## Friday, April 18, 2008

### Carnival of Mathematics #31

Welcome to Carnival of Mathematics #31 !

Step right up, folks, and behold the amazing properties of 31. Now, you all know that 31 is a prime number. But that's not all -- 31 is also a Mersenne prime, that is, a prime number of the form 2p - 1. There are currently only 44 such primes known, and nobody knows if there are infinitely many. If I had to bet, I'd wager that the question of whether there are infinitely many will not be resolved in my lifetime.

So 31 = 25 - 1. And 231 - 1 = 2147483647 is a pretty interesting number, too. It's another Mersenne prime, and it's also the largest integer representable in 32-bit signed arithmetic. Because of that, when programs die because of integer overflow, you might end up with 2147483647 in some unexpected places, such as this video game, where a Toyota GT achieves the faster-than-light speed of 2147483647 mph.

So if 25 - 1 is prime, and 2 25 - 1 - 1 is prime, is 22 25 - 1 - 1-1 also prime? Regrettably no. We currently know 4 different prime divisors of 22147483647 - 1, and they can be found here, on the line labeled M ( M (31) ).

Oops, I got sidetracked there. Let's go back to 31. One more property of 31 (which I got from the very cool book by François Le Lionnais, Les Nombres Remarquables) is that it is the only prime number known that can be written in two different ways in the form (pr - 1)/ (pd - 1), where p is a prime and r, d are integers with r ≥ 3 and d ≥ 1. One such representation for 31 is (25 - 1)/(2-1). Can you find the other?

OK, enough about 31 ... back to the carnival!

Step right up, and learn how to factor monic quadratic integer polynomials at Life Jelly. Too easy for you, my friend? Then you can go directly to factoring arbitrary quadratics.

Polynomials not your game? Then how about figurate numbers? The simplest example of a figurate number is the total number of balls in an equilateral triangle, like a rack of pool balls. A rack of pool balls has 5 rows of balls with 1,2,3,4, and 5 balls, for a total of 15, so 15 is the 5th triangular number. Denise, at Let's Play Math, has a gentle introduction to these numbers. Question: what's the smallest number ≥ 1 that is both triangular and square? Can you prove there are infinitely many? (See Beiler, Recreations in the Theory of Numbers, Chapter 18, for more about these numbers.)

I'm allergic to cats, but don't let that stop you from visiting Calculus for Cats and the Prime Number Theorem over at catsynth.com.

What's that you say? Not hard enough? Then drop on over to Charles Daney at Science and Reason for an introduction to the factorization of prime ideals in extension fields. You might want to scan his previous entries if you're entirely lost.

Now, wait, I understand, you've had enough algebra. It's time for geometry. Visit 360, the informal blog of the Nazareth College Mathematics Department, for some real life nonagons. Were the Beatles really closet Bahais?

It took me a while to figure out why David Eppstein's blog is called 11011110; maybe there was a hex on me. I guess he's lucky his parents didn't name him George. But his blog always contains deep and beautiful results explained in simple ways, and this contribution about biclique covers is no exception. Step right up!

Our last geometric contribution comes from Praveen Puri at Math and Logic Play, who offers a puzzle based on the square. Free admission!

Next up, we have a bevy of logical beauties. While we're waiting for Jason Rosenhouse to finish his book on the Monty Hall problem, you can think about this variation from Magpie Tangent. Almost Philosophy gives us an introduction to propositional logic. Quan Quach at blinkdagger gives us a macintosh mystery with a prize for the winner. And Presh Talwalkar at Mind Your Decisions recounts a classic, the hat problem.

After all that logic, it's time for some illogic. Visit my own blog to see how a mathematics educator abuses mathematics for Jesus .

That's all there is folks, there isn't any more. Until Carnival of Mathematics #32, that is.

catsynth said...

Hi there. I think my article got missed - submitted this morning.

entry here

BTW, I'm enjoying the other entries - and your "critique" of The Faith Equation.

Jeffrey Shallit said...

Sorry, catsynth, I didn't get your submission.

But I added you in now.