I haven't been around very much lately. I've been a graduate student in mathematics the whole time you all have known me, but now I have an adviser, and when you have an adviser, your adviser gives you a ton of shit to do, and that really cramps the whole making-jokes-on-the-internet scene. It's like he's never thought even once that I might want to try to make a bunch of anonymous porn addicts laugh with a three-paragraph joke about Nerlens Noel's haircut.
So over the past couple of months, instead of writing jokes at a rate of fifteen words per hour, I have been doing math. "Well, that's fucking stupid," you say, as you refresh the interactions tab on your Twitter. And sometimes, yeah, when I've been working on the same problem for two hours and have made no progress, it feels stupid. But you know what it isn't? It isn't the awful bullshit you learn in grade school, taught to you largely by halfwits, that instills a deep-seated hatred of mathematics that typically lasts your entire life. It isn't memorizing a bunch of rules. It isn't following a recipe. It is solving puzzles, and playing games, and making your brain do things that at first seem impossible, and, yes, it is staring at the same problem for two hours and making no progress, and then figuring out a special case of the problem, and then seeing how the logic you used in the special case can be adapted in general, and feeling what can only be described as joy radiate outward from the base of your skull to your extremities as pure mathematical realization dawns on you like the sun.
Is it good for anything? Probably not. Who cares?
What I want to do right now, and hopefully on a regular basis, is describe the stuff I'm working on in plain English. I want to do this for three reasons: 1) Explaining mathematics to others, especially without specialized notation, is kind of hard and something I want to get better at; 2) explaining mathematics to others, especially without specialized notation, forces me to think about it in ways I might not have previously, and thus makes me a better mathematician; and 3) I truly believe that the stuff I'll be talking about is very interesting and cool and awesome, and I want to see if anyone here agrees, because you guys are my e-friends and e-friends tell each other about interesting and cool and awesome things. I'm very excited to make this one post and then not make another one ever again once I receive absolutely no feedback whatsoever, and also because I'm lazy.
There's a very special set of numbers called the natural numbers, sometimes called the counting numbers, because you count with them. The natural numbers consist of 1, 2, 3, ... off to infinity, and they might include 0 depending on who you ask, and whether or not 0 is a natural number is one of those silly things some mathematicians like to argue about, usually loudly and in public, because they think it makes them look weird and quirky, and for some reason that is desirable to them. I am not friends with such mathematicians.
Contained within the natural numbers are the prime numbers. A number is prime if, among the natural numbers, it is divisible only by itself and 1. By convention, 1 is not prime. But 2 is! 2 is divisible by itself, and 1, and there are no other natural numbers that could possibly divide it. A moment's consideration leads you to the fact that 2 is the only even prime number; indeed, any other even number is, obviously, divisible by 2, and thus not prime. 3 is also prime, as is 5, 7, 11, 13, 17, ... off to...
"Surely," you say to yourself, "there are infinitely many primes. After all, every natural number is just some collection of primes multiplied together, and since there are natural numbers as big as I want, if there were a biggest prime number, well, it just doesn't seem likely that I could obtain every number bigger than that by multiplying some subset of the finitely-many prime numbers together, even with repetitions, without missing a few."
Sure. But can you prove it? You can't? Well, Euclid could, and he died like 2300 years ago, moron. His proof (or at least, what is his proof at heart) of the infinitude of the primes is probably the first proof I ever learned in my life. The argument is simple and elegant and beautiful, and in spite of (or maybe because of) its simplicity, it remains one of my all-time favorites. It hinges on a concept we all know and love: that of the "plus one."
Take any finite list of primes. You think you've got them all? Guess what, fucko: Multiply 'em all together, and add one. What do you have? Well, you've got a number with a curious property: No matter which prime on your list you divide it by, you get a remainder of one. Indeed, the prime will evenly divide the product, and the "plus one" will be left over. What does this mean? It means you've created a number that is not divisible by any prime on your list. But every natural number is divisible by a prime! This means your arbitrary finite list of primes is incomplete; it is missing, at least, a prime dividing the number you created. In the words of Euclid, "Prime numbers are more than any assigned multitude of prime numbers." QED. You may now dazzle any number of ladies at the club with this incredible proof.
The prime numbers are amazing, and mysterious, and if I wanted to say it as dramatically as possible I would say that I am devoting my life to studying them. We know a lot about the behavior of the primes, but some of the things we don't know, some things for which a proof would win you a professorship anywhere in the world, can be stated very simply. One such question: Two primes are called twin primes if their difference is 2. For example, 3 and 5 are twin primes, as are 11 and 13, 29 and 31, 41 and 43, and so on. Now, here's a question that is both easy to understand and seemingly impossible to answer: Are there infinitely many twin primes? Most mathematicians think the answer is "yes," but so far, not one of the great geniuses that have considered the problem has been able to prove it. Good luck!
There is much more I want to say about the primes, but I'll save it for future Mathspins. Next time I might talk about "how many" primes there are, in the following sense: As I mentioned above, 2 is the only even prime. Since natural numbers are alternately even or odd, no more than half of the natural numbers could possibly prime. So what "fraction" of the natural numbers are prime? The answer is perhaps a bit surprising. For now, I'll finish by saying what I always say to my math classes after I show them something cool, words which are inevitably met with silence and eye rolls and even outright shakes of the head:
Wasn't that neat?