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

The Beal Conjecture asserts that if Aˣ+Bʸ=Cᶻ with positive integers $A,B,C$ and x,y,z≥ 3, then (A,B,C)>1.We organize some classical factorization techniques around a fixed base A and the quantity D=(Aˣ,Bʸ,Cᶻ).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 (y,z), and use them to show that Aˣ+B³=C³ has no solution with (A,B,C)=1 and x≥ 3 forA∈\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 43³+B⁴=C⁵ has no coprime solution;and (iv) reduce 2⁵+B³=C⁷ to a family of systems of Thue-type equations in Z[√[3]2], which we do not solve.The methods are case-by-case, and no proof of the Beal Conjecture is claimed.