Fixed-Base Factorization Methods \\for the Beal Equation: A Case Study of Shared-Exponent and Quadratic-Field Reductions

The Beal Conjecture asserts that if $A^x+B^y=C^z$ with positive integers $A,B,C$ and $x,y,z\ge 3$, then $\gcd(A,B,C)>1$.We organize some classical factorization techniques around a fixed base $A$ and the quantity $D=\gcd(A^x,B^y,C^z)$.We (i) recall the explicit infinite families with $D>1$ obtained from B\'{e}zout's identity;(ii) prove elementary lemmas for the case $D=1$ when an odd prime divides $\gcd(y,z)$, and use them to show that $A^x+B^3=C^3$ has no solution with $\gcd(A,B,C)=1$ and $x\ge 3$ for$A\in\{14,20,21,22,23,29,30,41,42,45,52,63,65,78,222\}$, the base $222$ requiring an argument in the Eisenstein integers;(iii) describe a reduction over imaginary quadratic fields when exactly one exponent is even and the class number is a power of $2$, and use it to show that $43^3+B^4=C^5$ has no coprime solution;and (iv) reduce $2^5+B^3=C^7$ to a family of systems of Thue-type equations in $\mathbb{Z}[\sqrt[3]{2}]$, which we do not solve.The methods are case-by-case, and no proof of the Beal Conjecture is claimed.