  In this paper, we define $X$-base Fibonacci-Wieferich prime which is a
generalized Wieferich prime where $X$ is a finite set of algebraic numbers. We
are going to show that there are infinitely many non-$X$-base
Fibonacci-Wieferich primes assuming the $abc$-conjecture of
Masser-Oesterl\'{e}-Szpiro for number fields. We also provide a new conjecture
concerning the rank of free part of abelian group generated by all elements in
$X$, and we will use the arithmetic point of view and geometric point of view
to give heuristic.
