Questions tagged [binary-quadratic-forms]
A binary quadratic form is a quadratic form in two variables.
66
questions
2
votes
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 \...
4
votes
0answers
98 views
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 + ...
4
votes
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
votes
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 ...
14
votes
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....
1
vote
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
...
2
votes
0answers
178 views
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$....
6
votes
1answer
462 views
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 $...
1
vote
0answers
66 views
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_{\...
5
votes
0answers
154 views
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 $\...
11
votes
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 ...
5
votes
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 ...
6
votes
0answers
190 views
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....
1
vote
0answers
77 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})$, ...
6
votes
2answers
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}$$
always has integer solutions, where $(\frac p3)$ is the Legendre symbol. ...
3
votes
0answers
121 views
Which integers are represented by fourth powers (with respect to Gauss composition) of binary quadratic forms?
Let $m$ be a positive integer. We say that $m$ can be represented by a binary quadratic form
$$\displaystyle f(x,y) = ax^2 + bxy + cy^2, a,b,c \in \mathbb{Z}$$
if there exist integers $u,v$ such ...
2
votes
1answer
239 views
Representation of two related integers by the same binary quadratic form
Let $f(x,y) = ax^2 + bxy - cy^2$ be an indefinite, irreducible, and primitive binary quadratic form. That is, we have $\gcd(a,b,c) = 1$ and $\Delta(f) = b^2 - 4ac > 0$ and not equal to a square ...
3
votes
0answers
113 views
Does $(p-1)^2$ divide $\det[(\frac{i^2+cij+dj^2}p)]_{0\le i,j\le p-1}$ when $(\frac dp)=-1$?
Let $p$ be an odd prime. As in my paper, for $c,d\in\mathbb Z$ let us define
$$[c,d]_p:=\det\left[\left(\frac{i^2+cij+dj^2}p\right)\right]_{0\le i,j\le p-1},$$
where $(\frac{\cdot}p)$ is the Legendre ...
3
votes
0answers
96 views
Similar reduced integral matrices
Let $d\geq 1$ be an integer. I'm not assuming anything here about $d$ (it is not necessarily squarefree, for example), but if necessary I'm willing to reduce to the case where $d$ is squarefree and $-...
2
votes
1answer
198 views
On the determinant $\det[(\frac{i^2+dj^2}p)]_{0\le i,j\le(p-1)/2}$ with $(\frac dp)=-1$
Let $p$ be an odd prime. For $d\in\mathbb Z$ we define
$$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 ...
2
votes
0answers
38 views
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
votes
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
votes
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 ...
2
votes
0answers
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
votes
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 ...
6
votes
0answers
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 ...
2
votes
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
votes
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
votes
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(...
0
votes
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+...
0
votes
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|$$ and though likely non-...
3
votes
1answer
92 views
A question on homogeneous quadratic vector fields
Let $B(-,-): \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$ be a symmetric bilinear map. I am interested in the system of ODEs:
$\frac{dx}{dt} = B(x,x)$
Assume there exists some inner product $g(...
1
vote
0answers
70 views
Reducible binary quadratic form
Let $f(x,y)=(ex+fy)(gx+hy); \ x,y,e,f,g,h \in \mathbb{Z}$ be a reducible integral binary quadratic form. Is there a criterion to determine if a number is represented by this form? In particular, does ...
9
votes
2answers
520 views
The mean value of $y \log{y}$ over the ordinates of the CM points
Let $-D < 0$ be a negative fundamental discriminant and let $y$ range over the values $y = y_Q = \frac{\sqrt{|D|}}{2a}$, as the values $(a,b,c)$ run through the reduced binary quadratic forms $Q = ...
4
votes
0answers
144 views
Local-to-global principle for equivalence of binary quadratic forms
Let $f,g$ be irreducible binary quadratic forms with integer coefficients. Define the twisted action of $\operatorname{GL}_2(\mathbb{R})$ on $f$ by
$$\displaystyle f_T(x,y) = \frac{1}{\det T} f(t_1 ...
-1
votes
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
$$\displaystyle S_f = \{n \in \mathbb{N} : \exists (x,y) \in \mathbb{Z}^2 \text{ s.t. } f(x,y) = n^2\}.$$
...
3
votes
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 (...
1
vote
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 ...
9
votes
0answers
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
$$...
8
votes
2answers
329 views
On the equivalence of a pair of binary quadratic forms
Let $f,g, u,v \in \mathbb{Z}[x,y]$ be binary quadratic forms with co-prime coefficients. We say that the pair $(f,g)$ and $(u,v)$ are $\operatorname{GL}_2(\mathbb{Z})$-equivalent if there exists $T = \...
1
vote
0answers
86 views
Counting 'admissible' binary quadratic forms
Let $f(x,y) = f_2 x^2 + f_1 xy + f_0 y^2$ be a primitive, positive definite, and reduced binary quadratic form. Put $k_f$ for the fundamental discriminant associated to $f$. That is, $k_f$ is square-...
9
votes
1answer
221 views
On binary quadratic forms which are not proper subforms of another binary quadratic form
Let $f(x,y)$ be a binary quadratic form with co-prime integer coefficients. We say that $f$ is a proper subform of $g(x,y)$ if there exists an integer matrix $A = \left(\begin{smallmatrix} a_1 & ...
3
votes
0answers
240 views
Cycle length and total count of Gauss reduced indefinite binary quadratic forms
Wikipedia lists two articles on maximum length for the continued fraction of $\sqrt n,$ Hickerson 1973 and Cohn 1977. There is a mess in the references, Cohn is not visible, just the link.
Hmmm. ...
0
votes
0answers
214 views
MIQP formulation in L0 norm optimization
Consider the L0 norm compressed sensing problem:
$$\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
votes
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 ...
5
votes
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
votes
0answers
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
votes
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
votes
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
votes
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 ...