One of the more

surprising results in

mathematics, and a good example of a very

deep result which is nevertheless

simple enough that a six-year-old can understand its statement.

First, a succinct explanation for the mathematicians in the audience:

lim_{n->oo } pi(n)ln(n)/n = 1

And for the novices: for a start, go here to find out what a prime number is. Then go here to learn about logarithms, in particular those to base e. Now, what the prime number theorem says is this: to find out roughly how many primes there are less than a certain number, divide the number by its natural logarithm. The bigger the number, the more accurate the answer you get will be; in fact (to be very imprecise and annoy the pedantic readers) as the number gets "infinitely big", the approximation becomes "exact".

Amazed? Well, if you're not then you should be. Why the hell should the prime numbers' distribution have anything to do with e, which is defined using concepts which seemingly have nothing to do with number theory? The answer is long, and if I ever understand it I'll attempt a writeup.