# Value of n for which given expression is a perfect cube

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!

• It seems like $n=14$ is the only solution.. – Tintarn Sep 27 '15 at 14:26

Try to expand $(1+3^2)^3$ and compare with your expression.
• 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... – Tintarn Sep 27 '15 at 14:45