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Find a natural number n such that the expression $3^9+3^{12}+3^{15}+3^n$is a perfect cube. I converted expression to $3^9(1+3^3+3^6+3^{n-9}).$ Clearly if we prove the expression in bracket to be a perfect cube, we can prove the expression as a whole to be a perfect cube, but I cannot prove the expression in bracket to be a cube further. Please help!

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  • $\begingroup$ It seems like $n=14$ is the only solution.. $\endgroup$ – Tintarn Sep 27 '15 at 14:26
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Try to expand $(1+3^2)^3$ and compare with your expression.

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  • $\begingroup$ That's indeed useful if you want to find that solution $n=14$. But I don't see how it brings you closer to proving that this is the only solution... $\endgroup$ – Tintarn Sep 27 '15 at 14:45
  • $\begingroup$ Right, but the question didn't ask for ALL solutions. $\endgroup$ – Aretino Sep 27 '15 at 14:46
  • $\begingroup$ Oh I see. I did not read the question carefully. (+1) for your solution then. But nonetheless, it seems an interesting problem to prove that it's the only solution.. $\endgroup$ – Tintarn Sep 27 '15 at 14:48
  • $\begingroup$ I agree on that. $\endgroup$ – Aretino Sep 27 '15 at 14:49

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