On the Wikipedia page, the Beal´s conjecture is stated as:
If $A^x+B^y=C^z$, where $A,B,C,x,y,z$ are positive integers with $x,y,z>2$, then $A$,$B$, and $C$ have a common prime factor.
I think that it is within reach (possibly, this is already solved somewhere) to rephrase the conjecture so as to have $A,B,C,x,y,z >2$ as the assumption.
To clarify, the case when $A=1$ or $B=1$ is already solved and some of us call it Mihăilescu's theorem.
The case $C=1$ clearly does not have solutions.
So, the possible rephrasal needs for these cases to be considered and solved:
Of course, 1) and 2) can be considered as exactly the same cases, without any loss of generality.
So I am interested slightly in finding all of the solutions of two not extremely general Diophantine equations, the first task being:
Find all solutions of the equation $2^x+B^y=C^z$ where $B,C,x,y,z \in \mathbb N$ and $x,y,z>2$
and the second one being:
Find all solutions of the equation $A^x+B^y=2^z$ where $A,B,x,y,z \in \mathbb N$ and $x,y,z>2$
I know about Fermat–Catalan conjecture but this question is different, although much related.
So I would like to know are these two tasks somewhere solved and, if they are not, what is, approximately, the current progress made so far on them?