Integer solutions of $x^3+y^3=z^2$ Is there any integer solution other than $(x,y,z)=(1,2,3)$ for $x^3+y^3=z^2$?

If $n$ and $m$ are relatively prime positive integers,
there are infinitely many positive integral solutions to
$x^n + y^n = z^m$.
Proof: there are positive integers $u$ and $v$
such that $un - vm = -1$, or $un+1 = vm$.
Let $x = pa^u$, $y = q a^u$, and
$z = a^v$.
Then we want
$(p^n+q^n)a^{un} = a^{vm}
= a^{un+1}
$,
or $p^n+q^n = a$.
So, from $n$ and $m$ get $u$ and $v$.
Then choose any $p$ and $q$,
and compute $a = p^n+q^n$.
Then get $x$, $y$, and $z$ as above
($x = pa^u$, $y = q a^u$, and
$z = a^v$).
For this case,
with $n=3$ and $m=2$,
$u = 1$ and $v = 2$,
so choose $p$ and $q$
and let $a = p^3+q^3$
$x = p a$, $y = qa$, and $z = a^2$.
As a check,
$x^3+y^3 = (pa)^3 + (qa)^3
= (p^3+q^3)a^3
= a^4$
and $z^2 = (a^2)^2 = a^4$.
Of course, if you want $x$ and $y$ to be relatively prime,
it is harder.
