8.32 Find all natural numbers $n$ such that $2^{8}+2^{11}+2^{n}$ is a perfect square of a natural number.
(6th All-Russian Mathematical Olympiad, 1980)
［Solution 1］(1) When $n \leqslant 8$,
$$
\begin{aligned}
N & =2^{8}+2^{11}+2^{n} \\
& =2^{n}\left(2^{8-n}+2^{11-n}+1\right) .
\end{aligned}
$$

For $N$ to be a perfect square, $n$ must be even, i.e., $n=2,4,6,8$.
Upon verification, $N$ is not a perfect square for any of these values.
(2) When $n=9$,
$$
N=2^{8}+2^{11}+2^{9}=2^{8} \cdot 11,
$$

which is not a perfect square.
(3) When $n \geqslant 10$,
$$
\begin{aligned}
N & =2^{8}+2^{11}+2^{n} \\
& =2^{8}\left(1+2^{3}+2^{n-8}\right) \\
& =2^{8}\left(9+2^{n-8}\right)
\end{aligned}
$$

Thus, $9+2^{n-8}$ must be a perfect square and an odd perfect square. Let it be $(2 k+1)^{2}$, i.e.,
$$
\begin{aligned}
9+2^{n-8} & =(2 k+1)^{2}=4 k^{2}+4 k+1 . \\
2^{n-8} & =4 k^{2}+4 k-8=4(k-1)(k+2), \\
2^{n-10} & =(k-1)(k+2) .
\end{aligned}
$$

Since $k-1$ and $k+2$ are one odd and one even, we must have
$$
\begin{array}{l}
k-1=1, k=2 . \\
2^{n-10}=4=2^{2} .
\end{array}
$$

Therefore,
$$
n=12 \text {. }
$$
［Solution 2］Assume $2^{8}+2^{11}+2^{n}$ is a perfect square.

Then,
$$
\begin{array}{l}
2^{8}+2^{11}+2^{n}=m^{2} \\
2^{n}=m^{2}-\left(2^{8}+2^{11}\right) \\
=m^{2}-2^{8} \cdot 9 \\
=(m-48)(m+48)
\end{array}
$$

Since 2 is a prime number, let
$$
\left\{\begin{array}{l}
m-48=2^{s} \\
m+48=2^{t}
\end{array}\right.
$$

where $t, \quad s \in N, s<t, \quad t+s=n$.
Eliminating $m$ gives
$$
\begin{array}{l}
2^{t}-2^{s}=96, \\
2^{s}\left(2^{t-s}-1\right)=96=2^{5} \cdot 3 .
\end{array}
$$

Since $2^{t-s}-1$ is odd, we have
$$
s=5, \quad 2^{t-s}-1=3 .
$$

Thus,
$$
s=5, \quad t=7,
$$

Therefore,
$$
n=s+t=12 \text {. }
$$
［Solution 3］Let $2^{4}=x$, then
$$
2^{8}=x^{2}, \quad 2^{11}=2^{7} x,
$$

Thus,
$$
2^{8}+2^{11}+2^{n}=x^{2}+2^{7} x+2^{n} .
$$

If $2^{8}+2^{11}+2^{n}$ is a perfect square, then the equation
$$
x^{2}+2^{7} x+2^{n}=0
$$

has two equal real roots, so
$$
\begin{array}{l}
\Delta=\left(2^{7}\right)^{2}-4 \cdot 2^{n}=0 \\
n=12
\end{array}
$$