Quote:
Originally Posted by philmoore
I found the larger of the two known prime factors of M8191 in 2003 and did the computations at the time to show that the cofactor was not only composite, but was also not a power of a single prime factor, so we know that the cofactor has at least two distinct prime factors.

If we ever find a nonsquarefree Mersenne number (with prime exponent), it would make headlines. The factor in question would be the third known Wieferich prime.
I run a script to check for this every few days. Takes a fraction of a second. Very tiny effort, very huge payoff, astronomical odds.