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Let $\mathfrak{F}$ be the set of binary quadratic forms over $\mathbb{Z}$. Let $f(x, y) = ax^2 + bxy + cy^2 \in \mathfrak{F}$. Let $\alpha = \left( \begin{array}{ccc} p & q \\ r & s \end{array} \right)$ be an element of $SL_2(\mathbb{Z})$. We write $f^\alpha(x, y) = f(px + qy, rx + sy)$. Since $(f^\alpha)^\beta$ = $f^{\alpha\beta}$, $SL_2(\mathbb{Z})$ acts on $\mathfrak{F}$ from right.

Let $f, g \in \mathfrak{F}$. If $f$ and $g$ belong to the same $SL_2(\mathbb{Z})$-orbit, we say $f$ and $g$ are equivalent.

Let $f = ax^2 + bxy + cy^2 \in \mathfrak{F}$. We say $D = b^2 - 4ac$ is the discriminant of $f$. Let $m$ be an integer. If $m = ax^2 + bxy + cy^2$ has a solution in $\mathbb{Z}^2$, we say $m$ is represented by $ax^2 + bxy + cy^2$. If $m = ax^2 + bxy + cy^2$ has a solution $(s, t)$ such that gcd$(s, t) = 1$, we say $m$ is properly represented by $ax^2 + bxy + cy^2$.

Is the following proposition true? If yes, how do we prove it?

Proposition Let $ax^2 + bxy + cy^2 \in \mathfrak{F}$. Suppose its discriminant is not a square. Let $m$ be an integer. Then $m$ is properly represented by $ax^2 + bxy + cy^2$ if and only if there exist integers $l, k$ such that $ax^2 + bxy + cy^2$ and $mx^2 + lxy + ky^2$ are equivalent.

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    Yes, of course. Take the representation of $m$ as one column of a 2 by 2 matrix and solve for the other column so as to have determinant $1.$ Call that $R$, and call the Hessian matrix of the first form $H.$ The Hessian of the new form is $R^T H R.$2012-09-04
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    Why is this question so full of notation that isn't used at all in the formulation of the actual problem?2012-09-08
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    @t.b. Because we use the definitions in my other questions and I think it's convenient for the readers having the relevant definitions in the same place.2012-09-08
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    @Makoto: ... at the cost of obscuring *this* question. I also think it unlikely that readers of your other questions will find it convenient. I don't think such a style of presentation suits this type of medium -- it would be more appropriate for, say, a book, blog, or wiki.2012-09-08
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    @Hurkyl "at the cost of obscuring this question" I think the question is clear. As for your latter comment, I believe my questions concerning binary quadratic forms are interesting and useful for most people who are interested in number theory. So I think they are on topic in this site. Please also note that there are usually several different answers to a mathematical question. Even if I have an answer, it may well not be the only one, nor a correct one. Even my question may be negative.2012-09-08
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    @Makoto: You have the advantages of already knowing the context and having recently written the post, so that *you* can easily ignore everything that isn't the actual question. t.b.'s comment lets me put words to a problem I've had with your questions previously: a large fraction of your questions I never finish reading, because they drag on and on and I lose interest before I actually get to the content of the question.2012-09-08

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