Green-Tao Theorem

I have never been too crazy about number theory, but the Green-Tao Theorem (first proved in 2004) recently caught my attention. It states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. It is very surprising to me because my impression is that prime numbers get sparse very rapidly. You’d think evenly spaced points can’t possibly all land on prime numbers. Intuitively, you need something like geometric sequences.

The Green-Tao proof was an existence theorem. It doesn’t tell us how to find those sequences. I can’t imagine how but mathematicians actually managed to find some finite-length arithmetic sequences that are all primes. For example,

6171054912832631 + 366384*223092870*n, for n = 0 to 24

That’s it? I was hoping for something very exotic, like numbers so large you need specialized notations to express them.

PS: I am fascinated by Graham’s Number. It has nothing to do with Green-Tao Theorem. I just like the idea of a number so large you need a specialized notation.

PPS: The other brilliant mathematical hack is Bill Gosper’s algorithm for doing arithmetic with continued fractions. One of these days I’ll figure it out and write a program to demonstrate how it works. Here’s an implementation in c.

PPPS: The Mathematica Demonstration Project is doing a terrific job popularizing mathematics. I want to see two demonstrations: 1. Bill Gospher’s continued fraction algorithm. 2. The Helmholtz decomposition of vector fields. They are among the most beautiful things in mathematics that I’ve encountered.