# Questions tagged [binary-quadratic-forms]

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

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### heights of ideal classes and reduction theory for Bhargava cubes

Suppose $K$ is a quadratic imaginary field with discriminant $D$; let $S$ denote the ring of integers in $K$. For a fractional $S$-ideal $J$, define the height of $J$, denoted $H(J)$, to be the ...

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### A cubic equation, and integers of the form $a^2+32b^2$

I am trying to determine whether there are any integers $x,y,z$ such that
$$
1+2 x+x^2 y+4 y^2+2 z^2 = 0. \quad\quad\quad (1)
$$
It is clear that $x$ is odd. We can consider this equation as quadratic ...

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### Are there natural isomorphisms $S^{(2,1)}(k^{m+1})\cong k^2\otimes W$?

In this popular 2019 MO question, user მამუკა ჯიბლაძე asked:
The spaces $\operatorname{S}^2(k^n)$ and $\Lambda^2(k^{n+1})$ from the title have equal dimensions. Is there a natural isomorphism between ...

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### Positive definite quadratic form algorithm

Let $f(x,y)= ax^2+bxy+cy^2$, or similarly denote it by $(a,b,c)$. This question is about the case $(1,0,p)$ where $p$ is prime. Suppose I have one solution $\bar{x}_1=(x_0,y_0)$ for $f(x,y)=m$ for ...

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### asymptotics of numbers represented by certain indefinite binary quadratic forms

A postdoc wrote to me, asking about about the asymptotic number of elements represented integrally by, say, $x^2 - n^3 y^2$ in the set of numbers $-N$ to $N.$ Actually, he included $x^2 - n^5 ...

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### Solvability of two-variable quadratic equations with a parameter

(a) Prove that there exist infinitely many values of integer parameter $a$ such that equation
$$
2 x^2+a x y+y^2+1 = 0
$$
is solvable in integers $(x,y)$.
(b) The same question for a similar equation
$...

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### Is there an available English translation for Artin's "Quadratische Körper im Gebiete der höheren Kongruenzen"?

Otherwise, is it reasonable to work through the German edition with only a basic knowledge of German?

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### Eisenstein series evaluated at $2i$

Consider the real analytic Eisenstein series defined by
$$
E(z,s) := \sum_{\gamma\in\Gamma_\infty\setminus\Gamma} Im(\gamma z)^s
$$
where as usual, $\Gamma=SL(2,\mathbb{Z})$ and $\Gamma_\infty$ is the ...

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### Two conjectures for primes $p\equiv 1\pmod 8$

Motivated by my paper Quadratic residues and quartic residues modulo primes [Int. J. Number Theory 16 (2020), 1833-1858], here I pose two new conjectures for primes $p\equiv1\pmod8$ based on my ...

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### Integers representable as binary quadratic forms

It is known that odd prime $p$ can be represented as $p=x^2+y^2$ if and only if $p \equiv 1$ mod $4$, represented as $p=x^2+2y^2$ if and only if $p \equiv 1$ or $3$ mod $8$, represented as $p=x^2+3y^2$...

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### Correspondence between class group of binary quadratic forms and the narrow class group via Dirichlet composition: an elementary approach?

I have been trying to explore and learn about connections between the form class group and the ideal class group. To be on the same page, we define the form class group of a negative discriminant $D \...

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### Probability of quadratic real solutions

Solving geometric equations that sometimes lead to complex solutions made me
wonder how commonly this occurs.
I have two quadratic polynomial equations in $x$ and $y$,
$$a\, x^2 + b\, x y + c\, y^2 + ...

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### Clarification regarding a claim in Heilbronn’s 1934 paper

I was reading Heilbronn’s 1934 paper where he proves that $H(d) \to \infty$ as $d \to -\infty$, where $H(d)$ is the ideal class number of the imaginary quadratic field with discriminant $d$. I couldn'...

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### $x^2+7y^2=2^n$ and sums of four squares

Lagrange's four square theorem states that each $m\in\mathbb N=\{0,1,2,\ldots\}$ can be written as a sum of four squares.
Recently, I found that the diophantine equation $x^2+7y^2=2^n$ has certain ...

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### reference for: no finite set of positive (integer) binary quadratic forms represents all primes

This recent question asks for a set of forms (binary quadratic) representing all primes.
Set of quadratic forms that represents all primes
When the question was asked on MSE last month
https://math....

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### A conjecture for primes $p\equiv\pm1\pmod5$

For any prime $p\equiv\pm1\pmod5$, we can write $p$ uniquely in the form $x_p^2+3x_py_p+y_p^2$ with $x_p,y_p\in\mathbb Z$ and $x_p>y_p>0$.
I have the following conjecture.
Conjecture. We have
...

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### Conjectures for primes $p\equiv1\pmod3$

Let $p$ be a prime with $p\equiv1\pmod3$. It is well known that we can write $p$ uniquely as $a_p^2+a_pb_p+b_p^2$ with $a_p,b_p\in\mathbb Z$ and $a_p>b_p>0$.
Note that $a_b\not \equiv b_p\pmod3$....

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### Questions involving primes $p\equiv1\pmod4$

As claimed by Fermat and proved by Euler, any prime $p\equiv1\pmod4$ can be written uniquely as $s_p^2+t_p^2$ with $s_p,t_p\in\{1,2,3,\ldots\}$, $2\nmid s_p$ and $2\mid t_p$. For any positive integer $...

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### Number of representations by indefinite quadratic forms

Let $Q(x,y) = a x^2 + b x y + c y^2$ be a quadratic form with positive discriminant $d = b^2 - 4 a c > 0$. Is it true that the number of solutions to $Q(x,y) = n$ is bounded above by $\ll_{\...

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### Quadratic fields with moderately large fundamental units

Let $d > 1$ be a fundamental discriminant, and let $K_d = \mathbb{Q}(\sqrt{d})$. Denote by $\varepsilon_d$ the fundamental unit of $\mathcal{O}_{K_d}$, namely the smallest algebraic integer $\...

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### Infinite cyclic subgroups of $\text{SL}_2(\mathbb{Z})$

Let $\Gamma = \operatorname{SL}_2(\mathbb{Z})$ be the usual modular group. It is well-known that $\Gamma$ contains infinitely many distinct (non-conjugate even) subgroups which are isomorphic to the ...

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### Representation of integers by principal binary quadratic forms

Let $k > 1$ be a positive, square-free integer. Consider the quadratic form $f_k(x,y) = x^2 - ky^2$. When $k$ is composite, is it easy to determine for which proper divisors $k^\prime$ of $k$ does ...

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### some problems on sum of two squares

During my experiments with "Mathematica" I arrived to the following observations. My question is that are they interesting, known, solved or not. If they are known could you please give me a reference....

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### Equidistribution of integers and primes with respect to class group structure of imaginary quadratic fields

For a given positive integer $d$ such that $-d$ is a fundamental discriminant, put $h(-d)$ for the cardinality of the ideal class group of the imaginary quadratic field $K_d = \mathbb{Q}(\sqrt{-d})$, ...

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### Is the diophantine equation $3x^2+1=py^2$ always solvable for each prime $p\equiv 13\pmod{24}$?

In Question 337879, I conjectured that for any prime $p\equiv3\pmod4$ the equation $$3x^2+4\left(\frac p3\right)=py^2\tag{1}$$
always has integer solutions, where $(\frac p3)$ is the Legendre symbol. ...

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### Which integers are represented by fourth powers (with respect to Gauss composition) of binary quadratic forms?

Let $m$ be a positive integer. We say that $m$ can be represented by a binary quadratic form
$$\displaystyle f(x,y) = ax^2 + bxy + cy^2, a,b,c \in \mathbb{Z}$$
if there exist integers $u,v$ such ...

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### Representation of two related integers by the same binary quadratic form

Let $f(x,y) = ax^2 + bxy - cy^2$ be an indefinite, irreducible, and primitive binary quadratic form. That is, we have $\gcd(a,b,c) = 1$ and $\Delta(f) = b^2 - 4ac > 0$ and not equal to a square ...

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### Does $(p-1)^2$ divide $\det[(\frac{i^2+cij+dj^2}p)]_{0\le i,j\le p-1}$ when $(\frac dp)=-1$?

Let $p$ be an odd prime. As in my paper, for $c,d\in\mathbb Z$ let us define
$$[c,d]_p:=\det\left[\left(\frac{i^2+cij+dj^2}p\right)\right]_{0\le i,j\le p-1},$$
where $(\frac{\cdot}p)$ is the Legendre ...

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### Similar reduced integral matrices

Let $d\geq 1$ be an integer. I'm not assuming anything here about $d$ (it is not necessarily squarefree, for example), but if necessary I'm willing to reduce to the case where $d$ is squarefree and $-...

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### On the determinant $\det[(\frac{i^2+dj^2}p)]_{0\le i,j\le(p-1)/2}$ with $(\frac dp)=-1$

Let $p$ be an odd prime. For $d\in\mathbb Z$ we define
$$T(d,p):=\det\left[\left(\frac{i^2+dj^2}p\right)\right]_{0\le i,j\le(p-1)/2},$$
where $(\frac{\cdot}p)$ is the Legendre symbol.
By (1.17) of my ...

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### Distribution of binary quadratic forms in a given genus

It was brought up in this question (Distribution of 'square classes' of binary quadratic forms) that the objects I am interested in are actually binary quadratic forms in the principal genus ...

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### A class number estimate

Let $\mathcal{D} = \{D \in \mathbb{Z} : D \equiv 0, 1 \pmod{4}\}$ be the set of discriminants. It is well-known that each element in $\mathcal{D}$ is the discriminant of a primitive binary quadratic ...

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### Distribution of 'square classes' of binary quadratic forms

Let $f$ be a binary quadratic form with integer coefficients and non-zero discriminant $D$. Suppose for simplicity that $D$ is a fundamental discriminant (which in particular implies that $f$ is ...

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### Conjectural bound on gaps between values assumed by quadratic forms

Let $D$ be a discriminant, i.e., $D \equiv 1 \pmod{4}$ or $D \equiv 0 \pmod{4}$. Let $\mathcal{S}(D)$ be the set of positive integers for which there exists a binary quadratic form $f$ with integer ...

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### Equidistribution of CM points in the principal genus

It is well known that as the negative discriminant $-D$ goes to infinity, the number of quadratic forms of discriminant $-D$ belonging to the principal genus also goes to infinity. Can we say ...

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### From Bhargava to Gauss -- Why does correspondence of cubes and ideal classes imply Gauss correspondence?

In his seminal 2004 paper "Higher Composition Laws I" in the Annals of Mathematics, (doi:10.4007/annals.2004.159.217), Bhargava proves that for fixed $D \neq 0$, there is a bijective correspondence ...

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### Is the set of integers represented by a quadratic form of non-fundamental discriminant a subset of the rep. set of a form of fundamental discriminant?

I am currently working with positive-definite, reduced, primitive, integral binary quadratic forms, and I have noticed something interesting.
Conjecture:
Let $Q$ be a form of non-fundamental ...

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### 2-parts of class numbers of binary quadratic forms for non-fundamental discriminants

I need a formula for the 2-adic valuation of the number of proper equivalence classes of primitive positive definite binary quadratic forms of discriminant $-D$, call it $h_0(-D)$. I'm sure the answer ...

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### Primes of the form $x^2 + y^2 + 1$

There are infinitely many primes of the form $x^2+y^2+1$, as proved by Bredihin. Motohashi improved the result by showing that there were $\gg x/\log^2 x$ such primes up to $x$. But we expect $\Theta(...

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### Elementary description to count of perfect squares - II

What can we say about growth of smallest gap $g(a)$ which is the smallest $|x-y|$ where $0\leq x,y\leq\Big\lfloor\frac a2\Big\rfloor$ and $\sqrt{x(a-x)},\sqrt{y(a-y)}\in\Bbb Z$?
Is $g(a)=1\iff a=b^2+...

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### Elementary description to count of perfect squares - I

Is there an elementary description of $$N(a)=\Big|\Big\{x\in\{0,1,\dots,\Big\lfloor\frac a2\Big\rfloor-1,\Big\lfloor\frac a2\Big\rfloor\Big\}:\sqrt{x(a-x)}\in\Bbb Z\}\Big|$$ and though likely non-...

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### A question on homogeneous quadratic vector fields

Let $B(-,-): \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$ be a symmetric bilinear map. I am interested in the system of ODEs:
$\frac{dx}{dt} = B(x,x)$
Assume there exists some inner product $g(...

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### Reducible binary quadratic form

Let $f(x,y)=(ex+fy)(gx+hy); \ x,y,e,f,g,h \in \mathbb{Z}$ be a reducible integral binary quadratic form. Is there a criterion to determine if a number is represented by this form? In particular, does ...

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### The mean value of $y \log{y}$ over the ordinates of the CM points

Let $-D < 0$ be a negative fundamental discriminant and let $y$ range over the values $y = y_Q = \frac{\sqrt{|D|}}{2a}$, as the values $(a,b,c)$ run through the reduced binary quadratic forms $Q = ...

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### Local-to-global principle for equivalence of binary quadratic forms

Let $f,g$ be irreducible binary quadratic forms with integer coefficients. Define the twisted action of $\operatorname{GL}_2(\mathbb{R})$ on $f$ by
$$\displaystyle f_T(x,y) = \frac{1}{\det T} f(t_1 ...

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### Classifying binary quadratic forms by the set of representable squares

Let $f$ be a positive definite binary quadratic form with integer coefficients. Define
$$\displaystyle S_f = \{n \in \mathbb{N} : \exists (x,y) \in \mathbb{Z}^2 \text{ s.t. } f(x,y) = n^2\}.$$
...

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### Equivalence of binary quadratic forms over $\operatorname{GL}_2(\mathbb{Q}_p)$ or $\operatorname{GL}_2(\mathbb{Z}_p)$

Let $f(x,y) = ax^2 + bxy + cy^2$ be a binary quadratic form with integer coefficients and non-zero discriminant $\Delta(f)$. It is well-known that $f$ is $\operatorname{GL}_2(\mathbb{R})$-equivalent (...

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### On the notion of primary representation of a natural number by a quadratic form

This "discussion" has to do with some of the material we can find in pages 183-186 of the translation into English of the first part of E. Landau's Vorlesungen über Zahlentheorie (published by the ...

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### Cancellation in a sum of Möbius evaluated along a quadratic form

Let $Q(x,y)$ be an indefinite binary quadratic form. Suppose $0 < B < \sqrt{A} $ are such that $B \gg \sqrt{A}$.
Is it true one can save an arbitrary power of log from the trivial bound in
$$...

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### On the equivalence of a pair of binary quadratic forms

Let $f,g, u,v \in \mathbb{Z}[x,y]$ be binary quadratic forms with co-prime coefficients. We say that the pair $(f,g)$ and $(u,v)$ are $\operatorname{GL}_2(\mathbb{Z})$-equivalent if there exists $T = \...