Monthly Archives: September 2013

The million dollar conjecture you’ve never heard of…

Posted on by
4

Curating a blog like this one and writing about imaginary stuff like Fermat’s Lost Theorem means that you get the occasional comment of the form: I have a really short proof of a famous open problem in math. Can you check it for me? Usually, the answer is no. But, about a week ago, a reader of the blog that had caught an omission in a proof contained within one of my previous posts, asked me to do just that: Check out a short proof of Beal’s Conjecture. Many of you probably haven’t heard of billionaire Mr. Beal and his $1,000,000 conjecture, so here it is:

Let a,b,c and x,y,z > 2 be positive integers satisfying a^x+b^y=c^z. Then, gcd(a,b,c) > 1; that is, the numbers a,b,c have a common factor.

After reading the “short proof” of the conjecture, I realized that this was a pretty cool conjecture! Also, the short proof was wrong, though the ideas within were non-trivial. But, partial progress had been made by others, so I thought I would take a crack at it on the 10 hour flight from Athens to Philadelphia. In particular, I convinced myself that if I could prove the conjecture for all even exponents x,y,z, then I could claim half the prize. Well, I didn’t quite get there, but I made some progress using knowledge found in these two blog posts: Redemption: Part I and Fermat’s Lost Theorem. In particular, one can show that the conjecture holds true for x=y=2n and z = 2k, for n \ge 3, k \ge 1. Moreover, the general case of even exponents can be reduced to the case of x=y=p \ge 3 and y=z=q \ge 3, for p,q primes. Which makes one wonder if the general case has a similar reduction, where two of the three exponents can be assumed equal.

The proof is pretty trivial, since most of the heavy lifting is done by Fermat’s Last Theorem (which itself has a rather elegant, short proof I wanted to post in the margins – alas, WordPress has a no-writing-on-margins policy). Moreover, it turns out that the general case of even exponents follows from a combination of results obtained by others over the past two decades (see the Partial Results section of the Wikipedia article on the conjecture linked above – in particular, the (n,n,2) case). So why am I even bothering to write about my efforts? Because it’s math! And math equals magic. Also, in case this proof is not known and in the off chance that some of the ideas can be used in the general case. Okay, here we go…

Proof. The idea is to assume that the numbers a,b,c have no common factor and then reach a contradiction. We begin by noting that a^{2m}+b^{2n}=c^{2k} is equivalent to (a^m)^2+(b^n)^2=(c^k)^2. In other words, the triplet (a^m,b^n,c^k) is a Pythagorean triple (sides of a right triangle), so we must have a^m=2rs, b^n=r^2-s^2, c^k =r^2+s^2, for some positive integers r,s with no common factors (otherwise, our assumption that a,b,c have no common factor would be violated). There are two cases to consider now:

Case I: r is even. This implies that 2r=a_0^m and s=a_1^m, where a=a_0\cdot a_1 and a_0,a_1 have no factors in common. Moreover, since b^n=r^2-s^2=(r+s)(r-s) and r,s have no common factors, then r+s,r-s have no common factors either (why?) Hence, r+s = b_0^n, r-s=b_1^n, where b=b_0\cdot b_1 and b_0,b_1 have no factors in common. But, a_0^m = 2r = (r+s)+(r-s)=b_0^n+b_1^n, implying that a_0^m=b_0^n+b_1^n, where b_0,b_1,a_0 have no common factors.

Case II: s is even. This implies that 2s=a_1^m and r=a_0^m, where a=a_0\cdot a_1 and a_0,a_1 have no factors in common. As in Case I, r+s = b_0^n, r-s=b_1^n, where b=b_0\cdot b_1 and b_0,b_1 have no factors in common. But, a_1^m = 2s = (r+s)-(r-s)=b_0^n-b_1^n, implying that a_1^m+b_1^n=b_0^n, where b_0,b_1,a_1 have no common factors.

We have shown, then, that if Beal’s conjecture holds for the exponents (x,y,z)=(n,n,m) and (x,y,z)=(m,n,n), then it holds for (x,y,z)=(2m,2n,2k), for arbitrary k \ge 1. As it turns out, when m=n, Beal’s conjecture becomes Fermat’s Last Theorem, implying that the conjecture holds for all exponents (x,y,z)=(2n,2n,2k), with n\ge 3 and k\ge 1.

Open Problem: Are there any solutions to a^p+b^p= c\cdot (a+b)^q, for a,b,c positive integers and primes p,q\ge 3?

PS: If you find a mistake in the proof above, please let everyone know in the comments. I would really appreciate it!

Frontiers of Quantum Information Science

Posted on by
6

Just a few years ago, if you wanted to look for recent research articles about quantum entanglement, you would check out the quantum physics [quant-ph] archive at arXiv.org. Since 1994, quant-ph has been the central repository for papers about quantum computing and the broader field of quantum information science. But over the past few years there has been a notable change. Increasingly, exciting papers about quantum entanglement are found at the condensed matter [cond-mat] and high energy physics – theory [hep-th] archives.

I don’t know for sure, but that trend may have had something to do with an invitation I received a few months ago from David Gross, to organize the next Jerusalem Winter School in Theoretical Physics. David has been the General Director of the School for, well, I’m not sure how long, but it must be a long time. In the past, the topic of the school has rotated between particle physics, condensed matter physics, and astrophysics. Every year, a group of world-class scientists gives lectures on cutting-edge research for an enthusiastic audience of postdoctoral scholars and advanced graduate students.

David suggested that a good topic for the next school would be “quantum information, broadly envisaged — from quantum computing to strongly correlated electrons.” After some hesitation for family reasons, I embraced this opportunity to amplify David’s message: quantum information has arrived as a major subfield of physics, and its relevance to other areas of physics is becoming broadly appreciated.

I’m not good at organizing things myself, so I recruited two friends who are very good at it to help me: Michael Ben-Or and Patrick Hayden. As the local organizer at The Hebrew University, Michael has to do a lot of the hard work that I’m glad to avoid. We decided to call the school “Frontiers of Quantum Information Science,” and put together a slate of 10 lecturers, which I’m very excited about. The lectures will cover the core areas of quantum information, as well as some of the important ways in which quantum information relates to quantum matter, quantum field theory, and quantum gravity. Each lecturer will give three or four ninety-minute lectures, on these topics:

Scott Aaronson (MIT), Quantum complexity and quantum optics
David DiVincenzo (Aachen), Quantum computing with superconducting circuits
Daniel Harlow (Princeton), Black holes and quantum information
Michal Horodecki (Gdansk), Quantum information and thermodynamics
Stephen Jordan (NIST), Quantum algorithms
Rob Myers (Perimeter), Entanglement in quantum field theory
Renato Renner (ETH), Quantum foundations
Ady Stern (Weizmann), Topological quantum computing
Barbara Terhal (Aachen), Quantum error correction
Frank Verstraete (Vienna), Quantum information and quantum matter

The school will run from 30 December 2013 to 9 January 2014 at the Israel Institute for Advanced Studies at The Hebrew University in Jerusalem. If you are interested in attending, please visit the website for more information and fill out the registration form by November 1. I hope you can come — it’s going to be a lot of fun.

Rereading the first paragraph of this post, I got slightly nervous about whether the trend I described can be documented, so I have done a little bit of research. Going back to 2005, I plotted the number of papers with the word “entanglement” in the title on quant-ph, cond-mat, hep-th, and also the general relativity and quantum cosmology [gr-qc] archive. For 2013, I rescaled the data for the year up to now, taking into account that Sep. 22 is the 265th day of the year. I didn’t make any adjustment for papers being cross-listed on multiple archives.

Here is the data for quant-ph:quantph-plot-pdfIt’s remarkably flat. Here is the aggregated data for the other three archives:arxiv-plot-pdfIt’s pretty clear that something started to happen around 2010. I realize one could do a much more serious study of this issue, but since I was only willing to spend an hour on it, I feel vindicated.

Free Feynman!

Posted on by
9

Last Friday the 13th was a lucky day for those who love physics — The online html version of Volume 1 of the Feynman Lectures on Physics (FLP) was released! Now anyone with Internet access and a web browser can enjoy these unique lectures for free. They look beautiful.

Mike Gottlieb at Caltech on 20 September 2013. He's the one on the right.

Mike Gottlieb at Caltech on 20 September 2013. He’s the one on the right.

On the day of release, over 86,000 visitors viewed the website, and the Amazon sales rank of the paperback version of FLP leapt over the weekend from 67,000 to 12,000. My tweet about the release was retweeted over 150 times (my most retweets ever).

Free html versions of Volumes 2 and 3 are in preparation. Soon pdf versions of all three volumes will be offered for sale, each available in both desktop and tablet versions at a price comparable to the cost of the paperback editions. All these happy developments resulted from a lot of effort by many people. You can learn about some of the history and the people involved from Kip Thorne’s 2010 preface to the print edition.

A hero of the story is Mike Gottlieb, who spends most of his time in Costa Rica, but passed through Caltech yesterday for a brief visit. Mike entered the University of Maryland to study mathematics at age 15 and at age 16 began a career as a self-employed computer software consultant. In 1999, when Mike was 39,  a chance meeting with Feynman’s friend and co-author Ralph Leighton changed Mike’s life.

At Ralph’s suggestion, Mike read Feynman’s Lectures on Computation. Impressed by Feynman’s insights and engaging presentation style, Mike became eager to learn more about physics; again following Ralph’s suggestion, he decided to master the Feynman Lectures on Physics. Holed up at a rented farm in Costa Rica without a computer, he pored over the lectures for six months, painstakingly compiling a handwritten list of about 200 errata.

Kip’s preface picks up the story at that stage. I won’t repeat all that, except to note two pivotal developments. Rudi Pfeiffer was a postdoc at the University of Vienna in 2006 when, frustrated by the publisher’s resistance to correcting errata that he and others had found, he (later joined by Gottlieb) began converting FLP to LaTeX, the modern computer system for typesetting mathematics. Eventually, all the figures were redrawn in electronic form as scalable vector graphics, paving the way for a “New Millenium Edition” of FLP (published in 2011), as well as other electronically enhanced editions planned for the future. Except that, before all that could happen, Caltech’s Intellectual Property Counsel Adam Cochran had to untangle a thicket of conflicting publishing rights, which I have never been able to understand in detail and therefore will not attempt to explain.

Rudi Pfeiffer and Mike Gottlieb at Caltech in 2008.

Rudi Pfeiffer and Mike Gottlieb at Caltech in 2008.

The proposal to offer an html version for free has been enthusiastically pursued by Caltech and has received essential financial support from Carver Mead. The task of converting Volume 1 from LaTeX to html was carried out for a fee by Caltech alum Michael Hartl; Gottlieb is doing the conversion himself for the other volumes, which are already far along.

Aside from the pending html editions of Volumes 2 and 3, and the pdf editions of all three volumes, there is another very exciting longer-term project in the works — the html will provide the basis for a Multimedia Edition of FLP. Audio for every one of Feynman’s lectures was recorded, and has been digitally enhanced by Ralph Leighton. In addition, the blackboards were photographed for almost all of the lectures. The audio and photos will be embedded in the Multimedia Edition, possibly accompanied by some additional animations and “Ken Burns style” movies. The audio in particular is great fun, bringing to life Feynman the consummate performer. For the impatient, a multimedia version of six of the lectures is already available as an iBook. To see a quick preview, watch Adam’s TEDxCaltech talk.

Mike Gottlieb has now devoted 13 years of his life to enhancing FLP and bringing the lectures to a broader audience, receiving little monetary compensation. I asked him yesterday about his motivation, and his answer surprised me somewhat. Mike wants to be able to look back at his life feeling that he has made a bigger contribution to the world than merely writing code and making money. He would love to have a role in solving the great open problems in physics, in particular the problem of reconciling general relativity with quantum mechanics, but feels it is beyond his ability to solve those problems himself. Instead, Mike feels he can best facilitate progress in physics by inspiring other very talented young people to become physicists and work on the most important problems. In Mike’s view, there is no better way of inspiring students to pursue physics than broadening access to the Feynman Lectures on Physics!

The complementarity (not incompatibility) of reason and rhyme

Posted on by
8

Shortly after learning of the Institute for Quantum Information and Matter, I learned of its poetry.

I’d been eating lunch with a fellow QI student at the Perimeter Institute for Theoretical Physics. Perimeter’s faculty includes Daniel Gottesman, who earned his PhD at what became Caltech’s IQIM. Perhaps as Daniel passed our table, I wondered whether a liberal-arts enthusiast like me could fit in at Caltech.

“Have you seen Daniel Gottesman’s website?” my friend replied. “He’s written a sonnet.”

Quill

He could have written equations with that quill.

Digesting this news with my chicken wrap, I found the website after lunch. The sonnet concerned quantum error correction, the fixing of mistakes made during computations by quantum systems. After reading Daniel’s sonnet, I found John Preskill’s verses about Daniel. Then I found more verses of John’s.

To my Perimeter friend: You win. I’ll fit in, no doubt.

Exhibit A: the latest edition of The Quantum Times, the newsletter for the American Physical Society’s QI group. On page 10, my enthusiasm for QI bubbles over into verse. Don’t worry if you haven’t heard all the terms in the poem. Consider them guidebook entries, landmarks to visit during a Wikipedia trek.

If you know the jargon, listen to it with a newcomer’s ear. Does anyone other than me empathize with frustrated lattices? Or describe speeches accidentally as “monotonic” instead of as “monotonous”? Hearing jargon outside its natural habitat highlights how not to explain research to nonexperts. Examining names for mathematical objects can reveal properties that we never realized those objects had. Inviting us to poke fun at ourselves, the confrontation of jargon sprinkles whimsy onto the meringue of physics.

No matter your familiarity with physics or poetry: Enjoy. And fifty points if you persuade Physical Review Letters to publish this poem’s sequel.

Quantum information

By Nicole Yunger Halpern

If “CHSH” rings a bell,
you know QI’s fared, lately, well.
Such promise does this field portend!
In Neumark fashion, let’s extend
this quantum-information spring:
dilation, growth, this taking wing.

We span the space of physics types
from spin to hypersurface hype,
from neutron-beam experiment
to Bohm and Einstein’s discontent,
from records of a photon’s path
to algebra and other math
that’s more abstract and less applied—
of platforms’ details, purified.

We function as a refuge, too,
if lattices can frustrate you.
If gravity has got your goat,
momentum cutoffs cut your throat:
Forget regimes renormalized;
our states are (mostly) unit-sized.
Velocities stay mostly fixed;
results, at worst, look somewhat mixed.

Though factions I do not condone,
the action that most stirs my bones
is more a spook than Popov ghosts; 1
more at-a-distance, less quark-close.

This field’s a tot—cacophonous—
like cosine, not monotonous.
Cacophony enlivens thought:
We’ve learned from noise what discord’s not.

So take a chance on wave collapse;
enthuse about the CP maps;
in place of “part” and “piece,” say “bit”;
employ, as yardstick, Hilbert-Schmidt;
choose quantum as your nesting place,
of all the fields in physics space.

1 With apologies to Ludvig Faddeev.

Graphene gets serious

Posted on by
4

Imagine one marshmallow, 100 pieces of dried spaghetti, and a roll of masking tape lying on a large table. Next to the supplies are directions that read: “Elevate the marshmallow as high as possible using only the spaghetti and masking tape.” What was the first question that popped into your head? My assumption is your response had a disposition towards either “how can I do this?” or “why should I do this?” More precisely, your response probably could be whittled down to either a “how” or a “why.” If your first instinct was to ask yourself “how”, maybe an argument could be made that you are a natural problem solver, and that you welcome and genuinely are intrigued by challenges. If you asked “why”, then maybe you are someone who needs some good ‘ole fashioned incentive or a good extrinsic motive to perform well. Now imagine you were competing against three other people and the prize for the highest marshmallow was $100,000. Would the money motivate you to create a better structure, or would your relentless ambition towards excellence have been enough incentive for you to have placed your best foot forward from the outset? Undoubtedly, the money will make you think twice about your initial design ensuring your best effort, but my wish is to see more people performing at higher levels, not only due to monetary incentive, but also out of the sake of doing your best.

Chen-Chih Hsu & Benjamin Fackrell

Chen-Chih Hsu & Benjamin Fackrell

As humans, we are all naturally great problem solvers when compared, to say, any other known form of life on our planet. That is not to say, however, all humans choose to exercise those talents. Nonetheless, people do possess the ability to solve extremely complex problems, and I often wonder what makes some individuals face challenges head on with great heroism, while others whimper away with not as much as a grain of genuine interest or desire. I believe the reasons for different responses are connected with the way we individually have been taught to approach problems, and the amount of respect we have learned to award such methods. The attitude individuals possess when faced with a challenge can be shaped with encouragement from teachers and parents alike. When given an opportunity to educate students (of any age) regarding their attitude when faced with a problem, in that moment, we must teach absolute fearlessness. Attack the problem and take no prisoners, metaphorically speaking. Unfortunately, the prevailing attitude from the many students I work with daily is one of apathy and a play-it-safe approach with very little risk of making mistakes. For many students, forfeiting has greater power in protecting one’s reputation with peers and themselves than a courageous attempt that could end, in what they believe to be, an embarrassing mistake. I am always looking to instill a sense of honor, embracing a philosophy that a whole-hearted attempt merits infinitely more respect than a forfeit, and not to plan to fail, but prepare to stay the course in the case of an unfortunate event. My advice? Treat a failure like a fart; understand it’s sure to happen, try to find the humor in it, and keep moving forward. Mistakes can often indicate progress because if you are not making mistakes, per Albert Einstein, you must not be trying something new, consequently, you are not learning.
Continue reading →