Last week, in Fundamental Understanding of Mathematics LXII we graphed a function involving a couple of kids running a lemonade stand.
This week, for a change of pace, we are going to take a look at some ideas about prime numbers, and show how to demonstrate that one of them is true.
When we took a look at division, we found that there are some whole numbers that don't divide up evenly, that is, there is a remainder when you try to divide 15, say, by 4. In geometric terms, I can't make a rectangle that is four wide, out of 15 unit squares. I can't complete the last row.
But if I play around with the unit squares for a bit, I discover I can make a rectangle, if I use 3 squares for one of the sides instead of 4.
But, what if I start out with 17 unit squares? It turns out that there is no way I can make any rectangle with 17 squares, except for the trivial rectangle that I can make by lining all the squares up in one row.
Whatever I try, I'll either wind up with extra squares:
or something missing from the rectangle:
We call numbers that won't make rectangles Prime Numbers. The first few prime numbers are
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 and so on.
Now, here's where it gets interesting... In algebraic terms, instead of geometrical terms, we say that a prime number has no factors other than itself and one (factors are the lengths of the sides of a rectangle, the number itself is the area of the rectangle.)
Another way to put it is this:
If P is a prime number, then there are no integers (greater than 1) a and b such that ab = P
Here is the interesting bit: if we exclude 1 or 2, since they only form trivial rectangles,
every integer greater than 2 is either a prime number, or is the product of prime numbers.
So, we have two kinds of positive integers: prime numbers, and numbers that we can get by multiplying prime numbers.
Let's take a look at a few of them.
3, well, that's in our list of Prime Numbers
4, that's 2 x 2, and 2 is a Prime Number
5, in the list
6, 2 x 3, both 2 and 3 are Primes
and so on, and on, and on, we'll never do them all this way.
We may be able to convince ourselves that this is true by examining some numbers and not finding any cases where the rule isn't true, but we will never convince that skeptical guy who keeps asking "Yeah, but what about..." and throws us a positive integer higher than any we've done so far. Even when we show that one works, he just sneers and tosses an even larger one our way. We never did like him...
But he's got a point. He's never going to run out of still higher numbers for us to check, which means we will never be done checking. He pwns us. We don't like that, because we never did like him anyways, and it's pretty galling to be pwned by someone like that.
So we turn the tables. Let's suppose he's right.
Let's say there is some positive integer that is Not a prime and Not a product of primes.
Let's also say that, since all the small numbers that we checked followed the rule, that we found this particular number by checking numbers, one by one, until we came across it. In other words, this is the smallest positive integer that doesn't follow the rule (there has to be a smallest one, even the skeptic will agree.)
Now, since we say that this number is not a prime, it can form a rectangle (if it couldn't, it would be a prime.) so, there are two smaller numbers, call them a and b, that when multiplied form our special number, call it x.
So, if x is not a prime nor a product of primes, and it is the smallest of it's kind, there are still two integers a and b, where ab = x
Now, let's take a look at those numbers a and b. First off, we see they are both smaller than x. Which means they both follow the prime or product of primes rule.
How do we know this?
x is the smallest number which does NOT follow the rule. If either a or b did NOT follow the rule, then they would be the smallest number instead. Since we agreed that x was the smallest such number, any number smaller than x follows the rule.
So a is either prime, P, or the product of primes, P x P x P and so on ... and b is also a prime, P, or the product of primes, P x P x P... (we don't know how many primes are needed to produce a or b, but we do know that all the factors of a and b, other than 1, are prime numbers)
since x = a x b then we can substitute P x P x P x ... for both a and b and we get
x = P x P x P x P x P x ...
in other words, our special not-the-product-of-primes number x turns out to be a product of primes.
Since, logically, x can't be both a product of primes and not a product of primes, it doesn't exist. There is no such number
. Hah! Take that! Annoying skeptical person...
Have fun in the comments