If Beal's conjecture is proven true what would be the impact for Fermat–Catalan conjecture? If Beal's conjecture $A^x+B^y=C^z$ is proven true, then $A,B,C$ couldn't have different primes factors for $(x,y,z) > 2$. Then Fermat–Catalan conjecture which has a condition $A, B, C$ being positive coprime integers is then false ?

Let $A,B,C,x,y$ and $z$ be positive integers with $A,B,C$ coprime so that $A^x+B^y = C^z$.

The Beal conjecture states that this implies one of the three integers $x,y$ or $z$ is smaller or equal than two. The Fermat–Catalan conjecture states that this solution is one of finitely many if $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}<1$.

If Beal holds true the only thing that is implied for the Fermat-Catalan conjecture is that for any equation of the kind above one of the integers $x, y$ or $z$ is equal to 2. There is at least one integer equal to 2 because of Beal and at most one integer equal to 2 because otherwise $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}>1$.

It could still be possible that there are infinitely many solutions to this equation with one of the integer powers of $A,B$ or $C$ being two and still be possible that there are only finitely many of such solutions.