Counting on Monsters

28 September 2010

While we’re on the subject of books, here’s a book for the smaller mathematicians in your life.  The ones who can’t necessarily spell “mathematician”.

The book is You Can Count on Monsters by Richard Evan Schwartz (ISBN 1568815786), and it’s a picture book about prime number decompositions.  It’s much more colorful than you’re picturing, I promise.

One thing that makes this book so nice is that it doesn’t beat you over the head with anything.  There are some very basic remarks about primes and multiplication at the beginning  and some slightly deeper remarks at the very end, but the vast bulk of the pages have no text at all.

Each number from 1 to 100 gets a double page.  On the left is the the number and a configuration of that many dots (usually clustered into spirals or some such in an interesting way).  On the right is a whimsical (and strangely compelling) drawing of a monster.  For a prime number, the monster is some simple monster which smoehow embodies the nature of the number (the right number of teeth, or legs, or whatever).  For a composite number, the monster is in some way a conglomeration of the corresponding prime monsters.  (So the 70-monster incorporates the natures of the 2-, 5-, and 7-monsters.)

There’s plenty to stare at, plenty of patterns sitting right near the surface, and plenty more lurking underneath to be discovered over time.  An artistically-inclined child might try to invent prime monsters larger than 100, or to draw some composite monster.  (I’ll confess that when I bought the book, I entertained myself for quite some time trying to draw the 1001 monster.)  It’s just fun to look at, and it provides lots of interesting avenues for mathematical conversation with “grown-up” mathematicians.

Speaking as a number theorist, I love the way this book conveys the essential predictable/chaotic dual nature of prime numbers.  There are always more monsters, but you’re never quite sure when you’ll meet the next one.

How many prime numbers are there?

9 July 2009

“You can’t deal with my infinite nature.”

“That is so not true.  Wait, what does that even mean?”

So far, this blog has dealt with questions that are nontrivial to formulate but easy to answer.  Today I want to talk about a question about prime numbers which is not at all easy to answer.  Indeed, in many ways this is the starting point for the branch of mathematics where I live, number theory.

In number theory, we are especially interested in prime numbers.  (Quick and dirty definition in case it’s been a while: a prime number is an integer greater than 1 which cannot be written as a product of two integers greater than 1.)  The first few prime numbers are pretty easy to find just by trial and error.

$2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,\ldots$

So how many of these things are there?  Could we keep making this list as long as we want, or would we eventually run out of prime numbers?  In other words, which of the following alternatives is true?

Alternative A: There are infinitely many primes.  (There is no largest prime.)

Alternative B: There are only finitely many primes.  (There is a largest prime.)

I claim I know which of these statements is true and which is false.  I don’t just mean I have a persuasive guess, I mean I know.  One of those statements is right and one is wrong and there is no room for confusion about which is which.  Actually, this is a fundamental fact which should be known by any math major at any college in the world.

Notwithstanding what I just said, think about how preposterous it is to try to answer this question.  How could you ever know either of these things?  If you believe alternative A, then you have to believe that no matter how many prime numbers I have written down, you could find one larger than any of them.  If you believe alternative B, then you have to believe that there is a finite list of prime numbers, and no matter what number I say, however gigantic, all its prime factors are on the list.  Either way, we are claiming to know an infinite number of pieces of information.

You could get the faster supercomputer in the world to hunt for primes, but that wouldn’t help in the least.  No matter how many billion primes it found, that would be no guarantee there would be any more.  No matter how may years it went without finding another prime, that would be no guarantee it wouldn’t find one tomorrow.  So computational brute force is no good here.

It seems like a fool’s dream to want to settle the matter once and for all.  Nonetheless, we’ll answer it together after the cut.