50th Mathematical Olympiad in Ukraine, Fourth Round (March 23, 2010)

Consider our equation modulo $8$. For $k\ge 3$ the left hand side is divisible by $8$, but the right hand side is not $8$. So we have to consider $k\in \{0,1,2\}$.

If $k=0$ we have $n^2=16$ or $n=4$.

For $k=1$ we have $n^2=31$, that gives no solution.