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.

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Say it ain’t so … narcissism in the Natural Numbers?

Thanks to the digital age, many people are slowly beginning to believe that everything they do has some special significance.  Whether it is what they had for lunch, how many times they yawned in meeting or the number of licks it took to get to the center of a Tootsie Pop, every boring fact of people’s lives has somehow become important.  To put it simply, thanks to technology, narcissism is running wild.

So, where can people go to escape this growing epidemic?  For a long time, I took refuge in the wonderful world of mathematics.  There, I cannot be bothered by the all important mundane facts of people’s lives.  I can focus my thoughts and energies on the mysteries of numbers, free from narcissism … or so I thought.

Unfortunately, for me and many people like me, even in the world of mathematics, we cannot escape narcissism.  It turns out that our very own Natural Number system has a problem with narcissism, or what mathematicians call Narcissistic Numbers!

What is a narcissistic number, you ask?  Well, the official definition is as follows:  a n-narcissistic number is an n-digit number that is the sum of the nth powers of its digits.

For example, take the number 1634.  Here, n = 4.  So, if 1634 is a narcissistic number, then it follows that:

1634 = 14 + 64 + 34 + 44

or

1634 = 1 + 1296 + 81 + 256

Now, before you give up on escaping narcissism and start posting your own important facts involving the number of hairs you have on your kneecap, there is still hope!  As it turns out, there are only a total of 88 narcissistic numbers in our Natural Number system.  Further, it appears that there is no special mathematical significance to these numbers … narcissistic, indeed!  So, if you can find a way to avoid these numbers, you can still enjoy many hours of narcissistic-free entertainment in the world of mathematics.

(Thanks to MathWorld for the official definition of a Narcissistic Number: Weisstein, Eric W. “Narcissistic Number.” From MathWorld–A Wolfram Web Resource.http://mathworld.wolfram.com/NarcissisticNumber.html)

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!

A little fun with the Erdős-Mordell Inequality Theorem

One of my favorite subjects in mathematics is Geometry.  You would think that, after 2,000 years or so, we would know all there is to know about the subject.  Of course, you would be horribly wrong.  Take, for example, this fun theorem that usually isn’t mentioned in a high school Geometry course.  Originally proposed by the legendary Paul Erdős in 1935, it was later proven by in 1937 by Louis Mordell and D. F. Barrow.

Erdos-MordellTheorem

The Erdős-Mordell Inequality Theorem:  If P is a point inside of ΔABC, and PA, PB and Pare the feet of the perpendiculars from P upon the respective sides BC, CA, and AB, then

 PA+PB+PC>=2(PP_A+PP_B+PP_C).

Here’s the fun part, can you prove it?  It doesn’t hurt to start with a simpler case.  Consider an equilateral triangle where P is the circumcenter.  Consider an isosceles triangle where P is the circumcenter.  Can you prove it for these simpler cases?  Yes?  Good.  Now, can you prove it for the general case?  It isn’t as easy as it looks.

Note: diagram and Inequality provided at :Weisstein, Eric W. “Erdős-Mordell Theorem.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/Erdos-MordellTheorem.html

Finally … a museum dedicated to math!

Well, it’s been a long time coming.  In a country that has museums dedicated to SPAM®, funerals and hobos, we finally have one dedicated to something much more interesting … mathematics!

Opening on 12/15/12 in New York City, the Museum of Mathematics will offer “dynamic exhibits and programs [that] will stimulate inquiry, spark curiosity, and reveal the wonders of mathematics. The museum’s activities will lead a broad and diverse audience to understand the evolving, creative, human, and aesthetic nature of mathematics.”

If you are interested in learning more about the museum or its founder, check out one of these websites:

I hope you get a chance to check it out!