<s>[INST] Variation of Pythagorean triplets: $x^2+y^2 = z^3$ I need to prove that the equation $x^2 + y^2  = z^3$ has infinitely many solutions for positive $x, y$ and $z$.
I got to as far as $4^3 = 8^2$ but that seems to be of no help.
Can some one help me with it?
 [/INST] Take any Pythagorean triplet $(a,b,c)$.
$$\begin{align*}
a^2+b^2 &=c^2\\
a^2\cdot c^4+b^2\cdot c^4&=(c^2)^3\\
(ac^2)^2+(bc^2)^2 &=(c^2)^3
\end{align*}$$
Multiplying $c^{6k-2}$, where $k$ is a natural number.
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