A binary quadratic form is a quadratic form in two variables.

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### Lower bound on class number of binary quadratic forms of discriminant of the form n^2+4

While searching for a use for the "sum invariant" of indefinite binary quadratic forms of discriminant $D = n^2 + 4$ (see https://cs.uwaterloo.ca/journals/JIS/VOL17/Smith/smith5.html), I believe I ...

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### Recursions for some binary theta series in characteristic 3

Define $A(0), A(1), A(2) \dots$ in ${\bf Z}/3[[x]]$ as follows. For $n$ in $\bf N$ let $s=3^{2n+1}$. Then $A(n) = \sum a_kx^k$ where $a_k$ is the mod 3 reduction of the number of representations of ...

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### Does the Diophantine equation $(x^2+ay^2)(u^2+bv^2) = p^2+cq^2$ admit a complete solution?

In this MSE question/thread, I have been discussing the equation
$$
(x^2+ay^2)(u^2+bv^2) = p^2+cq^2, \tag{$\star$}
$$
where $x,a,y,u,b,v,p,c,q$ are integers. I posed a conjecture which turned out to ...

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### References for Gauss Composition using Galois Cohomology

Note: I have already posted this on stackexchange, but have not yet gotten a response.
What are some good references for the Gauss composition law on binary quadratic forms in terms of Galois ...

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### Representing primes explicitly with binary quadratic forms

This is probably quite naïve, maybe even stackexchange-worthy.
Consider a quadratic form such as $Q(x,y) = 3x^2+y^2$. We know that, for primes $p \equiv 1 \pmod{3}$, there exist integer solutions to ...

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### primes represented by indefinite quadratic forms

Let $Q$ be an indefinite binary quadratic form with discriminant $D$ and one class per genus (keep the example $x^2 - 2y^2$ in mind). If one asks about the set $P = \{ p : p \text{ prime and } p = ...

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### Classification of these Binary Quadratic Forms

What are necessary and sufficient conditions on a binary quadratic form $ax^2+bxy+cy^2$, with integer coefficients and solution set in integers, to be equivalent to $x^2-y^2$, and separately to ...

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### Question about Gauss composition law over PID.

Let $m$ be a square free integer, $\mathbb{Q}(\sqrt{m})$ a quadratic field extension of $\mathbb{Q}$, $\Delta$ is its discriminant and $O_{\mathbb{Q}(\sqrt{m})/\mathbb{Q}}$ its ring of integers. We ...

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### Heegner Points and Binary Quadratic Forms

I've been trying to read Gross' paper on Heegner points on $X_0(N)$ and I am stuck on a few details. The definition he is working with is that a heegner points is a pair $y=(E,E')$, where $E$ and $E'$ ...

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### Number of solutions of a binary quadratic form.

Given a binary quadratic form with negative discriminant, such as $3x^{2} + y^{2} = k$, is there an efficient algorithm to compute a value $k$ (or all the values) for which the form has exactly $n$ ...

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### Representations by positive definite binary quadratic forms

It's known that the number of representations of an integer $k$ by sum of two squares is
$$
4\;\sum_{d|k}\left(\frac{-4}{d}\right)
$$
or
$$
4\sum_{d|k,\; d \textrm{ odd}} (-1)^{\frac{d-1}{2}}= ...

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### What is the identity class of the set of equivalence classes of binary cubic forms of discriminant $D$ ?

Let $\Delta = \sigma + 4 m$ be the fundamental discriminant of a quadratic field, where $\sigma \in \{ 0, 1 \}$. The binary quadratic form $Q(x, y) = A x^2 + B x y + C y^2$ of discriminant $\Delta$ ...

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### If the discriminant of a binary quadratic form has high valuation, is the form “almost a square”.

For a binary quadratic form $ax^2+bxy+cy^2$ over a field (characteristic not 2), the discriminant $b^2-4ac$ is 0 if and only if that form is the square of a linear form. I am curious about an ...

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### Canonical form for a pair of quadratic forms

Could anyone recommend a reference to a canonical form for a pair of quadratic forms over R (not necessarily positively definite)? This is probably related to the Weierstrass elementary divisors (but ...

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### Symmetric Power and Gauss quadratic form

Gauss in his book "Disquisitiones arithmeticae" considered only forms $ax^2+bxy+cy^2$ where $b$ is even, apparently because he had some notion of integral matrix in his mind even though he did not ...

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### representability of consecutive integers by a binary quadratic form

I have two related questions on the representability of integers
by quadratic forms in two variables :
(1) Let $f: {\mathbb Z} \times {\mathbb Z} \to {\mathbb Z} $ be such a quadratic
form, i.e. we ...

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### Binary Quadratic Forms in Characteristic 2

One of the reasons why the classical theory of binary quadratic forms
is hardly known anymore is that it is roughly equivalent to the theory
of ideals in quadratic orders. There is a well known ...