Questions tagged [binary-quadratic-forms]

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

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0answers
64 views

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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1answer
326 views

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'...
2
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1answer
474 views

$x^2+7y^2=2^n$ 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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1answer
566 views

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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1answer
283 views

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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1answer
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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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1answer
436 views

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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2answers
270 views

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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717 views

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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1answer
239 views

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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113 views

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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1answer
198 views

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 ...
7
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1answer
240 views

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 ...
6
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1answer
204 views

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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110 views

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 ...
14
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2answers
485 views

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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605 views

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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1answer
330 views

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 ...
2
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1answer
153 views

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 ...
8
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1answer
567 views

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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1answer
102 views

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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1answer
122 views

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-...
3
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1answer
92 views

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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0answers
70 views

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 ...
9
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2answers
520 views

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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1answer
100 views

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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1answer
203 views

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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2answers
246 views

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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265 views

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 $$...
8
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2answers
329 views

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 = \...
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0answers
86 views

Counting 'admissible' binary quadratic forms

Let $f(x,y) = f_2 x^2 + f_1 xy + f_0 y^2$ be a primitive, positive definite, and reduced binary quadratic form. Put $k_f$ for the fundamental discriminant associated to $f$. That is, $k_f$ is square-...
9
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1answer
221 views

On binary quadratic forms which are not proper subforms of another binary quadratic form

Let $f(x,y)$ be a binary quadratic form with co-prime integer coefficients. We say that $f$ is a proper subform of $g(x,y)$ if there exists an integer matrix $A = \left(\begin{smallmatrix} a_1 & ...
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240 views

Cycle length and total count of Gauss reduced indefinite binary quadratic forms

Wikipedia lists two articles on maximum length for the continued fraction of $\sqrt n,$ Hickerson 1973 and Cohn 1977. There is a mess in the references, Cohn is not visible, just the link. Hmmm. ...
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214 views

MIQP formulation in L0 norm optimization

Consider the L0 norm compressed sensing problem: $$\eqalign{ & \min \quad {x^T}Qx + {c^T}x + {\mu\left\| x \right\|_0} \cr & s.t:\quad Ax \le b \cr} $$ Suppose I do want to solve this ...
3
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0answers
107 views

On covering with Idoneal integers

$d\in\Bbb N$ is an idoneal integer if $N\in\Bbb N_{>1}$ can be written uniquely as $N=x^2\pm dy^2$ then $N=2^mp^n$ where $p$ is odd prime and $n\geq0$ and $m\geq0$ holds. Let the $65$ known ...
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1answer
286 views

Ray class groups through binary quadratic forms

(Cross-posted from https://math.stackexchange.com/questions/2029407/ray-class-groups-through-binary-quadratic-forms) If $d$ is the discriminant of a quadratic number field, then the primitive classes ...
3
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125 views

An asymptotic formula involving the $2$-torsion subgroup of the class group of real quadratic fields

Let $R$ be an order in some number field $K$ (not necessarily maximal). Then the class number $\text{Cl}(R)$ is equal to the cardinality of the Picard group of $R$, which is the group of equivalence ...
13
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2answers
1k views

Upper bound on answer for Pell equation

A user on MSE, @martin , asked https://math.stackexchange.com/questions/1611411/pell-equations-upper-bound about an upper bound for $x$ in $x^2 - p y^2 = 1,$ when $p$ is prime. I checked, it appears ...
3
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0answers
122 views

Binary Quadratic Forms with coefficients in $F_q[T]$

I aim to study the binary forms $ax^2 + bxy + cy^2 = (a,b,c)$ where $a,b,c \in {F_q}[T]$ (charasteristic of $F_q$ not 2) in particular those such that the discriminant $D = b^2 - 4ac \in F_q[T]$ has ...
8
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1answer
395 views

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 ...