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Let $ax^2 + bxy + cy^2$ be a binary quadratic form over $\mathbb{Z}$. Let $D = b^2 - 4ac$ be its discriminant. It's easy to see that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). Let $a$ be a positive odd integer. Then the Jacobi symbol $\left(\frac{D}{a}\right)$ is defined.

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

Proposition Let $D \ne 0$ be a non-zero integer such that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). Let $a$ and $b$ be positive odd integers such that $a \equiv b$ (mod $D$). Then $$\left(\frac{D}{a}\right) = \left(\frac{D}{b}\right)$$

  • 5
    What is the point in copying results from textbooks and asking for proofs?2012-09-09
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    @franzlemmermeyer I don't think I'm copying textbooks. Since the subject is not entirely new, it's hard to be 100% original, though.2012-09-09
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    I'm curious to know what textbooks Mr. Lemmermeyer was talking about. Does anybody know them?2013-10-27

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