Baltic Way 1992 · Problem 3

Solution: For any natural number $n$, we have $n^{2} \equiv 0$ or $n^{2} \equiv 1 \pmod{4}$ and $n^{3} \equiv 0$ or $n^{3} \equiv \pm 1 \pmod{9}$. Thus $\{36n + 3 \mid n = 1, 2, \ldots\}$ is a progression with the required property.