his famous lectures.

Grigori Perelman. Not a Russian middle-schooler anymore, but still pretty smart.

Last Friday, Aleksander Kubica (pronounced Koobitsa), a Polish graduate student in the Theory group of IQIM and a first prize winner of the 2009 European Union Contest for Young Scientists, decided to test our ingenuity by giving us a problem that Russian middle-schoolers are expected to solve. If you haven’t met any Russian middle-schoolers, well, let’s just say that you should think carefully before accepting a challenge that is geared towards their level of mathematical maturity. What was the challenge?

Problem 1: Show that: \frac{1}{2}\cdot \frac{3}{4}\cdot \frac{5}{6}\cdot \frac{7}{8}\cdot \frac{9}{10}\cdots \frac{99}{100} < \frac{1}{10}.
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All those different things called Gauge theories.

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Generations of kids had their first encounter with the explanation to the question: What does matter consist of? Atoms -> protons -> quarks. There is a certain pleasure in explaining this, regardless of your physics background. Just like a soldier polishes every new medal he gets, the story about elementary particles gets more exciting with every new ‘magic number’ introduced. Indeed, why 3 colors for quarks? A lay person would think that the inspiration came from the RGB (red-green-blue) colors in a TV tube. However, what physicists really did was try to say a new word in the language of gauge theories. And colored quarks was one of the first words one can say when learning to speak such a language: U(1), SU(2), SU(3)! The very third word.

So why don’t we try to share a scientific excitement about gauge theories?
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Redemption: Part II

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Last week, a journey began to find the solution to a problem I could not solve as a seventeen year-old boy. That problem became an obsession of mine during the last days of the International Math Olympiad of 1997, the days when I also met the first girl I ever kissed. At the time, I did not have the heart to tell the girl that I had traveled across the Atlantic to compete with the best and brightest and had come up short. I told her that I had solved the problem, but that the page with my answer had been lost. I told my parents the same thing and to everyone at school who would ask me why I did not return with a medal from the Math Olympics. The lie became so powerful that I did not look at that problem again until now. So, you may be wondering why a blog about Quantum Information Science at Caltech includes posts on problems from Math Olympiads. And why I would open the book on the page with that one problem after fifteen years… Continue reading →