2
votes
1answer
22 views

Terminology: what is the “generic character” of a ternary quadratic form?

The title says it all: What is the "generic character" of a ternary quadratic form? Motivation: I'm reading a really old paper, and the author refers to this terminology without any further ...
0
votes
1answer
13 views

Find $\alpha , \beta$ s.t. $\forall s_i\in\mathbb{Z} ,\frac{\alpha^2}{\beta}\neq\frac{(s_1-s_2)^2+(s_3-s_4)^2+…}{(s_1+s_2)^2+(s_3+s_4)^2+…}$

Let us assume that $\alpha,\beta , s_i\in\mathbb{Z}$ , for $i=1,...,8$. is it possible to choose $\alpha,\beta$ such that for all $s_i\in\mathbb{Z}$ the following equation is $never$ ...
3
votes
3answers
122 views

Find $m_1 , m_2,m_3,m_4\in\mathbb{Q}$ s.t. $\forall a_k,b_k\in\mathbb Z,\,m_1(a_1^2+a_2^2)+m_2(a_3^2+a_4^2)\neq m_3(b_1^2+b_2^2)+m_4(b_3^2+b_4^2)$

Let us assume that $a_1 , a_2 , a_3 ,a_4,b_1,b_2,b_3,b_4\in\mathbb{Z}$. If $m_1 , m_2,m_3,m_4\in\mathbb{Q}$, then how can I choose $m_1,m_2,m_3,m_4$, such that the following equation is $never$ ...
2
votes
1answer
28 views

Using continued fractions to well-approximate a quadratic form?

Continued fractions are the "best rational approximation" of other numbers. For a real number $\alpha$ the continued fraction algorithm produces a sequence of integers $\alpha = [a_1, a_2, \dots, ...
0
votes
1answer
30 views

Representation of integers by ternary quadratic form $x^2+y^2-z^2$

Let $Q$ be the ternary quadratic form $Q(x,y,z)=x^2+y^2-z^2$. Since $Q(0,p+1,p)=2p+1$ and $Q(1,p+1,p)=2p+3$, we see that for every integer $k$, the equation $E_k:Q(x,y,z)=k$ always has a solution. Is ...
0
votes
0answers
31 views

Show that for any given d<0, the primitive positive definite quadratic forms of discriminant d all have the same number of automorphs.

Show that for any given d<0, the primitive positive definite quadratic forms of discriminant d all have the same number of automorphs. I think we should let f(x,y) = $ax^{^{2}} + bxy + ...
0
votes
0answers
31 views

Rank four quadratic form with trivial discriminant

Is there an example of a field $k$, quadratic form $\varphi$ of rank four, which is anisotropic over $k$, has trivial discriminant and is not a Pfister form? In case of rank six one can use Albert ...
1
vote
1answer
65 views

Signed determinant of quadratic forms over Q_p

Let $W(k)$ be the Witt-Ring of the field $k$. in this script http://math.uga.edu/~pete/quadraticforms2.pdf at the bottom of page 2 the signed determinant is introduced by $d^\pm (q) = ...
0
votes
1answer
29 views

Quaternary Quadratic Forms

What is a Quaternary Quadratic Form? I've looked for a definition online and cannot find a precise clear definition. I am not taking a course. Just reading about quadratic forms. Thank you.
1
vote
0answers
31 views

Classification of quadrtic forms over Q_p

I need some one to recap the topic with me and correct me when i am wrong. There are basically just a few questions at the end,but its important to also show what i know and not just what i dont. ...
9
votes
3answers
390 views

A Pell equation inside a Pell equation

While working on another problem (see http://mathoverflow.net/questions/143599/solving-the-quartic-equation-r4-4r3s-6r2s2-4rs3-s4-1), I found the following equation to be solved: $$ ...
0
votes
0answers
25 views

Stuck in quadratic forms and discriminats problem

So I'm stuck in a pretty easy question about discriminants and quadratic forms of equations. I have already proved one side of the problem: we suppose that $x_0, y_0$ are the solutions to the ...
2
votes
2answers
109 views

how to find rational numbers satisfying the binary quadratic equation $x^2+3xy+5y^2=4$

I am looking for a generalisation of the solution of $x,y$ wich are rational numbers,they could be infinite,how can i find such solutions,integer solutions are obvious I have found that ...
0
votes
0answers
12 views

how to write a lattice $[\alpha,\beta ]$ in the form [$a,b+c\omega _7$]

$\fbox{1}$ if we write [$2-\sqrt{7},5+3\sqrt{7}$] in the form [ $a,b+c\omega _7$],what is the value of $a,b,c$ $\omega=\sqrt{7}$,since $ 7\equiv 3\mod 4$ $N(2-\sqrt{7})=4-7=-3$ $N( ...
1
vote
2answers
68 views

number theory of coefficients in an infinite sequence of polynomials

EDIT: equivalent formulation by Hurkyl in comments: if $n$ is odd and $p^\nu \parallel n$ and $n > 2k,$ then $$ p^{(\nu + 2 + 2 k - n)} \; | \; \sum_j \left( \begin{array}{c} n \\ 2j \end{array} ...
0
votes
0answers
17 views

Positive definite integral quadratic form with minimal orthogonal group?

Are there explicit examples in every rank of positive definite integral quadratic forms with orthogonal group $\pm 1$?
0
votes
1answer
62 views

Counting function for sums of three squares

Legendre showed that an integer is the sum of three squares if and only if it is not of the form $4^n(8m + 7)$ for some nonnegative integers $n$ and $m$. However, I have been unable to find any ...
1
vote
0answers
60 views

Enumerating integer solutions to quadratic equations

Consider a quadratic equation with integer coefficients in two variables. $$ax^2+bxy+cy^2+dx+ey+f=0$$ I would like to know how to find the number of integer solutions $(x,y)$ to this equations. Is ...
2
votes
0answers
49 views

Connection between class number and the theory of Ideals/Quadratic Fields

I've been studying the classic results in integer binary quadratic forms, mainly the equivalence and reduction of quadratic forms and the class number $H(d)$ (the definition I got for $H(d)$ is the ...
2
votes
0answers
37 views

Definition of the term 'generic' in context of quadratic forms.

In Proposition 3.3 of the paper: A. Lubotzky, R. Phillips and P. Sarnak, Ramanujan Graphs, Combinatorica 8(1988), the authors use a result obtained by Malisev : "Let $f(x_1,\ldots,x_n)$ be a ...
2
votes
1answer
93 views

Can the equation $ax^2+by^2=cz^2$ be solved in integers (excluding trivial solutions)?

Suppose $a,b,c\in\mathbb{N}$ and are each squarefree. Is there a general solution for this equation? I found that for this equation to be soluble in integers there are three necessary and sufficient ...
0
votes
3answers
121 views

What integers can be represented by the quadratic form $4x^2 - 3y^2 - z^2$?

Actually, I need to find if $4x^2 - 3y^2 - z^2 = 12$ is solvable. But I somehow feel that applying theory of integer representation by quadratic forms in three variables would yield quicker results... ...
1
vote
1answer
172 views

The importance of Binary Quadratic Form [closed]

Can anyone explain to me the importance of Binary Quadratic forms in Number theory, and why do so many mathematicians think that it's interesting? I cannot understand why mathematicians in the past ...
0
votes
1answer
56 views

Find all $(a,b,c)\in\mathbb{Z}^3$ such that $b^2-4ac=-20$, and $-|a|<b\le|a|<|c|$, or $0\le b\le|a|=|c|$.

Find all $(a,b,c)\in\mathbb{Z}^3$ such that $b^2-4ac=-20$, and either of the following is true: $-|a|<b\le|a|<|c|$, or $0\le b\le|a|=|c|$. We see that if $(a,b,c)$ is a solution, then so is ...
1
vote
3answers
59 views

What restrictions are on th sum of two fourth powers?

I've got an equation of the form $$ a^4+1=2b. \qquad(\star) $$ By well-known results regarding the sum of two squares, $b$ must be the sum of two squares. But does $(\star)$ force any other ...
0
votes
2answers
153 views

How to solve an equation of the form $ax^2 - by^2 + cx - dy + e =0$?

I am trying to find out how to solve $ax^2 - by^2 + cx - dy + e = 0$ to get integer solutions, failing this the rational solutions. Thanks!
1
vote
0answers
83 views

About quadratic form and its discriminant

There are 3 parts of the problem. Let d be a perfect square, possibly 0. Show that there is a quadratic form $ax^2+bxy+cy^2=0$ of discriminant d for which a=0. Let a,b,c be integers with $a\ne0$. ...
2
votes
1answer
74 views

A problem about the discrete logarithm

suppose there are a multiplicative cyclic group $F_p^*(p \;is\;big\; prime)$, and $G=\langle g \rangle(g \;is\; a\; generator)$ is a subgroup of it and $G$'s order is $q(q\;is\;big\;prime \;and ...
2
votes
1answer
72 views

finding zeroes of a quadratic form

Let $a,b\in\mathbb Z$ be squarefree with $a>0$. Suppose that I know that there exist $(0,0,0)\neq (x,y,z) \in \mathbb Z^3$ s.t. $x^2-by^2-az^2=0$. Is there any known algorithm to find any such a ...
2
votes
0answers
33 views

Holzer reduction of solutions of quadratic ternary forms

Suppose $(x_{0}, y_{0}, z_{0})$ is a solution to the equation $ax^2 + by^2 + cz^2 = 0$. The solution is said to be Holzer reduced if $x_{0} < \sqrt{|bc|}$, $y_{0} < \sqrt{|ac|}$ and $z_{0} < ...
3
votes
1answer
142 views

Any integer can be written as $x^2+4y^2$

If $n$ is a positive integer with $(n,8)=1$ and $-4$ is square $mod$ $n$ then $n$ can be written in this form: $n=x^2+4y^2$. I was using that there are x, y integers satisfying $x^2+4y^2=kn$ where ...
7
votes
1answer
925 views

What's a BETTER way to see the Gauss's composition law for binary quadratic forms?

There is a group structure of binary quadratic forms of given discriminant $d$: Let $[f]=[(a,b,c)], [f']=[(a',b',c')],$ where $d=b^2-4 a c=b'^2-4 a' c'.$ The composition of two binary quadratic ...
3
votes
1answer
90 views

Question about the definition of representability of a quadratic form

Say I have an integral quadratic form $q$ on $\mathbb{Z}^r$ and another integral binary quadratic form $\tilde{q}$. What does it mean for $q$ to primitively represent $\tilde{q}$? I can't seem to find ...
0
votes
1answer
77 views

Generating vectors of the face-centered cubic lattice

I can't find a reference for a set of generating vectors for the Tetrahedral-octahedral honeycomb lattice. I would like to know the "canonical" set and if possible a more general set described by ...
9
votes
2answers
499 views

Generalizing the 290 theorem.

I have only just come across the remarkable theorem of Conway about universal quadratic forms over $\mathbb{Z}$; namely that in determining whether a integer coefficient, positive definite quadratic ...
-3
votes
2answers
139 views

Lucky Lattice Points

How many lattice points lie on the sphere given by following equation ? $$x^2+y^2+z^2=2013$$ Hint: A lattice point has integer coordinates.
3
votes
6answers
376 views

Integral solutions of hyperboloid $x^2+y^2-z^2=1$

Are there integral solutions to the equation $x^2+y^2-z^2=1$?
1
vote
2answers
129 views

Quadratic Forms and Congruences

How does one prove (the non-trivial direction) that, for $n \in \mathbb{N}$, $x^2 + y^2 + z^2 = n$ solvable $\iff$ $x^2 + y^2 + z^2 \equiv n\ (m)$ solvable for all $m$? In particular, is there a ...
7
votes
0answers
137 views

What numbers are integrally represented by $4 x^2 + 2 x y + 7 y^2 - z^3$

EDIT: a selection of material relevant to this problem is available at INHOMOGENEOUS In December 2010 my question appeared in the M.A.A. Monthly, show that $4 x^2 + 2 x y + 7 y^2 - z^3 \neq \pm 2 ...
1
vote
1answer
88 views

Solution count of quadratic form congruence over $\Bbb Z / 8 \Bbb Z$

Let $(L,q)$ be an even Lorentzian lattice of signature $(1,l-1)$, i.e., $q(\lambda) \in \Bbb Z$ for all $\lambda \in L$ and the (non-degenerate) quadratic form $q$ is of type $(1,l-1)$ (the ...
1
vote
0answers
133 views

solution count of quadratic form congruences

Let $(L,q)$ be an even Lorentzian lattice of signature $(1,l-1)$, i.e., $q(\lambda) \in \Bbb Z$ for all $\lambda \in L$ and the (non-degenerate) quadratic form $q$ is of type $(1,l-1)$ (the ...
4
votes
3answers
184 views

The quadratic form $x^2 + ny^2$ via prime factors

Elementary algebra shows that the product of two numbers in the form $x^2 + ny^2$ again has the same form, since if $p = (a^2 + nb^2)$ and $q = (c^2 + nd^2)$, $$pq = (a^2 + nb^2)(c^2 + nd^2) = (ac ...
3
votes
2answers
94 views

When does a binary quadratic form represent 1 or -1

Let $a,b,c$ be integers. Is there a reasonably concise condition on $(a,b,c)$ which ensures that $$ax^2+bxy+cy^2=\pm 1$$ has a solution in integers $x,y$? In addition to direct answers I would also ...
4
votes
2answers
285 views

Equivalence of quadratic forms over p-adic fields.

There is a theorem that states that two quadratic forms over $\mathbb{Q}_p$ are equivalent iff they have the same rank, discriminant and the same $\epsilon$ invariant. (The last is defined as ...
8
votes
1answer
144 views

On the set of integer solutions of $x^2+y^2-z^2=-1$.

Let $$ \mathcal R=\{x=(x_1,x_2,x_3)\in\mathbb Z^3:x_1^2+x_2^2-x_3^2=-1\}. $$ The group $\Gamma= M_3(\mathbb Z)\cap O(2,1)$ acts on $\mathcal R$ by left multiplication. It's known that there is ...
1
vote
1answer
114 views

Quadratic form over the dyadic numbers

I would like to know whether $q=\langle 3,3,11\rangle$ (a diagonal ternary form) represents $2$ over $\mathbb{Q}_2$ (i.e. whether there exist $x,y,z\in\mathbb{Q}_2^\times$ such that $q(x,y,z)=2$). I ...
1
vote
1answer
319 views

Find all solutions of this diophantine equation of the second degree in three variables

Consider the Diophantine equation $Q(x,y,z)=1$, where $Q(x,y,z)$ is the quadratic form $x^2+y^2-z^2$. Let $S \subseteq {\mathbb Z}^3$ denote the set of all solutions. It is rather easy to find several ...
5
votes
0answers
167 views

Solutions to equation in matrix form

Suppose $x=[x_1, x_2, \cdots, x_n]^t$, $b=[b_1,b_2,\cdots,b_n]^t$ with $b_i\in K$ and $A\in M_n(K)$, where $K$ is a field. There are well known criteria for the system of equations $Ax=b$, by ...
2
votes
2answers
190 views

A question about integral quadratic forms

Hi Would you please advise me? Consider the equation below: $$ ax^2+bxy+cy^2=n $$ in which $a, b, c$ and $n$ are integers. We then suppose that $a, b, c$ are constant. Is there any way to find the ...
5
votes
1answer
238 views

Universal quadratic forms

A quadratic form is a polynomial $p(x_1,\dots,x_n)$ of the form $$ p(x_1,\dots,x_n)=\sum_{i \leq j}a_{ij}x_ix_j. $$ For example, $p_1(x,y,z,w)=x^2+y^2+z^2+w^2$ and $3x^2-5y^2$ are quadratic forms. I'm ...