6. Find all positive integers n such that $latex (3^{2n} + 3 n^2 + 7 )$ is a perfect square.
Solution:
We use the fact that between square of two consecutive numbers there exist no perfect square. That is between $(k^2 )$ and $((k+1)^2 )$ there is no square.
Note that $(3^{2n} = (9^n)^2 )$ and $(9^n + 1)^2 )$ are two consecutive perfect square and $(3^{2n} + 3 n^2 + 7 )$ is always a number between them for $n > 2$ (easily proved by induction).
Hence the only solution is $n = 2$.
