Cheesehead thanks for your detailed reply!
OK, so (3) the lowest factor of composite Mersenne numbers is not a GIMPS goal.
And I appreciate the points you make so succinctly that the goals are perfectly verifiable with increasing computer power, plus that all cutting edge Mathematics is hard to verify in any case and errors have been made throughout history. The Four Color theorem was indeed a nice example of an early (in computer history) very difficult proof carried out using brute force software attack based initially on clever Mathematics and programming. Also a good example of previous flawed attempts as I read in the link you give. I appreciate that GIMPS and other distributed computing projects are not really any different in principle, just larger scale because of widespread participation. Thanks again for your lucid help.
Brian.
