Exercise 3.7.2 (Improper representations of $c$)

Question

Let $f(x,y) = ax^2 + bxy + cy^2$ be a reduced positive definite form. Show that improper representations of $c$ may exist.

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Hint. Consider the form $x^2 + xy + 4y^2$.

Richard Ganaye · Accepted Answer

\begin{proof} Put $f(x,y) = x^2 + xy + 4y^2$. Then
$$4 f(x,y) = (2x+y)^2 + 15 y^2,$$
so the representations of $c = 4$ by $f$ are given by
$$(2x+y)^2 + 15 y^2 = 16.$$
Then $15y^2 \leq 16$ thus $y = 0$ or $y = \pm 1$. The representations of $c = 4$ are 
$$(-1,1),(0,-1),(0,1)(1,-1),(2,0),(-2,0).$$
But $2 \wedge 0 = 2$, so the representation $(2,0)$ of $c$ by $f$ is not a proper representation.
\end{proof}

Exercise 3.7.2 (Improper representations of $c$)

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Exercise 3.7.2 (Improper representations of $c$)

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