Suppose, a third Wieferich prime will be found.
Can we estimate the chance that this new Wieferich-prime will refute the conjecture that a Mersenne number $\ 2^q-1\ $ with $\ q\ $ prime is always squarefree ?
It is well known that $\ p^2\mid 2^q-1\ $ with primes $\ p,q\ $ implies that $\ p\ $ must be a Wieferich prime and that the two known Wieferich primes cannot satisfy $\ p^2\mid 2^q-1\ $.
If $\ p^2\mid 2^q-1\ $ , then the order of $\ 2\ $ modulo $\ p^2\ $ must be $\ q\ $.
Hence a third Wieferich prime $\ p\ $ would refute the above conjecture if and only if the order of $\ 2\ $ modulo $\ p^2\ $ is a prime number , which is not the case for the known Wieferich-primes. What is the chance this is the case , if the third Wieferich prime has , lets say , $\ 20\ $ digits ?
Also : How is the situation in the case of Fermat-numbers ? Can we hope that a third Wieferich prime will refute that Fermat-numbers are always squarefree ?