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Powers Of Three, Oh Well!

Can anyone please help me solve this problem? I have been stuck with this for a long time and am unable to solve it.

The expression:

\(3^{9} + 3^{12} + 3^{15} + 3^{n} \)

is a perfect cube for some natural number \(n\). Find the value of \(n\).

Note by Nilabha Saha
11 months, 1 week ago

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\(3^{9} + 3^{12} + 3^{15} + 3^{n} = 3^9(1 + 3^3+3^6+3^{n-9})\).
\((a+b)^3=a^3 + 3 a^2 b + 3 a b^2 + b^3\).

\((3^k+1)^3 = 3^{3k} + 3^{2k+1}+3^{k+1}+1 = 1+ 3^3+3^6+3^{n-9}\)
It works for \(k=2, n=14\).

Maria Kozlowska - 11 months, 1 week ago

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Nice factorization! Are there other values of \(n\) that would work?

I can show that \( n = 3m \) would not work, but the other cases seem somewhat hairy.

Calvin Lin Staff - 11 months, 1 week ago

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I think it can be shown that the number to be cubed needs to be in a form \((9n+1)3^3\). Wolfram alpha shows just one solution. For a proper proof you might need some number theory experts on Brilliant.

Maria Kozlowska - 11 months, 1 week ago

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Thank you very much. You literally releived me from an enthralling problem!

Nilabha Saha - 11 months, 1 week ago

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I'd start by thinking that that looks like a trinomial expansion. So let's take our cubed result and write it as 3^3 + a.

So we have 3^9 + 3^12 + 3^15 + 3^n = (3^3 + a)^3

=> 3^9 + 3^12 + 3^15+3^n = 3^9 + 3a3^6 + 3a^2 3^3 + a^3

=> 3^9 + 3^12 + 3^15+3^n = 3^9 + a3^7 + a^2 3^4 + a^3

Which fits if a = 3^5, and n = 14.

You can show that those are the only values by considering the graphs (of y = (fixed point + 3^x) and y=x^3) - they will only cross at one point.

Katie Finlayson - 9 months, 1 week ago

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Comment deleted 9 months ago

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While it is true that the graphs of \( y = C + 3^x \) will only cross the graph of \( y = x^3 \) at one point, when we require the expression to be a perfect cube, it need not just be that \( C + 3^x = x^3 \). It is actually \( C + 3^x = n^3 \) for all integers \(n\).

Calvin Lin Staff - 9 months, 1 week ago

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Yes, I was thinking of comparing something like an exponential with a cubic progression rather than solving both for x, but you're right, that doesn't fly where there is no relationship between the two integers. Interesting.

Katie Finlayson - 9 months, 1 week ago

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3^9,3^12,3^15.....9+3=12+3=15+3=18...so,I hope the answer is 18....

Md Mainu - 11 months, 1 week ago

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The answer is not 18. It is 14. The answer can't be found by assuming a geometric progression.

Nilabha Saha - 11 months, 1 week ago

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yeap i understand

Md Mainu - 11 months ago

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