From the Wikipedia:
The Collatz conjecture is an unsolved conjecture in mathematics. It is named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, as the Ulam conjecture (after Stanislaw Ulam), or as the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers, or as wondrous numbers.
From the Collatz Conjecture site:
Collatz Conjecture is a research project that uses Internet-connected computers to do research in mathematics, specifically testing the Collatz Conjecture also known as 3x+1 or HOTPO (half or triple plus one). Collatz Conjecture is based in Wood Dale, Illinois, USA and continues the work of the previous 3x+1@home BOINC project which ended in 2008.
Collatz Conjecture homepage:
Collatz has clients for Windows, Linux, Macintosh, NVidia CUDA & and ATI CAL. The CAL & CUDA clients have a huge speed advantage over the CPU apps. The ATI CAL client is now easy to set up due to the newly improved ATI support in BOINC. Unlike MilkyWay, even many lower end ATI cards can run this project (I ran it on the IGP on one of my AMD based MBs, a 790GX).
Join the fun and don't forget to sign-up for the Ars Technica team!
Edited to account for the new ATI CAL support in BOINC. You'll need to be using BOINC v6.10.x (I recommend v6.10.57 at the moment for best GPU support). For NVidia CUDA in Windows you'll need to be using v190.xx or newer drivers for v2.3 CUDA support. For ATI use driver v9.12 or newer.