## Numbers: Perfect, Prime... Fresh and Juicy!

3/10/2011

A little while ago, we talked about factors: The factors of 6 are 1, 2, 3, and 6 since those are the only numbers that divide evenly into 6. When I was six years old, I told everyone my favorite number was 6.  Most adults thought I was cute and that when I was seven, my favorite number would be 7.  Well, unfortunately, 7 is not nearly as cool as 6, and here's why: 6 is a perfect number, and it is the first perfect number.

## Perfect Numbers

Perfect numbers are numbers whose factors add up to the number itself.  Since the factors of six (not including six itself, of course) are 1, 2, and 3, and 1+2+3=6, six is a perfect number.  We can pretty quickly check the numbers up to six to see that it's also the first perfect number.  This is why 6 is still my favorite number, even though I'm not 6 years old anymore.

The next perfect number is 28, and the next one after that is 496, and after that, they keep getting bigger and bigger--as you can imagine it's really hard to find perfect numbers because it would take us forever to just go around checking the sums of all the factors of all the numbers!

But fear not! Mathematicians have found a way to generate perfect numbers, and if you're curious about that method, you can read the Wikipedia article on Mersenne primes.  It has a section about the generation of perfect numbers.  The mathematician Euler proved that this formula generates all even perfect numbers.  As of yet, we haven't found an odd perfect number, but we don't know for sure that there aren't any.  So if you find one, you could be famous!  (At least in the math world...)

## Prime Numbers

Now we can move on to my second favorite number: 2.  It's a small number, but it's a really special one.  Two is the first prime and the only even prime.

Prime numbers are numbers who have only two factors: 1 and the number itself.  Given this definition, 1 is not a prime because it has only one factor: 1.  This means that 1 is actually in a class all by itself because it's the only number with only one factor!

So 2 (with factors 1 and 2) is the first prime number.  It is also the only even prime because every even number after 2 has 2 as a factor--that's the definition of being even.

After two, we have a lot of primes early on: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43...  And the prime numbers go on forever.  We'll show the proof for that in one of our Elegant Math posts.

Another curious thing about prime numbers is that even though for the most part they seem to keep getting farther apart, we keep seeing primes that are right next to each other (except for an even number in between): 11 and 13, 17 and 19, 29 and 31, 41 and 43...  No one knows if these "twin primes" keep going on forever.  If you can figure that one out, you'll be rich and famous!

The only triplet primes are 3, 5 and 7.  Can you figure out why?

## Composite Numbers

So, we've got these prime numbers.  All other whole numbers (besides 1) are called composite numbers.  They're called composite because they can be made by multiplying primes together.  This is a really cool theorem in mathematics, and we'll probably prove that one in yet another Elegant Math post.
The fun thing to do with composite numbers is to find their prime factors.  We can do this by creating a factor tree like the one on the left.  I found all the prime factors for 320.  They're at the ends of the branches of my tree: 2, 2, 5, 2, 2, 2, 2.  If we multiply all those numbers together, we get 320.  The greatest thing about prime factorization, is that every whole number has a unique prime factorization.  The numbers 2, 2, 2, 2, 2, 2, and 5 are the only prime factors we could ever find, no matter how we made our factor tree for 320.

## What's the big deal?

A friend recently asked me why prime numbers are such a big deal.  He said, "They're cool, but why do mathematicians care so much about them?"  Well, one answer to that is that prime numbers are the building blocks for all other whole numbers.  If we know the prime numbers, we can make every other whole number, and that's pretty powerful.

I guess another answer to his question (and probably the true answer) is that mathematicians just like to look at numbers and how they behave, and prime numbers behave in very interesting ways!