No, it's not a distributed-computing thing; raw compute isn't the bottleneck. (That would only help if there were a need to check many machines that halt after a tractable-but-nontrivial number of steps; that's a narrow sweet spot, given the superexponential nature of the problem, and few machines of interest are believed to be in it.) Rather, it's a collaboration of human (mostly amateur) mathematicians chipping away at different parts of the problem.
In the old days we used to just chop wood, and burn it to keep warm. Then sit down and watch the sportsball game on TV to waste time.