Can $7n + 13$ ever equal a square? If not, why not? Can it be proved? And if it can be proved that it does equal a square (which I doubt), what is the smallest value for which this occurs?  

\begin{align}
1^2 &\equiv 1\pmod 7 \\
2^2 &\equiv 4\pmod 7 \\
3^2 &\equiv 2\pmod 7 \\
4^2 &\equiv 2\pmod 7 \\
5^2 &\equiv 4\pmod 7 \\
6^2 &\equiv 1\pmod 7 \\
7^2 &\equiv 0\pmod 7 \\
8^2 &\equiv 1\pmod 7 \\
9^2 &\equiv 4\pmod 7 \\
10^2 &\equiv 2\pmod 7 \\
11^2 &\equiv 2\pmod7\\
12^2 &\equiv 4\pmod7 \\
& \space\space\space\space\space\space\space\vdots
\end{align}  
It repeats, so a perfect square can only be 0,1,2,4 mod 7    
$7n+13 \equiv 6 \mod 7$  which is clearly not a perfect square.
