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Prove that for all $n \in \mathbb N$ the quotient of $\frac{3^{2^n}}{2^n}$ is an even number. Maybe we have to check a more general case, namely $\displaystyle\frac{3^k}{k}$. What would you do here? Thanks!

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    Now that there are two answers - may I assume that the real question was to show that $\frac{3^{2^n}-1}{2^n}$ is even?2012-09-28

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Hint: $3^{2^n}-1^{2^n}=(3^{2^{n-1}}+1^{2^{n-1}})(3^{2^{n-1}}-1^{2^{n-1}}).$ Continue.

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    nice and simple :-) Thanks!2012-09-28
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From Euler's theorem, $3^{2^{n-1}} \equiv 1 \pmod {2^n}$

Hence, $3^{2^{n-1}} = M2^n + 1 \implies 3^{2^n} = \left(3^{2^{n-1}} \right)^2 = \left(M2^n + 1 \right)^2 = 2^n \left( M^2 2^n + 2M\right) + 1$

Hence, $\dfrac{3^{2^n}}{2^n} = 2 M \left(2^{n-1} M + 1 \right) + \dfrac1{2^n}$