Is it true that $10^n + 1$ is never a perfect square? If so, how can you prove it

Question

I'm working on a problem and right now I want to prove that $10^n + 1$ is never a perfect square. I know that for even values of $n$, the expression is not a perfect square since two positive perfect squares cannot differ by $1$ and both $1$ and $10^{2m}$ are perfect squares. I just don't know how to prove it for odd $n$s.

Thanks in advance!

Answer 1

Considering squares in $\bmod 9$, we can see that every perfect square must be $0,1,4,7\mod 9$ (these are the quadratic residues $\bmod 9$). However, $10^n+1\equiv 2\mod 9$, so it can never be a square.