# positive integer value of n

#### jacks

##### Well-known member
If $3^{n} +81$ is a perfect square, find a positive integer value of $n$.

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

#### MarkFL

Staff member
I would write:

$$\displaystyle 3^n+81=m^2$$

$$\displaystyle 3^n=(m+9)(m-9)$$

Now, observing that:

$$\displaystyle 18+9=27=3^3$$ and $$\displaystyle 18-9=9=3^2$$

What value do we obtain for $n$?

#### mente oscura

##### Well-known member
If $3^{n} +81$ is a perfect square, find a positive integer value of $n$.
Hello.

$$3^n+81$$, It cannot be a perfect square, For being, both, odd numbers.

Demostration:

$$(2n-1)^2+(2m-1)^2=4(n^2+m^2)-4(n+m)+2=$$

$$=2[2(n^2+m^2)-2(n+m)+1]$$

The square root, of the latter expression, has to be an irrational number, for being divisible, only once for "2".

Regards.

#### mathbalarka

##### Well-known member
MHB Math Helper
mente oscura said:
$3^n+81$, It cannot be a perfect square
False. I can find a precise value $n$ for which the above is a perfect square.

#### mathbalarka

##### Well-known member
MHB Math Helper
Hint : Prove that $18$ is divisible by $m - 9$.

#### mente oscura

##### Well-known member
False. I can find a precise value $n$ for which the above is a perfect square.
Hello.

I am sorry. Really, I have considered "n", even number and, it is not possible, but yes it can be an odd number.

Regards.