# Questions tagged [quadratic-forms]

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356
questions

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votes

**1**answer

120 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 ...

**5**

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**1**answer

93 views

### Evaluating a binary quadratic form at convergents

We use the notation
$$\displaystyle [a_0; a_1, \cdots, a_n] = a_0 + \cfrac{1}{a_1 + \cfrac{1}{\ddots + \cfrac{1}{a_n}}}$$
to denote a finite continued fraction, and for a given real number $\alpha$, ...

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249 views

### On positive integers of the forms $x^2+5y^2$, $x^2+6y^2$, $x^2+10y^2$

For a positive integer $n$ and a prime $p$, we let $\nu_p(n)$ denote the $p$-adic valuation of $n$ (i.e., the $p$-adic order of $n$).
On the basis of my computation, I formulated the following ...

**4**

votes

**2**answers

199 views

### Quadratic diophantine equations and geometry of numbers

Let (for concreteness) $a = 2$, $b = \sqrt{5}$ and $\varphi = (\sqrt{5}+1)/2$. I am interested in solutions $(w,x,y,z) \in \mathbb{Z}[\varphi]^4$ of the system
$$
w^2 - ax^2 -by^2 + abz^2 = 1
$$
$$
\...

**1**

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157 views

### Topological vs algebraic intersection forms

Let $X$ be a simply connected complex projective surface (hence a real $4$-manifold). Let $(H^2(X,\mathbb Z)/\mathrm{tors}, q_X)$, $(A^1(X)/\mathrm{tors},q_X')$ be the corresponding lattices in ...

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68 views

### Proper ideals are invertible

I am reading through Cox's book Primes of the form $x^2+ny^2$ and I am stuck with some proofs in Chapter 7 (I have the 2nd edition). There, the author presents the following Lemma:
Lemma 7.5: Let $...

**3**

votes

**1**answer

215 views

### Strong Approximation for sol'ns to quadratic diophantine equations

Can anyone either direct me to an relatively elementary proof in the literature--or show me why this (Conjecture 1 stated below) is true--or if I am mistaken and it is not true:
For any 4-tuple $\xi =...

**7**

votes

**3**answers

253 views

### Integer positive definite quadratic form as a sum of squares

Let $A$ be a symmetric $d\times d$ matrix with integer entries such that the quadratic form $Q(x)=\langle Ax,x\rangle, x\in \mathbb{R}^d$, is non-negative definite. For which $d$ does it imply that $Q$...

**2**

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**2**answers

191 views

### Solving diagonal simultaneous quadratic equations

A problem I am trying to solve has led to me to the following system of equations:
$$A(x^2) + Bx + c = 0$$
Where $A$ and $B$ are known matrices, $c$ is a known vector, $x$ is the vector of unknowns ...

**1**

vote

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87 views

### Norm quadrics and their motives

Let $k$ be a field of characteristic $\neq 2$ and $\langle\!\langle a_{1},\cdots,a_{n}\rangle\!\rangle$ a Pfister form over $k$. Denote by $Q_{\underline{a}}=Q_{a_{1},\cdots,a_{n}}$ the projective ...

**9**

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**1**answer

196 views

### Is it possible for the Witt group of a scheme to have non-trivial odd torsion?

Let $F$ be a field of characteristic not $2$. A well-known theorem of Pfister asserts that the torsion of $W(F)$, the Witt group of $F$, is $2$-primary.
Baeza [B, V.6.3] extended this result to Witt ...

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107 views

### A conjecture stronger than Gauss' triangular number theorem

A subset $A$ of $\mathbb N=\{0,1,2,\ldots\}$ is called an asymptotic additive basis of order $3$ if any sufficiently large integer can be written as the sum of three elements of $A$.
I have ...

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89 views

### Nonnegative integers written as $x(3x+1)+y(3y-1)+z(3z+2)+w(3w-2)$ with $x,y,z,w\in\{0,1,2,\ldots\}$

Recently, I formulated the following conjecture which is somewhat similar to Lagrange's four-square theorem.
Conjecture. Each $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as $x(3x+1)+y(3y-1)+z(3z+...

**3**

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146 views

### On sums of three squares

Let $\mathbb N=\{0,1,2,\ldots\}$. By the Gauss-Legendre theorem on sums of three squares, any $n\in\mathbb N$ with $n\equiv1,2\pmod4$ can be written as $x^2+y^2+z^2$ with $x,y,z\in\mathbb N$. Clearly, ...

**5**

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207 views

### Sums of two integer squares in arithmetic progressions

Is there an explicit formula in the literature for the number of representations of a positive integer $n$ as a sum of two integer squares, the second of which is divisible by $5$? So this means to ...