Questions tagged [binary-quadratic-forms]

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

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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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1answer
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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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2answers
192 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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57 views

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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1answer
567 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
146 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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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
182 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 ...
6
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1answer
208 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 ...
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1answer
151 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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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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2answers
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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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1answer
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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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1answer
129 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 ...
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1answer
469 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
101 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
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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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1answer
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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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0answers
63 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 ...
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2answers
490 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
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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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1answer
166 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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220 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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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 = \...
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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-...
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1answer
208 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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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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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 ...
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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
240 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 ...
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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 ...
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963 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 ...
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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 ...
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1answer
341 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 ...
6
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1answer
194 views

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 $k$...
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6answers
878 views

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 ...
6
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0answers
138 views

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 ...
3
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1answer
256 views

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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346 views

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 = Q(...