Interesting Numbers

There's a well-known (to mathematicians) story about a famous mathematician, Srinivasa Ramanujan. The story goes that Ramanujan was taking a taxi ride with another mathematician, Godfrey Hardy. Their taxi was number 1729, and Hardy commented that this was rather an uninteresting number. Ramanujan replied that it was in fact quite interesting, as it is the smallest whole number expressible as the sum of two cubes in two different ways.

Sure enough, it's true: 1729 = 10³ + 9³ = 12³ + 1³. This is the kind of thing mathematicians love, and Ramanujan is very well-respected, so this story is popular, and 1729 has even come to be known as the Hardy-Ramanujan number. Mathematicians also enjoy generalizing, so there's now a whole set of taxicab numbers, having to do with summing up powers like the cubes in this example.

There's another thing about this story that I like to think about - the notion of an interesting number. It seems an obvious and expected thing that some numbers are interesting and others aren't; there's even a Book of Curious and Interesting Numbers. This fact about 1729 makes it interesting (at least to me), and I can imagine some other number that has no similar interesting properties.

However, something odd happens if you try to actually find an uninteresting number. Let's just look at whole numbers, starting with zero. 0 is interesting because it is the additive identity, among other reasons. 1 is interesting because it's the multiplicative identity. 2 is the first prime number. 3 is the first odd prime number. 4 is 2². We can continue this way until we find our first uninteresting number. But wait! The first uninteresting number seems like an interesting property for our number to have, so it turns out to be interesting after all.

This seems like a bit of a trick, and maybe it is. There's some kind of paradox or weird self-reference at work here; our decision that a number is uninteresting is what makes it interesting. You can come up with more tricks like this without too much trouble. Here's another quick one, just to make the point: what is the smallest whole number that is not describable in twenty or fewer words?

With 20 words, you can describe a lot of numbers. One hundred; fifty million and three; three googol squared; Steve Wozniak's bank balance. However, there aren't an infinite number of english words, so there aren't an infinite number of 20-english-word combinations. That means there are more whole numbers than 20-word combinations, so some numbers are not describable in 20 words, and there must be a smallest one of these. But wait! The smallest whole number that is not describable in twenty or fewer words is only 13 words long, so this number is describable in less than 20 words after all.

Ok, it's another nice little trick. These are fun little diversions, and they've been known for a long time. They just seem like curiosities, though, without much real meaning or importance in more concrete matters. You wouldn't think that this idea of self-reference could undermine the foundation of all mathematical thought. It did, but that's a topic for a future post.