A Wieferich prime has the property $$2^{p-1}\equiv 1\mod p^2$$ We only know two Wieferich primes $1093$ and $3511$ , a further Wieferich prime must exceed $2^{64}$.
It is conjectured that there are infinite many Wieferich primes. I wonder whether this gives a reasonable chance that a Mersenne number with prime index or a Fermat number can be non-squarefree.
For this , the Wieferich prime must have an order with respect to base $2$ that is either prime or a power of $2$. If the order is prime, we have a Mersenne number with prime index not being squarefree. If the order is a power of $2$ , we have a Fermat-number that is not squarefree.
Can we expect the existence of such a Wieferich-prime although it is known to have be huge ? Or is there a big likelihood that such a Wieferich prime does not exist ?