<s>[INST] Proving that $3^{n+2}$ does not divide $2^{3^n} +1$ for any positive integer $n$ Proving that $3^{n+2}$ does not divide $2^{3^n} +1$ for any positive integer $n$
I am thinking of applying inductionon $n$ but am not being successful. 
 [/INST] Suppose $\left.3^{n+1}\,\middle|\,2^{3^n}+1\right.$ and $\left.3^{n+2}\not\,\middle|\,\,2^{3^n}+1\right.$; that is $2^{3^n}+1$ has exactly $n+1$ factors of $3$. Then
$$
\begin{align}
\left(2^{3^n}+1\right)^3
&=2^{3^{n+1}}+3\cdot2^{2\cdot3^n}+3\cdot2^{3^n}+1\\
&=2^{3^{n+1}}+3\cdot2^{2\cdot3^n}\left(2^{2\cdot3^n}+1\right)+1
\end{align}
$$
Therefore,
$$
2^{3^{n+1}}+1=\overbrace{\left(2^{3^n}+1\right)^3}^{\substack{\text{exactly }3n+3\\\text{ factors of }3}}-\overbrace{3\cdot2^{2\cdot3^n}\left(2^{2\cdot3^n}+1\right)\vphantom{\left(2^{3^n}\right)^3}}^{\substack{\text{exactly }n+2\\\text{ factors of }3}}
$$
has exactly $n+2$ factors of $3$. Since this holds for $n=0$, for all $n\ge0$,
$$
\left.3^{n+1}\,\middle|\,2^{3^n}+1\right.\quad\text{and}\quad\bbox[5px,border:2px solid #C0A000]{\left.3^{n+2}\not\,\middle|\,\,2^{3^n}+1\right.}
$$
</s>