Assume for a second the Beal's conjecture is correct. It's then true that $A=pa$,$B=pb$, $C=pc$ and the exponent on each gives back $$p^xa^x+p^yb^y=p^zc^z$$ but this shows that if one of $x,y,z$ is minimal, we get back that two terms have $p$ still after division, but the third won't defying distributivity of multiplication over addition/subtraction. The only way to eliminate $p$ completely is if exponents on $p$ in the factorization contain the other exponents ( think parenthesized exponentiation rules) .
Is this ever used to speed up searching ?