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?

For any integer $a,$
$$a\equiv0,\pm1,\pm2,\pm3\pmod7\implies a^2\equiv0,1,4,9\equiv2$$
$$\implies a^2\not\equiv3(=7-4),5(=7-2),6(=7-1)\pmod7$$
