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?

First, notice that $[n\text{ is a perfect square}]\implies[n\equiv\color\green{0,1,2,4}\pmod7]$:


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*$k\equiv0\pmod7 \implies k^2\equiv0^2\equiv0\pmod7$

*$k\equiv1\pmod7 \implies k^2\equiv1^2\equiv1\pmod7$

*$k\equiv2\pmod7 \implies k^2\equiv2^2\equiv4\pmod7$

*$k\equiv3\pmod7 \implies k^2\equiv3^2\equiv2\pmod7$

*$k\equiv4\pmod7 \implies k^2\equiv4^2\equiv2\pmod7$

*$k\equiv5\pmod7 \implies k^2\equiv5^2\equiv4\pmod7$

*$k\equiv6\pmod7 \implies k^2\equiv6^2\equiv1\pmod7$


Second, observe that $[n\text{ is an integer}]\implies[7n+3\equiv\color\red3\pmod7]$.
