23. (USA 4) Find all natural numbers $n$ for which $2^{5}+2^{11}+2^{n}$ is a perfect square.
23. It is easily verified that no solutions exist for $n \leq 8$. Let us now assume that $n>8$. We note that $2^{5}+2^{11}+2^{n}=2^{5} \cdot\left(9+2^{n-8}\right)$. Hence $9+2^{n-8}$ must also be a square, say $9+2^{n-8}=x^{2}, x \in \mathbb{N}$, i.e., $2^{n-8}=x^{2}-9=$ $(x-3)(x+3)$. Thus $x-3$ and $x+3$ are both powers of 2 , which is possible only for $x=5$ and $n=12$. Hence, $n=12$ is the only solution.