Why is it gay? Because I'm doing it.So I'm reading this cool book about some of Leonhard Euler's better known papers, and two nights ago -- while plodding my way through the tedious, repetitive nature of Euler's signature style -- I was exposed for the first time to Goldbach's conjecture. It completely blew my mind. All even numbers greater than two can be expressed as the sum of two primes. Whoa.
Learning something like this for the first time is like finding out two completely different people you've known for years somehow know each other. It's quite disorienting.
It turns out that Goldbach and Euler exchanged letters on the subject, and that Euler wound up offering up the version of the conjecture that we're familiar with today. Neither one has been proven, but it's hard to prove that something is infinitely true, I suppose. No matter, it's still one of Euler's most accessible and intuitive ideas.
When I first thought about this conjecture I was reminded for some reason of the stunning relationship between odd numbers and perfect squares. I remember being knocked right on my ass the day I learned that the distance between consecutive squares is exactly the consecutive odd numbers. For example, the first few squares are 1, 4, 9, 16, 25 and 36. The distance between these numbers are, respectively, 3, 5, 7, 9 and 11. And apparently it goes on like this forever.
(I don't know if there is a similar significance for cubes, as in 1, 8, 27, 64, 125, 216 and 343. The difference between these numbers are 7, 19, 37, 61, 91 and 127, which apparently are the centered hexagonal numbers. I have no idea if that means anything.
So here's where I took it. The counting numbers start as 1, 2, 3, 4, etc. But the odd numbers that separate the squares begin with three. Isn't 1 the first odd number? The solution seems to be to start counting at 0, as the distance between 0 and 1 is 1, and both numbers squared are the same. So you would start counting the perfect squares as 0, 1, 4, 9, 16, 25, 36, etc. That seems unnatural for some reason, but I suppose it lends credence to the idea that you always start counting with zero, i.e. the "what year does the millenium start?" ridiculousness).Anyway, these relationships got me thinking about the nature of the numbers 0, 1, and 2. They seem to be the most mysterious of all counting numbers, and here's why: 0 is the only number that is neither negative nor positive, 1 is the only number that is only divisible by itself and nothing else, and 2 is the only even prime. I found myself watching questions roll around in my head like, is zero a prime? Is one? What?
By the way I'm writing my noodling meanderings here without doing any research, because I want my thoughts to be fresh. Later I'll go look it up and find out that, duh, I wasn't even close. But how awesome is it to think about things like this? You can look up the answers in any old book. Trying to come up with the answers first is the fun part.
The other thing about zero that blew my mind was the close relationship it seems to share with infinity, a subject into which I shan't delve too deeply. (Nor for that matter dare I wade into the madness of Riemann.) But I remembered that a shortcut between the two can be found in an expression involving the limit as n goes to infinity of n and 1/n. Just putting n in the denominator instead of the numerator switches infinity to zero, like it ain't no thing.
I guess I always knew that calculus was the study of the infinite and the infinitesimal, but I never really thought much about how easy it is to go from zero to infinity. Does this mean zero and infinity are somehow the same thing? No, that's gay.
(BTW I checked and 1 is indeed not a prime number, which makes all the sense in the world to me. A prime is any number that is divisible by itself and one. And one. Not "or one.")
Love it.






