Find all natural numbers $n$ such that $2^{n}+3$ is a perfect square.
For $n=0,2^{n}+3=2^{2}$ is a perfect square. We verify that $n=1$ is not a solution. Therefore, we can assume $n \geq 2$ and then $2^{n}+3 \equiv 3[4]$. However, a square is congruent to 0 or 1 modulo 4, so for $n \geq 2, 2^{n}+3$ is not a perfect square.

Proposition 150. If $n$ is an integer, then $n^{3} \equiv -1, 0$ or $1[7]$.