there does not exist a perfect square of the form $7\ell+3$ I have been trying to prove that there does not exist a perfect square of the form $7\ell+3$.  I've tried using $n$ as even or odd, and I'm getting stuck.  Can someone put me on the path?  Is this an equivalence class problem?

If $n^2\equiv3\mod7$, then $(n\bmod7)^2\equiv3\mod7$ as well.
Indeed, $n\bmod7=n-7q$, so that $(n\bmod7)^2=(n-7q)^2=n^2+7(-2n+7q^2)$.
So yes, it is an equivalence class problem.
