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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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  • $\begingroup$ Here's what Wolfram says... $\endgroup$ – draks ... May 14 '14 at 11:06
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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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