eon

Posted by tpc at August 27th, 2010

Feature column on AMS website this month, using FFT to quickly multiply two large integers. Yet another useful application of Fourier transform.

Posted by tpc at June 6th, 2010

Yesterday (5th June) was World Environment Day. I wouldn’t say I’m a green fanatic but I do try my best to recycle, use the air conditioner only when it is unbearable and bring my own non-plastic shopping bags whenever I know I was going shopping. According to this webpage, a total 17,291 species are known to be facing extinction.

I wonder what is the source of the number. If you google World Environment Day 17291, 17 of the first 20 webpages has a variation of that same sentence. Out of these 17 only this one suggests that the number is not exact. I quote (emphasis mine)
“In totality, there are roughly 17,291 species that are on the threatened list”

The number 17291 stood out for me. It turns out to be prime! In fact wolfram alpha tells me it is a twin prime. No prizes for working out which is its twin. Of course, 1729 is the famous Ramanujan Taxicab number. Perhaps that’s why it looked familiar.

Update 1: Google’s spider is amazing. I just googled 17291 and this post has been indexed within 13 minutes.

Update 2: The source seems to come from The IUCN Red List of Threatened Species and this blog post has a breakdown. So the number does appear to be exact at least at a certain point in time.

Posted by tpc at May 24th, 2010

Martin Gardner passed away last week on 22 May, aged 95. Wikipedia is a good place to read about his contribution in bringing mathematics to the public. My favourite article of Gardner’s is Six Sensational Discoveries that Somehow or Another have Escaped Public Attention, Sci. Amer. 232, 127-131, Apr. 1975. (Also published in Time Travel and Other Mathematical Bewilderments.) Inside, Gardner announces six discoveries among which a counter-example to the four colour theorem. Before you jump off your seat, the article was dated 1st April 1975. Yes, it’s another very clever hoax.

The best among the six is the claim that

is exactly an integer and this fact was found by Ramanujan. The attribution to Ramanujan was clever not because of Ramanujan’s remarkable prowess of calculation but that constant is actually an evaluation of the modular j-invariant

and of course has class number one.

Posted by tpc at May 21st, 2010

If for some reason, you wanted the values of certain mathematical constants to a few billion digits, numberworld.org might be a site for you. Well known ones like are all there to billions of digits. Not much is written about the what and the how of the computations so you’ll have to take the author’s word for it. Incidentally do not make the mistake of confusing A.J. Yee, computer scientist and author of the website with A.J. Yee, the combinatorist.

Posted by tpc at May 8th, 2010

Being down under results in the privilege of receiving my copy of the Notices of AMS five months late. The December 2009 copy showed up in my mail last week and the best article of all is about Legendre. It seems that despite being such a famous figure in mathematics, the mathematical world had been using a wrong portrait of him until recently. Everything came to light only when a computer search revealed that two different man named Legendre shared the same portrait. Full story available online.

Posted by tpc at May 5th, 2010

is a book by Niels Lauritzen that I just checked out. My initial impression is that it is well written and contains many interesting gems. It certainly looked like a good book to teach from, although the topics covered are a little broad and thus I suspect not in enough details for a student struggling to learn abstract algebra.

One of the gems was how a computational number theorist, Thomas R Nicely, nicely found a flaw in Pentium’s floating point unit. The book provided this link to the discoverer’s site and as usual, wikipedia has a nice coverage.

Other gems include Sam Loyd’s 15-puzzle and of course the last chapter on Grobner Bases.

Posted by tpc at March 6th, 2010

The conjecture states that starting with any positive integer, n, map this it n/2 if n is even, or map n to 3n+1 otherwise. Iterating will eventually lead to 1.

This was a subject of today’s clever xkcd comic. It’s quite remarkable that if you google it, xkcd is the second hit after wikipedia. Pretty quick indexing, and I guess xkcd has a pretty high pagerank - I can’t check because I don’t have (like) the google toolbar.

Wiki has some good references on the problem but a really good source would be Jeff Lagarias. I remembered being interested in the problem at one point and printed out several papers to read. Of course, I had absolutely no good idea on how to tackle this. I would think that the ergodic theory guys would be in a good position to prove this.

Posted by tpc at January 15th, 2010

Announcement here https://documents.epfl.ch/users/l/le/lenstra/public/papers/rsa768.txt.
It was done on 12 Dec 2009 using NFS.

If you visit the webpage of Laboratory for Crytologic Algorithms, part of the team that accomplished this feat, you’ll see that they actually test algorithms on a PS3 cluster. I wonder if the experiment results are ever undermined by students trying to hijack the PS3s to play games.

Posted by tpc at October 7th, 2009

for

Genus = 0
N= 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18 or 25
This corrects an error in Schoeneberg’s book. See Sloane’s sequence A091401.

Genus = 1
N= 11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, 49 (presumably complete.)
See Sloane’s sequence A091403. Schoeneberg listed 24 as genus 3.

Genus = 2
N= 22, 23, 26, 28, 29, 31, 37, 50 (presumably complete.)
See Sloane’s sequence A091404.

A list for N=1 to 102 is also available here.

Posted by tpc at May 11th, 2009

The International Congress in 2010 will be held in Asia again, in Hyderabad, India. I love the conference logo.

To the uninitiated, the picture is about hyperbolic geometry and is related to the equivalent fundamental regions for a modular form and the inequality is the Ramanujan’s Conjecture for the Fourier coefficients of the delta function - the cusp form of weight 12 for the modular group.

Posted by tpc at April 17th, 2009

A classical and computational introduction is a new book by L.J.P. Kilford. New enough that it even has reference to the resolution of Serre’s conjecture. But this book is really an introduction to the classical aspects of the theory of modular forms and it does a great job. I took a few days to read through the book (of course, ignoring the details and proofs) and I would say it is very enjoyable. Kilford adds in lots of funny and quirky anecdotes, most of which I’ve read from different places but it’s nice to have everything collected in one book. For example, he mentioned Lang’s famous foreword:

“It is possible to write endlessly on elliptic curves. (This is not a threat.)”

I remembered being so tickled when I first saw this in Lang’s book.

Back to this book. Even with the subject of modular forms it is the same. He knows he can’t possibly explain everything and so is not afraid to be a little vague at times and cite the various references to where more in depth discussions can be found. Thus, he is able to accomplish much in this modest sized (200+ pages) book. My one small complaint is the title should not include “and computational”. I found the last chapter on computational aspects too brief. He highlighted some history, discussed MAGMA and SAGE, giving some examples of the codes used, but I believe that this is not enough for someone interested in computing modular forms to get started on. And the appendixes on MAGMA and SAGE codes are each one page long with two longish lines of commands.

Posted by tpc at April 8th, 2009

I chanced upon this neat number puzzle devised by William Wallace, via wild about maths. It’s form is certainly recognizable, but still the effort deserves praise.

I remember a similar puzzle played with a deck of cards. You lay out the cards in rows and ask the participant to tell who which row his card is in. The trick is to collect the cards back horizontally, but deal out the next four rows vertically, this way after a few deals, you can tell which card is it.

This happen to tie in with a class I’m teaching tomorrow, maybe I’ll print out the cards and try it in class.

Posted by tpc at December 13th, 2008

I don’t believe I have seen the following identity before, although it is pretty easy to prove it by induction.

I should mention I saw it in Edward Barbeau’s book Power Play.

Posted by tpc at September 23rd, 2008

Yes, it is plural because two such primes were found within two weeks. See the
press release from GIMPS. I guess we can’t really comprehend how big a 10 million digit number is. My ruler tells me that the normal font size of this blog gives 6 characters per cm, hence 600 characters or digits per metre. 10 million / 600, gives about 16.6 km. That’s how long the number would be written out in a line. It would take me almost 2 hours to run 17km. But that’s just writing it out! We haven’t even talked about doing arithmetic on it, let alone checking that it is indeed a prime number!

UCLA who found the larger of the two primes first is getting about $50,000 of the prize money. There goes my dream of making an impact in mathematics and getting some useful cash out of it. And no, I’m realistic enough to not dream about solving Riemann’s hypothesis or the BSD. In my younger days, I dreamt about Goldbach’s conjecture, which is still open but without cash prizes. Even that may be knocked down within the next few years. We do live in exciting times.

Still it is momentous to break the 10 million mark, important enough for me to brush the dust off this blog and write something. Join GIMPS and keep searching. 100 million is not very far away.

Posted by tpc at March 16th, 2008

supposedly used by Russian Peasants as claimed by A. Posamentier and I. Lehmann in chapter 6 of their book The (Fabulous) Fibonacci Numbers.

Suppose you want to multiply 23 to 41. What you do is to write the numbers in two columns. In the first column you successively half (round down) while in the second column you double. By the time you reach 1 in the first column, look at those odd numbers in the first column and add the corresponding numbers of the second to get the answer.

23* x 41
11* x 82
5* x 164
2 x 328
1* x 656

The odd numbers are marked by *, hence the answer is
41 + 82 + 164 + 656 = 943.

The interesting question is why does this work? It boils down to binary numbers.

Posted by tpc at December 7th, 2007

If you’ve watched Disney’s High School Musical, you might remember a scene where the female lead corrected the teacher “shouldn’t the second equation read sixteen over pi?”

What was written on the board looked vaguely familiar, and so it got me trawling over the world wide web looking for details to no avail. I later found out from one of the world’s renown expert on that indeed the equation is one of three series that appeared in Ramanujan’s work “Modular equations and approximations to ” Naturally, I went back to the web and this time hit the jackpot. Two screencaps:


Ramanujan’s series

Now if you want to watch the video, here’s the link.
It happens in the first minute. So you don’t have to wait too long.

Posted by tpc at July 31st, 2007

Cryptography article on Nova

Posted by tpc at April 15th, 2007

My favourite sum:

To Euler,
The master of infinity, who summed with impunity!

PS: A nice article on infinity, explaining concepts up to Cantor’s work.
via http://www.mathed.org via amazon.

Posted by tpc at April 7th, 2007

How many times have you tried to search for a particular item and find that the only thing missing is the one that you want?

Two years back, I was searching for a 1995 article in the Rocky Mountain Journal of Mathematics, Volume 25 no. 4. I couldn’t believe my bad luck when I discovered the library had every volume, every issue for the past 35 years, since volume 1 in 1971, except the very issue that I was looking for! I was so amazed that I even went to ask the librarian about it.

A similar thing happened again today. The local newspaper had a 100 word column with the headlines Untrained S’porean’s maths Ideas on print. It’s about a Mr Bertrand Wong who claims to have proven the Twin Prime conjecture. His work was published in a peer reviewed journal known as the International Mathematical Journal. Some googling revealed that this is a relatively new journal started in 2002 and since 2006 has changed its name to International Mathematical Forum. Some 2006 articles are available online.

Wong’s article appeared in vol 3 no 8 pg 873-886. It got me piqued because just last week I was looking for some material on the twin prime problem to share with some students. Unfortunately, our very well stocked library does not carry this journal, so I tried to go into Math Reviews to find out more about the article. But no matter how I search, I could not find a review. When I pulled out all the reviews for Int. Math. J vol 3 no 8, I discovered that out of 12 articles in that issue, all 11 have review entries except the very last article. Another coincidence?

On a related note, I found this page on exceptional Math Reviews.

Posted by tpc at September 2nd, 2006

Came across a funny little problem last week. Explain the following:

For primes , the decimal expansion of is non-terminating. Is there a formula for the number of repeating digits?

6 digits

2 digits

6 digits

16 digits

and my favourite twin primes

5 digits

21 digits