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My Trail:: When $n\leq 4,$ then easy to know that $3^{n} +81$ is not a perfect square.

Now let $\displaystyle n = k +4 (k\in \mathbb{Z^{+}}),$ then $3^{N} +81 = 81 (3^{k} +1).$

So $3^{N} +81$ is a perfect square, and $81$ is square,

there must be a positive integer $x,$ such that

$3^{k}+1 = x^2\Rightarrow 3^k = (x-1)\cdot (x+1)$

Now How can i solve after that

Help me

Thanks