Music of the primes - website related to book of the same name. A history of attempts to prove the Riemann hypothesis (all non-trivial solutions of the zeta function lie on the line with real component 0.5), including a surprising link to quantum theory.
Great Internet Mersenne Prime Search - hunt for the largest known prime, of form 2**p-1 (p prime). Lots of interesting theory about prime numbers and algorithms for finding them.