<s>[INST] Prove that for ${a^3} = {b^2} + {c^2}$. where $a$, $b$ and $c$ are positive integers that there are an infinite number of values for $a$. Prove that for ${a^3} = {b^2} + {c^2}$. where $a$, $b$ and $c$ are positive integers that there are an infinite number of values for $a$.
 [/INST] Let suppose first that exist $u,v\in\mathbb{Z}$ such that $a=u^2+v^2$ then $a^{3}=(au)^2+(av)^2.$ That is, every number $a$ that is sum of two squares is solution to the problem. Now, $5=2^{2}+1^2$ so every number of the form $5n^2$ can be written as $(2n)^2+n^2$ so there we have infinite possible values to $a.$  For example, taking $n=1$ we have $5=2^2+1^2$ so $5^3=10^2+5^2,$ for $n=2$ we have $20=4^2+2^2$ so $20^3=80^2+40^2,$ etc...
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