All Questions
Tagged with quadratic-forms nt.number-theory
231 questions
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Notion of length in projective space over function field
Given a projective space $\mathbb{P}^n(\mathbb{R})$ and two points $x, y \in \mathbb{P}^n(\mathbb{R})$, the distance between $x$ and $y$ is defined as
$$
d(x, y) = \frac{\|v_x \wedge v_y\|}{\|v_x\| \|...
- 87
11
votes
1
answer
598
views
How to prove this problem about ternary quadratic form?
Is this right? And how to prove it ?
For $n \equiv 1,2 \bmod 4$
$$ \Bigg|\ \mathbb Z^3\cap\Big\{(a_1,a_2,a_3)\ \Big|\
a_1^2+a_2^2+a_3^2=n \Big\}\Bigg| \\
= \frac12\Bigg|\mathbb Z^3\cap\Big\{(a_1,...
- 113
5
votes
1
answer
259
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Reduced form of Bhargava cubes
Let $A \in \mathbb{Z}^{2} \otimes \mathbb{Z}^{2} \otimes \mathbb{Z}^{2}$ be a Bhargava cube. We have a natural action of $\mathrm{SL}_{2}(\mathbb{Z})^{3}$ on the space of Bhargava cubes, and I wonder ...
- 2,215
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62
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Isotropic vectors of quadratic forms over number fields
By Meyer's theorem, an indefinite quadratic form $Q$ over $Z$ has an integral isotropic vector is the dimension is at least $5$ and this bound is tight. Indeed, in dimension $4$ there are indefinite ...
2
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0
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156
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Sets represented by integral quadratic forms
Let $f(x) = x^\intercal A x$ be a positive definite integral quadratic form on $d$ variables. A positive integer $n$ is said to be represented by $f$ if $f(x) = n$ for some $x \in \mathbb Z^d$. A set ...
- 974
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1
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127
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Automorphism groups in class sets of ternary lattices
Let $\Lambda$ be an integral lattice in some definite ternary quadratic space $(V,Q)$ over $\mathbb{Q}$.
Consider the usual class set $\text{Cl}(\Lambda) = O(V)\backslash\text{Gen}(\Lambda)$, i.e. the ...
- 562
6
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380
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A possible variant of Zagier's one-sentence proof for Fermat's sum of two squares theorem?
Is it possible to modify Zagier's one-sentence proof of Fermat's sum of two squares theorem (see here) to prove certain non-trivial cases of Jacobi's four-square theorem (see here)?
Let $p$ be a prime ...
- 81
1
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0
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136
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On the equation $q(\mathbf{x}) = 1$ for $q$ a quadratic form
Let $q(\mathbf{x}) = q(x_1, \cdots, x_n)$ be a quadratic form with integer coefficients. For $n \geq 3$, is there a reasonable theory for the set of integer solutions to the equation
$$\displaystyle q(...
- 26.9k
8
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4
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870
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A cubic equation, and integers of the form $a^2+32b^2$
I am trying to determine whether there are any integers $x,y,z$ such that
$$
1+2 x+x^2 y+4 y^2+2 z^2 = 0. \quad\quad\quad (1)
$$
It is clear that $x$ is odd. We can consider this equation as quadratic ...
- 7,182
2
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1
answer
171
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Integers $8k+3>0$ not represented by $2x^2+4y^2+4yz+9z^2$ over the integers
Oeis A306970 lists positive integers of the form $8k+3$ which are not reprented by
$$f(x,y,z):=2x^2+4y^2+4yz+9z^2$$
over the integers as $3,43,163,907$. It says this list may not be complete and ...
- 458
0
votes
0
answers
145
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Positive definite quadratic form algorithm
Let $f(x,y)= ax^2+bxy+cy^2$, or similarly denote it by $(a,b,c)$. This question is about the case $(1,0,p)$ where $p$ is prime. Suppose I have one solution $\bar{x}_1=(x_0,y_0)$ for $f(x,y)=m$ for ...
0
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1
answer
213
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number of representations by sums of three squares (with coefficients)
There are formulas for counting the number of representations of a positive integer $N$ as a sum of three integer squares. What is a reference for
$$
\#\{(x,y,z)\in \mathbf{N}^3: 5^4 x^2+y^2+z^2=N\}
?$...
- 3,062
4
votes
1
answer
328
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Fields in which $ -1 $ can't be written as sum of two square elements
We say a field $F$ has the property $*$ if the equation $x^2 + y^2=-1$ has no solution in $F$. For an example if $F$ is a subfield of real numbers then $F$ satisfies $*$. On the other hand if $ F $ is ...
- 923
2
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1
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258
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An arithmetic problem involving a system of equations
Fix a positive integer $r$. Describe the solutions to the system of equations given by:
$$\begin{equation}\sum_{1\leq i\leq r}X_i^2\equiv0\pmod{X_k}(1\leq k\leq r)\end{equation}$$
Example: In the case ...
- 1,433
5
votes
0
answers
175
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Epstein zeta function for non-fundamental discriminant to L-series
Let $Q(x,y) = ax^2+b xy + cy^2$ be a primitive integral positive-definite quadratic form, with associated number field $K$. If $D=b^2-4ac$ is a fundamental discriminant, then it's well-known that $$\...
- 528
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1
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202
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Schur multiplier of a Chevalley group of type $D_5$
$\DeclareMathOperator\EO{EO}\DeclareMathOperator\SO{SO}\DeclareMathOperator\St{St}\DeclareMathOperator\Sp{Sp}$This is sort of a follow up question to my post here regarding the commutator subgroup of $...
- 545
6
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1
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549
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Computing a Commutator Subgroup
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$I’m studying the group $\O(5,5,\mathbb{Z})$, the indefinite orthogonal matrices with integer entries. In particular, I ...
- 545
7
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0
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259
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K3 surfaces with no −2 curves
I seem to remember that a K3 surface with big Picard rank always
has smooth rational curves.
This question is equivalent to the following question about integral quadratic lattices. Let us call a ...
- 9,237
0
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0
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179
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Points at which a polynomial becomes reducible
Let $n \geq 10$ and set $\mathbf{y} = (y_1,\ldots,y_n)$. Let $Q_1(\mathbf{y}),\ldots,Q_5(\mathbf{y})$ be non-zero quadratic forms with integer coefficients such that the cubic form $x_1Q_1(\mathbf{y})+...
- 105
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1
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274
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Equivalence of quadratic forms over $p$-adic integers vs over localisation at $p$
To discern whether two integral quadratic forms are equivalent over the $p$-adic integers, one can compute a Jordan decomposition at $p$ and read off some invariants. Restricting to $p\ne2$ for ...
- 323
1
vote
2
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401
views
Integral solutions of quadratic equation $5 X² − 14 XY + 5 Y² = n$
Solve for all integers $x$ and $y$ the quadratic form $5 X² − 14 XY + 5 Y² = n$ for some integer n. I know that for some cases there are recurrence solutions, but I'm not sure how to solve these ...
- 1,109
2
votes
0
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59
views
Equidistribution of lattice points on quadratic forms without certain values
I have recently been studying some results about equidistribution of lattice points on positive definite quadratic forms $Q$ with $3$ or more variables. Concretely the article Duke and Schulze-Pillot, ...
- 313
10
votes
1
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908
views
How to describe all integer solutions to $x^2+y^2=3z^2+1$?
The question is in the title. Here is a short motivation. The general quadratic Diophantine equation is
$$
x^TAx+bx+c=0,
$$
where $x$ is a vector of $n$ variables, $A$ is $n \times n$ matrix with ...
- 7,182
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1
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131
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Does this quadratic system admit an integral or a rational solution?
Let $a,b$ be coprime and say $0<a<b<2a$.
Consider the quadratic system:
$$\alpha\delta-\beta\gamma=1$$
$$(\alpha^2-(\alpha\delta+\beta\gamma))a^2b+\beta^2b^3+(2\alpha\beta-\beta\delta)ab^2-\...
- 13.9k
2
votes
1
answer
148
views
Primitive representation of integers by some form on the genus of a quadratic form
Some time ago, I asked a question about equidistribution on a paper of Duke and Schulze-Pillot that was usefully answered.
However, on the answer there was a statement that was unimportant for me back ...
- 313
4
votes
1
answer
199
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Computing spinor equivalence for positive definite forms
Given an integral positive-definite rank $n$ quadratic form $f$, one can use the algorithm in Conway and Sloane (Chapter 15, SPLaG) to efficiently determine if the genus of $f$ contains more than one ...
- 323
2
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1
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148
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Does the F-unitary group isomorphism arises from a conformal isometry?
Let $K$ be a CM-field with totally real subfield $F$. Let $(V_1, h_1)$ and $(V_2, h_2)$ be two $n$-dimensional $K$-vector spaces with nondegenerate Hermitian forms, where $n\geq 3$.
Question 1 Does ...
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2
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257
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Generalization of the geometric series representation of the Kronecker delta for arbitrary lattices
In On construction of holomorphic cusp forms of half integral weight by Shintani, in the last equation of page 96, an identity for the Kronecker delta for elements of $L^*/L$ is defined. Here, $L$ is ...
anon
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0
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153
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Polynomial parametrization for solutions of quadratic Diophantine equations
A previous Mathoverflow question asks if there is an algorithm that would determine all integer solutions to a given quadratic Diophantine equation.
To make this question more formal, we need to agree ...
- 7,182
4
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1
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411
views
Calculating the explicit constant – Siegel zeros and class numbers
Let $\chi$ denote the Legendre symbol of conductor $q$. A Siegel zero for the $ L $ series associated to $ \chi $, which we denote by $ L(s,\chi) $ is a real zero $ \sigma $ satisfying $ 1-\frac{c}{\...
- 589
3
votes
1
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230
views
Duke and Schulze-Pillot condition for equidistribution
When regarding a ternary quadratic form $Q(x,y,z)$, is a classic question to consider which integers $n$ can be represented by $Q$. It is also classic to wonder how "well distributed" are ...
- 313
1
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0
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99
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Integral zeros of quadratic forms
Suppose I have a definite quadratic form over $\mathbb{Z}$ in $n$ variables
$$Q(\mathbf{x}) = \sum_{i,j} a_{i,j}x_ix_j$$
I'm looking to find whether an integer $m$ is representable by $Q$. Everyone ...
- 901
2
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0
answers
118
views
Computing coefficients of theta functions associated to quadratic forms
If we take an integral positive definite quadratic form $Q$ and set $\Theta_Q(z) = \sum_{k\geq 0}R_Q(k)e^{2\pi ikz}$, what are the most efficient algorithms to compute the $R_Q(k)$? I am aware e.g. of ...
- 323
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2
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574
views
Existence of solution for a system of quadratic diophantine equations / symmetric quadratic froms
I am interested in solving, or even just deciding the existence of a solution, for a system of quadratic diophantine equations.
Let $p$ be a prime congruent to 1 modulo 8, so $ p =17$ is the first ...
- 91
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1
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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 $p=x^2+3y^2$...
- 7,182
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143
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Can $12n+5$ be written as $2x^2+5y^2+9z^2+xyz$ with $x,y,z$ nonnegative integers?
Let $\mathbb N=\{0,1,2,\ldots\}$. By the Gauss-Legendre theorem on sums of three squares, for any $n\in\mathbb N$ we can write $4n+1$ as $x^2+y^2+z^2$ with $x,y,z\in\mathbb N$.
Motivated by this, here ...
- 15.6k
2
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1
answer
269
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Can each natural number be represented by $2w^2+x^2+y^2+z^2+xyz$ with $x,y,z\in\mathbb N$?
It is well known that each $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as $2w^2+x^2+y^2+z^2$ with $w,x,y,z\in\mathbb N$. Furthermore,
$$\{2w^2+x^2+y^2:\ w,x,y\in\mathbb N\}=\mathbb N\setminus\{4^k(...
- 15.6k
0
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1
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199
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Maximum dimension of a simultaneous anisotropic subspace of quadratic forms over $ \mathbb{Q} $
Let $(V,q )$ be a quadratic space over $ \mathbb{Q} $. A subspace $ U $ is called totally isotropic if $ q(x) = 0 $ for all $ x \in U $ and a subspace $ U $ is called an anisotropic subspace if $ ...
- 923
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276
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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 \...
- 739
1
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0
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93
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Uniqueness of decomposition for positive-definite integral bilinear forms?
Let $\Lambda$ be a lattice, that is a free finitely generated abelian group with a symmetric bilinear form.
In general, decomposition of lattices into indecomposable orthogonal sublattices is not ...
- 2,300
2
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0
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111
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Possible values of hyperbolic quadratic forms
$\newcommand\Z{\mathbf{Z}}$Given an integral quadratic form $q$ of signature $(n-1,1)$ and a value $\lambda \in \Z$ is there an algorithm that can determine whether there exist $x\in \Z^n$ such that $...
8
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0
answers
263
views
Positive integer solutions of $ab+ac+ad+bc+bd+cd=n$
Consider a quadratic form $Q(a,b,c,d)=ab+ac+ad+bc+bd+cd$ on $\mathbb Z^4$.
For some reason I am interested in the number of solutions $(a,b,c,d)\in\mathbb Z_{> 0}^4$ of $Q(a,b,c,d)=n$ as a function ...
- 5,053
3
votes
2
answers
246
views
Number of solutions of quadratic equation from a perfect pairing over $\mathbb{Z}/p^n\mathbb{Z}$
Let $p>2$ be a prime number, $V=\left(\mathbb{Z}/p^n\mathbb{Z}\right)^{2k+1}$. The bilinear form $$B:V\times V \rightarrow \mathbb{Z}/p^n\mathbb{Z}$$
is a perfect pairing. That is, mapping $x\in V$ ...
- 453
1
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0
answers
36
views
Values on minkowski square of rational ellipsoid
Let $q(x)$ be positive-definite rational quadratic form. Let $C_q$ be the following set:
$$C_q = \{q(x + y) \quad | \quad q(x) = q(y) = 1\}$$
Is it true or not that set $C_q$ defines the equivalence ...
2
votes
1
answer
177
views
Simultaneous diagonalization of two rational forms
It is known that any two real quadratic forms are equivalent iff they have the same signature. If we consider rational quadratic forms, they are $\mathbb{Q}$-equivalent iff the have the same signature,...
3
votes
0
answers
308
views
Is there an integer-valued quadratic polynomial $P(x,y)$ such that $\{P(x,y)+2^k:\ x,y\in\mathbb Z\ \text{and}\ k\in\mathbb N\}=\mathbb Z^+$?
I seek for very sparse representations of positive integers. Let
$$\mathbb N=\{0,1,2,\ldots\}\ \ \ \text{and}\ \ \ \mathbb Z^+=\{1,2,3,\ldots\}.$$
Recall that a polynomial $P(x,y)$ is integer-valued ...
- 15.6k
4
votes
1
answer
340
views
Computing the genus of certain ternary indefinite lattices
For $k$ a positive integer, let us consider the rank 3 ternary indefinite lattice $L=L_k$ with quadratic form
$$6kx^2-2(y^2+yz+z^2).$$
Its discriminant group has length $2$.
Question. Is this lattice ...
- 91
4
votes
1
answer
317
views
Classification of vector spaces with a quadratic form and an order n automorphism
Introductory general nonsense (for motivation: feel free to skip): Let $G$ be a finite group and $k$ be a field of characteristic $0$. Consider the set $\mathcal{S}$ of isomorphism classes of finite ...
- 32.5k
1
vote
0
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195
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finding automorphisms of binary hermitian forms
Let $K$ be an imaginary quadratic field and $\mathcal{O}_K$ be its ring of integers. Write $\mathcal{H}(\mathcal{O}_K)$ for the set of hermitian 2 x 2 matrices with entries in $\mathcal{O}_K$. Now, we ...
- 33
5
votes
1
answer
414
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Binary quadratic forms order four in the form class group not having desired coefficients
I have been looking at binary quadratic forms for a question on MSE, If a binary quadratic form primitively represents $n$
and $n^3$, must it be the identity form?, about forms representing a prime (...
- 25.7k