<s>[INST] Expressing $2^n$ as sum of five rational cubes For which positive integers $n$ can $2^n$ be written as a sum of five non-zero rational cubes ? 
For which positive integers $n$ can $2^n$ be written as a sum of five positive rational cubes ? 
 [/INST] Notice that if $2^n$ can be written as the sum of five rational cubes, then so can $2^{n + 3}$ and $2^{n - 3}$.  The same goes for the sum of five positive rational cubes.
Allowing negative cubes, we then write
\begin{align*}
1 &= 1^3 + 1^3 + 1^3 - 1^3 - 1^3 \\
2 &= 3^3 - 2^3 - 2^3 - 2^3 - 1^3 \\
4 &= 2^3 - 1^3 - 1^3 - 1^3 - 1^3
\end{align*}
Without negatives, it appears to be much harder: a bit of work finds
\begin{align*}
64 &= 3^3 + 3^3 + 2^3 + 1^3 + 1 ^3 \\
\end{align*}
and one experiences difficulty finding other solutions.  However, it turns out to be the case that every rational number is the sum of three positive rational cubes (1,
2)
so a solution certainly exists for every power of two.  I imagine simple solutinos for, say, $32$ and $16$ could be attained with a bit of patience or programming.  This would finish the problem.
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