Does Fermat's Last Theorem or Euler's proof for the case n = 3 imply that $x*a^3+y*b^3 = c^3$ has no solutions in integers for $a,b,c$ positive integers and integers $x$ and $y$?
My question comes from me trying to provide an elementary proof of Beal's Conjecture $(A^x + B^y = C^z$ iff $gcd(A,B,C) > 1$ for positive integers $A,B,C,x,y,z$ and $x,y,z >2$). My attack was to assume that $gcd(A,B,C) = 1$ which implied that A,B,C were pairwise mutually coprime. Otherwise, they all share a prime factor (easy to show). Thus, $gcd(A,B) = 1$ implies $gcd(A^x,B^y) = 1$. Therefore, by Bezout's Identity there exists integers $z_1$ and $z_2$ such that,
(1) $$A^x*z_1 + B^y*z_2 =1.$$
Since $x,y > 2$, we have $x= 3 + s$ and $y =3 + r$. Thus, let $m = C^3*A^s$ and $n = C^3*A^r$, then (1) becomes,
(2) $$m*A^3 + n*B^3 = C^3.$$
if (2) has no solutions in positive coprime integers A,B and integers m and n, then there would be a contradiciton, and $gcd(A,B) > 1$ implying $gcd(A,B,C) >1$ and Beal's conjecture would follow. However, you all and others in private emails have provided a way to construct infinitely many counterexamples to my reasoning. Can this line of reasoning be fixed? Maybe an examination of Elliptic Curves and Modular Forms, etc. Thank you.