<s>[INST] Pairs of integers $ (x,m)$ for which $\sqrt[3]{\sqrt[3]{x-2}+m}+\sqrt[3]{-\sqrt[3]{x-2}+m}=2$ hold? 
Find all pairs of integers $(x,m)$ for which $$\sqrt[3]{\sqrt[3]{x-2}+m}+\sqrt[3]{-\sqrt[3]{x-2}+m}=2$$ hold.

I have used this property :
Property:
if $$a+b+c=0 \implies  a^3+b^3+c^3=3abc, $$  I come up to the following equation:
$(2m-8)^3=-216(m^2-(x-2)^{2/3})$ , such that $a=-2, b=(\sqrt[3]{x-2}+m)^{1/3}, c=(-\sqrt[3]{x-2}+m)^{1/3}$, I can't solve the last equation however i tried $x$ as a paramater instead of $m$ , The solution from wolfram alpha are $(x,m)=(2,1),(66,4)$, Any Help ?
 [/INST] Generally solutions $(x,m)$ can be split into three cases:

*

*$(x,m)=\left(\frac{1}{243} \left(486-\sqrt{3} (m-1)^{3/2} (m+8)^3\right),m\right)$

*$(x,m)=\left(\frac{1}{243} \left(486+\sqrt{3} (m+8)^3 (m-1)^{3/2}\right)),m\right)$

*$(x,m)=(2,1)$
In the first two cases $m\ge2$ and we choose $m$ such that the resulting term for $x$ is an integer. For example if we choose $m=4$, the first case yields the solution $(x,m)=(-62,4)$ and from the second case we obtain $(x,m)=(66,4)$.
You can generate infinitely such solution pairs by choosing a $m$ where $m-1$ is an odd cubic, e.g. $m=3^1+1,3^3+1,3^5+1,\ldots$ or, more precisely, as shown by @Aqua where $m=3c^2+1$ for natural $c$.
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