Happy 150th anniversary to the Riemann Hypothesis, one of the most important math problems ever!

Proposed by Bernhard Riemann in 1859, the Riemann Hypothesis deals with prime numbers. You may recall that a prime number is a positive whole number that has only two positive whole number divisors: one and itself. The first of them are 2, 3, 5, 7, 11, 13, in order.

This hypothesis would be able to provide a better estimate than ever before of a special function denoted as Pi(x). Pi(x) represents the number of prime numbers that are no bigger than x, where x is a positive number. For example, Pi(14) would be 6, because there are six prime numbers (2, 3, 5, 7, 11, 13) no bigger than 14. That's probably the most understandable explanation you're going to get that doesn't involve "zeta functions" and other technical terms.

Given that many of the best mathematicians have tried and failed to provide a solution, the proof is probably not easy or obvious, says Peter Sarnak, professor of mathematics at Princeton University and an authority on the subject. “Most experts expect that a proof will require a major new insight into the structure of whole numbers and the prime numbers,” he said.

But if you can solve it, the Clay Mathematics Institute will give you $1 million.

A proof would have implications not only for mathematics, but also for cryptography and computer science, says Ramin Takloo-Bighash, associate professor at the University of Illinois at Chicago. Internet security protocols, after all, are largely based on prime numbers. Experimental and theoretical evidence has supported the truth of the Riemann Hypothesis, although there are a small number of naysayers who say it can’t be proven, Takloo-Bighash said.

Still, there’s enough confidence in the truth of the Riemann Hypothesis that mathematicians have established “conditional” theorems, which can never be validated until someone proves the 150-year-old problem, says Kenneth Ribet, professor of mathematics at the University of California, Berkeley.

Riemann's paper on the subject was first published in November 1859, but no one knows the day. So, the American Institute of Mathematics picked a Wednesday in the middle of November to celebrate the 150th anniversary, said Brian Conrey, executive director.

Intrigued? Stop by one of these lectures today.

Google has announced it is testing a new form of search architecture codenamed "Caffeine," and the company wants your help to examine the results.

According to a post on the Google Webmaster Blog:

For the last several months, a large team of Googlers has been working on a secret project: a next-generation architecture for Google's web search. It's the first step in a process that will let us push the envelope on size, indexing speed, accuracy, comprehensiveness and other dimensions.

Google's current search infrastructure relies primarily on hyperlinks. Pages that receive a large number of incoming links from external sites are given a higher PageRank and are more likely to appear near the top of Google's search results.

Google is unlikely to stray far from its successful PageRank system, but the possibility of new search results is a huge deal to companies that rely on Google-generated traffic or those who have invested heavily in search engine optimization (SEO).

Business Week claims "Caffeine may cause corporate jitters:"

Google suggested that Caffeine could change search results, which raises the prospect of companies' needing to change their search engine optimization (SEO) to protect their Google ranking.

To test Caffeine for yourself, visit: http://www2.sandbox.google.com/

Google admits "most users won't notice a difference in search results," but the company is still looking for "feedback on the differences between Google's current search results and our new system."

Did you notice any substantial differences in your searches with Caffeine? Were they more accurate than Google's current results?

Pi is the ratio of circumference to diameter of a circle. Celebrate it on Saturday, March 14!

In honor of Pi Day, March 14 (because it's 3/14, like 3.14), I came up with the basis for this song, inspired by Don McClean’s 1971 song “American Pie,” when I was in high school. Feel free sing along with that tune.

American Pi
by Elizabeth Landau

A long, long time ago, it was one December when I thought I'd found the end of pi.
And if I'd had some batteries, my efforts would have surely pleased
That math fair judge, and that's no lie.
But tests and papers made me shiver with every paper they'd deliver.
More bad news in chemistry,
Just needed that time to study.
I can't remember if I cried when I realized that I hadn't tried
Before something clicked off deep inside
And then, my calculator died.

CHORUS:
So why, why can't I calculate pi
I just want to see the numbers 3.1415
And if that's all, then let's keep it alive
'Cause my calculator seems to have died.
My calculator seems to have died.

Did you write the law of sines, or draw two perfect parallel lines
Without a ruler helping you?
Can you still use a protractor, or find just one imaginary factor
Of a polynomial of order 2?
Well I had a TI-83 when a great misfortune came to me
The screen began to blur - I couldn't even find my curve!
I was a lonely teenage computer freak,
My pencils were chewed but my glasses were sleek,
And the future was looking pretty bleak
The day my calculator died.

[CHORUS]

Books, eraser, graphing paper, I was just a young number chaser
Searching for the holy grail.
Like Archimedes, Ramanujan, and Gregory, Leibniz, van Ceulen,
I could approximate, but then I’d fail!
My calculator was in a daze,
It just wouldn’t run on new AAAs.
The math fair was in an hour
And I was still out of power.
I knew that I was running out of time
A new formula would have been sublime
But all that I could do was rhyme
The day my calculator died.

[CHORUS]

“We know it’s not the pi you eat, this pie’s a far more delicate treat
And it can be so much fun.
Well, Pi is a movie, Pi’s a perfume,
The digits of pi could fill a room,
But could ever find the very last one?”
And although this poem was nice and sweet
In my heart I knew it couldn’t compete.
The other kids had proof.
I felt like such a goof!
And, as I sat down I felt disgraced,
Like I had just gotten a pie in the face.
I knew that I had really lost the race
The day my calculator died.

[CHORUS]

I met a guy in Period 3 who seemed like he could help me
At least he didn’t turn away.
My calculator looked so sad, but I know this kid of course was glad
To make me feel worse on Pi Day.
He said, “I can’t believe you were so dumb.
All you need is a new lithium.”
I asked, “What do you mean?”
He wouldn’t even come clean
And tell me there’s another set of batteries besides the quartet.
So until I get some, and it resets
I think I’ll eat some pie.

So why, why can't I calculate pi
I just want to see the numbers 3.1415
And if that's all, then let's keep it alive
But my calculator didn’t really die!

–p.s. Tell us how you're celebrating Pi Day this year. It's also the birthday of Albert Einstein! And, check out the Science Channel's salute to pi for more information about one of the most important mathematical constants.

Square Root Day celebrates perfect squares like these. How many more can you name?

For years I have been celebrating March 14 (3/14) as Pi Day, which I have always considered the mathematical holiday of the highest importance - I mean, come on, it's Albert Einstein's birthday! - and never thought about commemorating any other day for its numerical beauty.

But today, I have been out-geeked. Today, it turns out, is Square Root Day. That's because it's 3/3/09, and 3 x 3 = 9, meaning 3-squared equals 9. I had never considered that this relationship between month, day, and year occurs so rarely. In fact, this "square root" coincidence in dates only happens nine times every century - the last one being 2/2/04.

How should you celebrate Square Root Day? According to CNET, a teacher in Redwood City, California, has organized a contest. Apparently, people will cut root vegetables into squares or make foods into the square root symbol shape.

In addition, perhaps you can listen to the song by Huey Lewis and the News, "Hip to be Square."

For more on Square Root Day and other math holidays, check out the Cybrary Man Web site.

And, stay tuned for my Pi Day tips next weekend.

We all know one fish and one fish make two fish, but recent evidence suggests that fish may know that too. In fact, mosquitofish can count up to four, according to results from a team of researchers led by Angelo Bisazza of the University of Padua in Italy.

In each test, a lone female mosquitofish had two options for shoals to join, each containing between two and eight other fish. The results showed the lone fish would usually choose the shoal that was larger by just one fish, consistently picking the shoal of four fish over the one with three, and the shoal of three fish over the one with two.

But when one of the choices was larger than four fish, the fish could no longer discriminate. Monkeys and one-year-old children exhibit the same limit, Bisazza said.

Experiments examining what the fish would do when confronted with larger numbers found that, for shoals of more than four fish, they could still tell the difference between the quantities if there was at least a 2:1 ratio. That is, they would choose a shoal of eight fish over the one with four, but they could not discriminate between a shoal of 12 and a shoal of eight. These results are consistent with mathematical abilities observed in birds and mammals.

Similar performance has also been observed in people who speak languages that contain limited vocabularies for numbers, Bisazza said. For instance, speakers of the Amazonian language Munduruku only have words for numbers from one to five, and do not have names for numbers beyond that. Though they are able to solve nonverbal number tasks involving quantities up to 70, in exact arithmetic they do not do well with numbers larger than four or five, he said.

Besides fish, other non-human creatures that have shown at least some rudimentary mathematical abilities in studies include chimpanzees, macaques, dolphins, dogs, parrots and pigeons.

"Many researchers are now convinced that mammals and birds may share common mechanisms to count objects and compare quantities," Bisazza said.

–Elizabeth Landau, Associate Producer, CNN.com

Remember in math class when you had to multiply the diameter of a circle by a number called “pi,” approximately equal to 3.14, to get the circumference? Maybe you’d rather not remember that. But don’t underestimate this number. Pi has captivated the human imagination for thousands of years, and will be celebrated during Pi Day, observed on March 14, 3/14.

Before you join the math fest, take note that you can’t calculate pi on your cellphone. There are approximations like 22/7, but no simple fraction will give the exact number. And while great mathematicians throughout history toiled over uncovering more digits, pi doesn’t stop at 3.1415926535897932384626. These digits continue infinitely in an apparently random pattern that will never repeat. More than a trillion digits have been found so far, according to the Web site for the indispensable book The Joy of Pi.

Pi Day offers a fun reason to learn more about this fundamental constant and, of course, eat more pie. The Exploratorium in San Francisco will hold pie tossing events, along with educational lectures, on Pi Day. Other Web sites like TeachPi.org give suggestions for pi activities in the classroom.

If you have a knack for remembering things, try memorizing pi. The number’s mysterious, infinite string of digits has inspired some people to take up memorization as a sport. The current world record holder for memorizing pi is Chao Lu of China, who recited 67,890 digits from memory in 2005, according to the Pi World Ranking List.

Pi has also inspired poets, artists, and musicians. A poem called Poe, E: Near a Raven encodes 740 digits of pi, where each word represents a digit based on the number of letters in that word. Try that technique: see, a nerd a month considers pi.

Before you start writing, memorizing or baking, you’d better look the part. Web sites such as ThinkGeek sell pi attire. Givenchy even markets a perfume called Pi.

And don’t forget to wish Albert Einstein happy birthday. The physicist would turn 129 years old on this year’s Pi Day.

–Elizabeth Landau, Associate Producer, CNN.com