Theorem: If x^3 + y^3 = z^4 then the the largest prime factor b of x divides y.
Let  x = ab. Assume b does not divide y so y = bk+r for some k and r < b.
Now ,
1.  x^3 + y^3 = (x+y)(x^2 - xy + y^2) = z^4 => z = (x+y)h for some h. 

So z^4 = (x+y)^4(h^4). Then from 1., 

2.  (x^2 - xy + y^2) = (x+y)^3(h^4). Adding -xy to both sides we get,

3. (x-y)^2 = (x+y)^3(h^4) - xy

Substituting x = ab and y = bk+r and expanding, we get

 (ab-(bk+r))^2 = (ab+(bk+r))^3(h^4) - ab(bk+r)  = 
bG + r^2 = bH + r^3 for some G and H. This implies b divides r a contradiction r<b. Therefore,
The largest prime divisor of x divides y.