Find the $a, n \geq 1$ such that $\left((a+1)^{n}-a^{n}\right) / n$ is an integer.
Suppose $n>2$. Let $p$ be the smallest prime factor of $n$. Then $p$ divides $(a+1)^{n}-a^{n}$. In other words, $((a+1) / a)^{n} \equiv 1 \bmod p$. Let $\omega$ be the order of $(a+1) / a$ modulo $p$. Then $\omega$ divides $n$. On the other hand, by Fermat's little theorem, $((a+1) / a)^{p-1} \equiv 1 \bmod p$ so that $\omega$ divides $p-1$. Given the condition on $p$, necessarily $\omega=1$. This implies $a+1 \equiv a \bmod p$, which is absurd.

The solutions are therefore $n=1$ and $a$ arbitrary.