
2. Find all pairs of integers $(m, n)$ that satisfy the equation $n^{n-1}=4 m^{2}+2 m+3$.

(Tomáš Jurík)


Solution. Clearly, the right-hand side is an odd integer, hence $n^{n-1}$ is also an odd integer which implies that $n$ is odd, $n-1$ is even and finally $n^{n-1}$ is a square of an integer.

For $m>1$, the inequalities $(2 m)^{2}4 m^{2}+2 m+3>(2 m+1)^{2}$, so there are no solutions in this case either.

The only remaining options are $m \in\{-1,0,1\}$; by checking each of them we find the only solution $(m, n)=(1,3)$.
