ABC implies there are infinitely many non-Fibonacci-Wieferich primes (Q2307462)

The prime number $p$ is called an $\alpha$-Wieferich prime if $\alpha$ is a rational number and $\alpha^{p-1} \equiv 1 \pmod{p^2}$. The authors generalized these primes, introducing the concept of an $X$-base Fibonacci-Wieferich primes, where $X$ is a finite set of algebraic numbers. It is proved that if the abc-conjecture of Masser-Oesterlé-Szpiro is valid, then there are infinitely many non-$X$-base Fibonacci-Wieferich primes. In the paper, a new conjecture is formulated and discussed, concerning the rank of the free part of the abelian group generated by all elements in $X$.