Google honors Euler!

Euler doodle

Kudos to Google!

Today marks the 306th birthday of Leonhard Euler and, thanks to Google, millions of non-math people are being exposed to some of his incredible achievements through this great doodle.

If you are interested in reading more about Euler, here are some great resources:

Happy Birthday, Euler!


Happy 100th Birthday to Paul Erdős!

Happy 100th!

Happy 100th!

March 26 is the birthday of one of the greatest mathematicians of all time, Paul Erdős.  Considering this fact, it should be easy to write some sort of tribute, right?  Well, maybe not.  When I originally wrote this post a few years ago to celebrate his birthday, I was very intimidated.  I was worried that, no matter what I wrote, I wouldn’t write enough to honor his memory.  I even wondered what I should write about.

Maybe I should write about the fact that he was gifted mathematician?  Erdős is said to rival Leonard Euler as the most prolific mathematician in history, having produced some 1500 mathematical papers, many with collaborators.

Maybe I should write about his quirks?  He could be known to appear at your doorstep, unannounced, for an extended visit, announcing that his “brain is open”.  Legend has it that he had trouble tying his shoes, buttering his toast and opening containers of orange juice.  He loved ping-pong.  Even his childhood was unique.

Maybe I should write about Erdős as the philanthropist?  Erdős had little need for money so most of the money he earned was donated … whether to charities, needy friends or to set up scholarships.  If there was someone, anywhere, who needed financial help, Erdős was there.

Or, maybe I should leave it up to a professional wordsmith?  In 1996, columnist Charles Krauthammer wrote a beautiful and touching tribute to Erdős, titled “Paul Erdős, Sweet Genius”.   I think I made the right choice.

If this isn’t enough and you are interested in learning more about Paul Erdős, you can read a more academic biography by clicking on this link.  If reading a book is more to your liking, here are three to consider.

  • The Man Who Loved Only Numbers by Paul Hoffman  (Click here to read my brief synopsis.)
  • My Brain is Open by Bruce Schechter  (Click here to read my brief synopsis.)
  • THE BOY WHO LOVED MATH: The improbable life of Paul Erdős by Deborah Heiligman – available in June 2013 (Click here to read the first review.  Click here to see some of the AMAZING illustrations and read an article about it in the New York Times.)

If you are interested in a few items that I have written about him, you can consider reading these.

Happy 100th Birthday, Paul!

Erdős and √2

paul erdos 2

I asked you to tell me at every step if you don’t understand something. You said nothing!

In honor of Paul Erdős’s 100th birthday (March 26), I wanted to share one of my favorite stories involving his attempt to prove the irrationality of the square root of 2 to a non-mathematician.

Now, before we get to the story, a quick mathematical refresher.  Remember that the proof uses the popular technique of “proof by contradiction” or “Reductio ad Absurdum“.  (For example, if you are trying to prove some claim “A” true, first assume instead that “the opposite of A” is true.  Then, show that the new assumption leads to some logical contradiction.  This contradiction means that “the opposite of A” is wrong and “A” must be true after all.  Tricky, isn’t it?)  I love how G.H. Hardy explains it, “it is one of a mathematician’s finest weapons. It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.”

(Still need a little more of a refresher with the proof?  Click here.  Don’t worry if you’re not a great mathematician – it doesn’t require a lot of math to understand it.  In fact, that’s what makes it so beautiful – or “from the book” as Erdős would say!)

Now, on to the story …

One afternoon, Erdős was visiting with his life-long friend and fellow mathematician Andrew Vazsonyi.  (This is the same friend who many years ago told Erdős that he was thinking of majoring in something other than mathematics.  Erdős responded by saying that if he chose that path, “I’ll hide, and when you enter the gate of the Technical University, I will shoot you!”)  Anyway, Erdős decided to explain the magic of mathematics to Vazsonyi’s non-mathematican wife, Laura, by proving the irrationality of the square root of 2.  Unfortunately for Erdős, things didn’t go according to plan!  As Vazsonyi tells the story:

One day, Erdos got reckless and told Laura, my wife, that he will prove to her the Pythagorean “scandal,” that the square root of 2 is irrational. (According to legend, the disciple of Pythagoras, who revealed the secret to laymen, was put to death.) He started with an almost blank sheet and started the proof .  “Laura, if you do not understand a step, let me know, so I will clarify the proof,” he said. Let us assume that the square root of 2 is rational, that is it equals a/b, where a and b are whole numbers. “OK?,” Laura agreed. Then he went down, step-by-step and reached a contradiction. “See, the assumption is wrong, the square root of 2 cannot be rational.

But Laura did not like the proof. Erdos got annoyed. “I asked you to tell me at every step if you don’t understand something. You said nothing.”

“Why didn’t you tell me at the beginning that this is all wrong?” said Laura. Erdos flipped his top.

I recalled that when Albert Einstein gave one of his last talks, at the end they unscrewed the black board and sent it to the Smithsonian. So I asked Erdos to certify the document, so I could keep it for history. He signed his name and p g o m a. d, signifying Poor Great Old Man Archeological Discovery. At age 70 he started to use LD for Legally Dead, and at 75 CD for Count Dead, for reasons unknown to me. 

Here is the actual “document” from that day:


Beautiful, isn’t it.  Happy 100th, Paul!  Thanks for all the great memories – and, of course, mathematics!

If you are looking to read a little more about Erdős, you can read any of these:

Here are the sources for this post::

New “largest” prime number discovered!

Hold on to your hats!  A new “largest” prime number has been discovered.  Meet

257,885,161 – 1 

How large is it?  Try to imagine this … a whopping 17,425,170 digits!  (Click here to see most of the number.)

This number is a special kind of prime number, called a Mersenne Prime.  First popularized by the French monk Marin Mersenne, primes of this form are generated using the formula 2p − 1 (where p is prime).  For example:  if p = 2, then 2– 1 = 3 or if p = 5, then 2– 1 = 31.  And, as you know, both 5 and 31 are prime numbers.

I know what you are thinking – I thought it was impossible to have a formula that generates primes.  Well, yes and no.  While there is no formula that will generate ALL prime numbers, there are many formulas that generate some primes.  Unfortunately, as with all prime formulas, even this formula doesn’t always work.  For example, if p = 11, then 211 – 1 = 2047.  2047 is a composite number with factors 23 and 89.

So, why bother with a formula that inconsistently generates primes?  Well, mathematicians are fun people.  And, like most people, they are attracted to big things – like big prime numbers.  Since this formula can generate some pretty massive numbers, the potential for monstrous-sized prime numbers exists.  And, what’s better than massive prime numbers?  Nothing!  In fact, some mathematicians are so obsessed with really big primes that they have started an internet search for big primes called GIMPS –  the Great Internet Mersenne Prime Search.

And, now, thanks to the GIMPS project, we have the 48th Mersenne prime … all 17,425,170 glorious digits.

Click here for the official press release.

A New Year’s Resolution … revisited

G. H. Hardy

” A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.” – G. H. Hardy

Well, it’s that time of year again.  The time of year millions of people resolve to change who they are, what they look like or what they do.  And, as it turns out, mathematicians are no different from anyone else.  Sometime in the 1940′s,G.H. Hardy sent the following list of New Year’s resolutions to a friend.

  1. To prove the Riemann Hypothesis
  2. To make a brilliant play in a crucial cricket match
  3. To prove the nonexistence of God
  4. To be the first man atop Mount Everest
  5. To be proclaimed the first president of the U.S.S.R., Great Britain, and Germany
  6. To murder Mussolini

Unfortunately, just like us mere mortals, G.H. Hardy never fulfilled any of his resolutions – what a shame, a proof of the Riemann Hypothesis would have been a great one.  Oh well.  At least he can inspire us to think big!