In 1937, Lothar Collatz proposed the following conjecture: Start with a positive integer n, then repeatedly iterate the following: If n is even, divide it by 2; if n is odd, compute 3*n+1. Collatz conjectured that for every starting value n, the result will invariably return to 1.
The Collatz conjecture has been studied by thousands of mathematicians and computer scientists. Portuguese mathematician Tomas Oliveira e Silva has verified the conjecture for all integers up to 5.76 x 10^18. But no proof has yet been found. Well-known mathematician Paul Erdos once characterized the Collatz conjecture as “Mathematics is not yet ready for such problems”. He offered a $500 reward for its solution.
Now German mathematician Gerhard Opfer (who coincidentally was a student of Collatz) has announced that he has proven the conjecture. A draft of his paper, which has been submitted to a journal for review, is available here.
Some additional information is available from this New Scientist article, from which some of the information above was excerpted. Additional information is given in an article by Jeffrey Lagarias, which is available here.
Added 3 Jun 2011: It now appears that the proof has flaws. See our post Quick tests for checking whether a new mathematical result is plausible.