I don't think there is a solution which doesn't use brute force. Looking at$\pmod 7$ and$\pmod 8$ for $n\geq 8$ we get that $n$ is even and odd respectively. Hence a contraction. So we check for $n<8$ to get the only solutions $n=\{1,2\}$. Motivation: $13^2-1=168=2^3\cdot 3 \cdot 7$.