Questions tagged [binary-quadratic-forms]
A binary quadratic form is a quadratic form in two variables.
75
questions
8
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4
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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 ...
7
votes
1
answer
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views
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 ...
0
votes
0
answers
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views
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 ...
3
votes
1
answer
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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 ...
1
vote
1
answer
186
views
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
$...
1
vote
0
answers
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views
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?
2
votes
0
answers
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views
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 ...
5
votes
0
answers
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views
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 ...
0
votes
1
answer
352
views
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 $p=x^2+3y^2$...
2
votes
0
answers
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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
0
answers
129
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
1
answer
346
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
1
answer
524
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
1
answer
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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....
2
votes
1
answer
320
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
...
3
votes
0
answers
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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$....
7
votes
1
answer
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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 $...
1
vote
0
answers
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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_{\...
6
votes
0
answers
186
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
1
answer
532
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
2
answers
346
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
0
answers
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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
0
answers
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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
2
answers
793
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}$$
always has integer solutions, where $(\frac p3)$ is the Legendre symbol. ...
3
votes
0
answers
127
views
Which integers are represented by fourth powers (with respect to Gauss composition) of binary quadratic forms?
Let $m$ be a positive integer. We say that $m$ can be represented by a binary quadratic form
$$\displaystyle f(x,y) = ax^2 + bxy + cy^2, a,b,c \in \mathbb{Z}$$
if there exist integers $u,v$ such ...
2
votes
1
answer
376
views
Representation of two related integers by the same binary quadratic form
Let $f(x,y) = ax^2 + bxy - cy^2$ be an indefinite, irreducible, and primitive binary quadratic form. That is, we have $\gcd(a,b,c) = 1$ and $\Delta(f) = b^2 - 4ac > 0$ and not equal to a square ...
3
votes
0
answers
119
views
Does $(p-1)^2$ divide $\det[(\frac{i^2+cij+dj^2}p)]_{0\le i,j\le p-1}$ when $(\frac dp)=-1$?
Let $p$ be an odd prime. As in my paper, for $c,d\in\mathbb Z$ let us define
$$[c,d]_p:=\det\left[\left(\frac{i^2+cij+dj^2}p\right)\right]_{0\le i,j\le p-1},$$
where $(\frac{\cdot}p)$ is the Legendre ...
3
votes
0
answers
108
views
Similar reduced integral matrices
Let $d\geq 1$ be an integer. I'm not assuming anything here about $d$ (it is not necessarily squarefree, for example), but if necessary I'm willing to reduce to the case where $d$ is squarefree and $-...
2
votes
1
answer
211
views
On the determinant $\det[(\frac{i^2+dj^2}p)]_{0\le i,j\le(p-1)/2}$ with $(\frac dp)=-1$
Let $p$ be an odd prime. For $d\in\mathbb Z$ we define
$$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
0
answers
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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
1
answer
253
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
1
answer
291
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
0
answers
115
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
2
answers
544
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
0
answers
691
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
1
answer
452
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
1
answer
194
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
1
answer
745
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
1
answer
103
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
1
answer
130
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|$$ and though likely non-...
4
votes
1
answer
114
views
A question on homogeneous quadratic vector fields
Let $B(-,-): \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$ be a symmetric bilinear map. I am interested in the system of ODEs:
$\frac{dx}{dt} = B(x,x)$
Assume there exists some inner product $g(...
1
vote
0
answers
94
views
Reducible binary quadratic form
Let $f(x,y)=(ex+fy)(gx+hy); \ x,y,e,f,g,h \in \mathbb{Z}$ be a reducible integral binary quadratic form. Is there a criterion to determine if a number is represented by this form? In particular, does ...
9
votes
2
answers
541
views
The mean value of $y \log{y}$ over the ordinates of the CM points
Let $-D < 0$ be a negative fundamental discriminant and let $y$ range over the values $y = y_Q = \frac{\sqrt{|D|}}{2a}$, as the values $(a,b,c)$ run through the reduced binary quadratic forms $Q = ...
4
votes
0
answers
148
views
Local-to-global principle for equivalence of binary quadratic forms
Let $f,g$ be irreducible binary quadratic forms with integer coefficients. Define the twisted action of $\operatorname{GL}_2(\mathbb{R})$ on $f$ by
$$\displaystyle f_T(x,y) = \frac{1}{\det T} f(t_1 ...
-1
votes
1
answer
112
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
1
answer
321
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
2
answers
280
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
0
answers
269
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
2
answers
380
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
0
answers
95
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-...