the april american mathematical monthly has an article "prime number patterns" by andrew granville with lots of interesting stuff in it. my favorite result from the article is that there are infinitely many primes p and q such that q are also prime. this is the first result like this that i've ever heard of. it's expected based on computations, of course, but i think it's pretty amazing that this can now be proven. unfortunately, things like this are still too difficult to prove: there are infinitely many primes p such that 2p + 1 is also prime there are infinitely many primes p such that p + 2 is also prime a preliminary version of the paper is here: http://www.dms.umontreal.ca/~andrew/PDF/PrimePattMonthly.pdf the result i quoted above is in section 5. bob baillie