Let $M_n$ denote the Mersenne numbers $M_n=2^n-1$.
As $n$ varies, must $M_n$ be divisible by arbitrary large prime $p$ with exponent one, i.e. $p \mid M_n, p^2 \nmid M_n$?
In other words, must the exponential diophantine equation $2^n-1 = A x^2 y^3$ for constant $A$ has only finitely many solutions $(n,x,y)$ and $n >1$?
Related question is this.
Also related question that might show there are infinitely many non-Wieferich primes is here.
Looking for unconditional results, abc easily implies it.
Added
The paper Remarks on Exponential Congruences and Powerful Numbers P. RIBENBOIM on p7.
(M') There exist infinitely many Mersenne numbers which are not powerful.
(M') implies B_2 implies infinitely many non-Wieferich primes.
One easy way to construct infinitely many non-powerful Mersenne numbers is to observe that 3 divides $M_{6n+2}$ with exponent one, so $M_{6n+2}$ is not powerful.
Doesn't this approach give infinitely many non-Wieferich primes?