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Results tagged with quadratic-forms
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user 11919
Algebraic and geometric theory of quadratic forms and symmetric bilinear forms, e.g., values attained by quadratic forms, isotropic subspaces, the Witt ring, invariants of quadratic forms, the discriminant and Clifford algebra of a quadratic form, Pfister forms, automorphisms of quadratic forms.
26
votes
Accepted
When does $axy+byz+czx$ represent all integers?
Here is a proof of the conjecture. I will refer several times to the book Cassels: Rational quadratic forms (Academic Press, 1978).
1. Let $p$ be a prime such that $p\nmid a$. Using the invertible li …
- 105k
21
votes
Accepted
two's and three's survive in gcd of Lagrange
The claim is certainly true for $n$ sufficiently large, and "sufficiently large" could be specified explicitly with more care.
We follow the suggestion of Fedor Petrov, and rely on the results of Br …
- 105k
20
votes
Accepted
A diophantine equation in $\mathbb{N}$
This is an elaboration of Emil Jeřábek's important comment, and contains no original contribution. The OP's problem was examined in depth by Borwein-Choi (1999), and their article is available for fre …
- 105k
20
votes
Accepted
Which quaternary quadratic form represents $n$ the greatest number of times?
Theorem. Let $Q(x_1,\dots,x_k)$ be a positive definite integral quadratic form in $k\geq 2$ variables. Then the number of integral representations $Q(x_1,\dots,x_k)=n$ satisfies
$$r_Q(n)\ll_{k,\epsil …
- 105k
20
votes
Many representations as a sum of three squares
Let me restrict to the number of primitive representations
$$r_3^\ast(n) = \left|\{(a,b,c)\in {\mathbb Z}^3 :\, a^2+b^2+c^2=n\ \text{and}\ \gcd(a,b,c)=1 \}\right|.$$
Note that $r_3(n)$ can be easily e …
- 105k
14
votes
Accepted
Set of quadratic forms that represents all primes
Every prime $p$ is represented by at least one of the following quadratic forms: $x^2+y^2$, $x^2+3y^2$, $3x^2-y^2$:
if $p=2$ or $p\equiv 1\pmod{4}$, then $p$ is represented by $x^2+y^2$;
if $p=3$ or …
- 105k
12
votes
Accepted
how to prove an equation involving sums of Kronecker symbol
The identity can be rewritten as
$$\sum_{\substack{|x|<p\\ 2|x}}\sum_{r|p^2-x^2}\left(\frac{-3}{r}\right)=p+2,$$
because for $x=0$ the inner sum is $1-1+1=1$. Writing $x=2c$, the identity becomes
$$\s …
- 105k
12
votes
Accepted
How to prove this problem about ternary quadratic form?
Let $G(n)$ denote the weighted number of $\mathrm{SL}_2(\mathbb{Z})$-equivalence classes of positive definite quadratic forms
$$[a,b,c]:=ax^2+bxy+cy^2$$
with $2\mid b$ and $b^2-4ac=-4n$, where the cla …
- 105k
11
votes
Accepted
Are lattice points in thin spherical shells uniformly distributed?
Yes, they are equidistributed as long as $\delta<11/16$ and $r=R^{-\delta}$ and $R\to\infty$. Without loss of generality, we shall assume that $\delta>-1$ (i.e. $r<R$).
To see this, let $\mathcal{F}\s …
- 105k
9
votes
Equivalence of binary quadratic forms over $\operatorname{GL}_2(\mathbb{Q}_p)$ or $\operator...
The number of classes of (regular) binary quadratic forms over $\mathbb{Q}_p$ equals $7$ when $p\neq 2$, and it equals $15$ if $p=2$. See Section IV.2.3 of Serre: A course in arithmetic.
The number o …
- 105k
8
votes
Accepted
The quadratic form $x^2+ny^2$ via prime factors
The answer is yes. To see this, consider the ring $R=\mathbb{Z}[\sqrt{-n}]$. If $z=p_1\dots p_k$ is the decomposition of $z$ into rational primes, then by assumption each $p_j$ decomposes in $R$ as $p …
- 105k
7
votes
A quadratic Diophantine equation
Working in the finite field $\mathbb{F}_p$ and applying a linear change of variables, the equation can be written as
$$a_1x_1^2+a_2x_2^2=1$$ with some nonzero coefficients $a_1,a_2\in\mathbb{F}_p^\ti …
- 105k
7
votes
How many solutions are there to $\sum_{i=1}^3 x_i^2+x_iy_i+y_i^2=k$?
This is a supplement to Noam Elkies' nice answer. The coefficients $s(k)$ can be expressed as
$$ s(k)=27\sum_{d\mid k}\chi(k/d)d^2-9\sum_{d\mid k}\chi(d)d^2, $$
hence the function $\varphi$ is a linea …
- 105k
6
votes
Accepted
Connection between quadratic forms and ideal class group
There is a concise account in the Appendix of these notes.
- 105k
5
votes
Accepted
Duke and Schulze-Pillot condition for equidistribution
I recommend that you study the paper by Duke and Schulze-Pillot, Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids (Inventiones, 1990) …
- 105k