Exercise 3.5.9 (If $n = f(x_0,y_0)$ then $4an$ is a square modulo $|d|$)

Question

Show that if a number $n$ is represented by a quadratic form $f$ of discriminant $d$, then $4an$ is a square modulo $|d|$.

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Hint. Use (3.3).

Richard Ganaye · Accepted Answer

\begin{proof} By equation (3.3),
$$
4a f(x,y) = (2ax + by)^2 - d y^2.
$$
If $n$ is represented by $f$, then $n = f(x_0,y_0)$ for some integers $x_0,y_0$. Then 
$$4an = f(x_0,y_0) = (2ax_0 + by_0)^2 - d y_0^2 \equiv (2ax_0 + by_0)^2 \pmod {|d|},$$
thus $4an$ is a square modulo $|d|$.
\end{proof}

Exercise 3.5.9 (If $n = f(x_0,y_0)$ then $4an$ is a square modulo $|d|$)

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Exercise 3.5.9 (If $n = f(x_0,y_0)$ then $4an$ is a square modulo $|d|$)

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