<s>[INST] What is known about $x^m + y^m = z^n$ over $\mathbb{N}$ when $m,n \geq 2$ and $m \neq n$? So Fermat's Last Theorem resolves the question of positive integer solutions to $x^m + y^m = z^n$ when $m = n \geq 3$. But what about if $m \neq n$ and $m,n \geq 2$? Is anything general known about when there are positive integer solutions? Clearly if $gcd(m,n) \geq 3$ then there are no solutions by Fermat's Last Theorem. So let's assume $gcd(m,n) \leq 2$.
 [/INST] I attempt to solve the equation $x^3+y^3=z^n, x,y,z, n\in N \textrm{  and }n>3.$
Noting that $(1,1,2,1)$ is a solution and multiplying it by $2^{3k}$, we have
$$1^3+1^3=2^1 \Rightarrow \left(2^k\right)^{3 }+\left(2^k\right)^3=2^{3 k+1}$$
Therefore there are infinitely many solutions
$$(2^k,2^k, 2,3 k+1),$$
where $k\in N.$

Also $(1,2,3,2)$ is a solution and multiplying it by $3^{3k}$, we have
$$1^3+2^3=3^2 \Rightarrow (3^k)^{3 }+\left(2 \cdot 3^k\right)^3=3^{3 k+2}$$
Therefore there are infinitely many solutions
$$(3^k,2(3^k), 3,3 k+2) \textrm{ or } (2(3^k) ,3^k, 3,3 k+2), $$ where $k\in N.$
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