why $x^2-7y^2 = 11$ cannot have integer solutions? I have a quadratic equation $x^2-7y^2 = 11$, I understand it's not a pell equation, because $7/11$ is not an integer, but how can I prove it has no integer solutions?

If there were integers $x,y$ such that
$$ x^2 - 7y^2 = 11 $$
we see that it would follow that:
\begin{align*}
x^2 - 7y^2 &\equiv 11 \pmod{4} \\
x^2 + y^2 &\equiv 3 \pmod{4}
\end{align*}
We see that the congruence classes mod 4 are $\{0,1,2,3\}$. Hence, the quadratic residues are $\{0,1\}$. Since $x^2, y^2 \in \{0,1 \} \pmod{4}$, it follows that:
$$ x^2 + y^2 \not\equiv 3 \pmod{4} $$ 
By contradiction,
$$ \text{there exists no integer solutions to }  x^2 - 7y^2 = 11$$
