What is the difference between factoring, double-checking, and Lucas-Lehmer testing?
Trial factoring is an effort to prove a Mersenne number composite (not prime) by finding a small factor. We do this in the hopes that a few hours of factoring will eliminate the need to run weeks of Lucas-Lehmer testing. Trial factoring cannot be used to find a new Mersenne prime. The Lucas-Lehmer primality test is the only way to prove these large numbers prime. Double-checking is the re-running of a Lucas-Lehmer primality test. Computers are not perfect and sometimes the original Lucas-Lehmer test was incorrect. If the original Lucas-Lehmer test was in error, then double-checking could find a new Mersenne prime.
