# 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 ?

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Here's what Wolfram says... – draks ... May 14 '14 at 11:06

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.
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.