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Results tagged with quadratic-forms
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user 3324
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.
7
votes
2
answers
711
views
Mass of spinor genus, positive integral quadratic forms
There seems to be general opinion that, for positive integral quadratic forms in at least three variables, spinor genera in the same genus all have the same mass (not representation measures of some n …
- 25.2k
3
votes
1
answer
394
views
Faux Mordell equation and positive binary quadratic forms
This is about the frequency of integral solutions to
$$ b^2 - 4 a^3 = \Delta, $$
when $\Delta < 0$ is a discriminant of positive binary quadratic forms such that the class number is divisible by 3. …
- 25.2k
1
vote
1
answer
184
views
probably Lagrange or Legendre, Pell variant
Evidently Legendre showed that, for positive primes, if $p \equiv 3 \pmod 8$ there is an integral solution to $x^2 - p y^2 = -2.$ Next, if $q \equiv 7 \pmod 8$ there is an integral solution to $x^2 - …
- 25.2k
4
votes
1
answer
169
views
optimal bound in diophantine representation question
Given that, with integers $t \geq 1$ and $q \geq 3,$ there are solutions to $$ x^2 - q x y + y^2 = - t q $$
with integers $x,y \geq 1,$ I was able to show that
$$ q \leq 1 + \frac{324}{25} t^2. $$
…
- 25.2k
1
vote
0
answers
183
views
Watson Transformation Squared reference request
See
http://www.numbertheory.org/obituaries/OTHERS/watson.html
George Leo Watson (1909-1988) wrote in a conversational manner, it is difficult to see when he switches from the trivial to the incredibl …
- 25.2k
5
votes
2
answers
293
views
Matrix version of number theoretic integral lattice claim
I know this not, everybody else seems to. This is from page 215 of Cassels, Rational Quadratic Forms, formula 4.1, or SPLAG, page 389 formula (36). quote:
If $f$ and $g$ are forms of
determinant …
- 25.2k
7
votes
0
answers
437
views
When is the set of numbers represented by certain quaternary quadratic forms completely mult...
Expired by this question
A quadratic form represents all primes except for the primes 2 and 11.
I would like to know some simple sufficient conditions for when the set of numbers integrally represe …
- 25.2k
9
votes
2
answers
2k
views
Does a positive binary quadratic form represent a set of primes possessing a natural density
In his answer to my question
The Green-Tao theorem and positive binary quadratic forms
Kevin Ventullo answers my initial question in the affirmative. What remains is the title questio …
- 25.2k
11
votes
1
answer
814
views
Primes $ 1 + x^2 + y^2$
EDIT, Saturday 11:32 am, March 24: a complete answer for $4 + x^2 + y^2$ and generally $4 m^2 + x^2 + y^2$ for fixed $m$ is given in Friedlander and Iwaniec page 282, Theorem 14.8. We might also expec …
- 25.2k
8
votes
3
answers
1k
views
Integral orthogonal group for indefinite ternary quadratic form
I have the indefinite quadratic form $q(x,y,z) = 19 x^2 + 5 y^2 - z^2.$ It's not my fault. I find, on reflection, that I have no idea how to describe the orthogonal group of this over the integers. Th …
- 25.2k
7
votes
3
answers
1k
views
Fricke Klein method for isotropic ternary quadratic forms
Preface: the most natural way to take one isotropic vector for an indefinite quadratic form and find others is to use stereographic projection. This gives a parametrization in the same $n$ variables a …
- 25.2k
11
votes
2
answers
995
views
Positive primes represented by indefinite binary quadratic form
Neil Sloane asked me about commands in computer languages to find the (positive) primes represented by indefinite binary quadratic forms. So I wrote something in C++ that works. This is for the OEIS, …
- 25.2k
11
votes
1
answer
714
views
Positive ternary quadratic forms in the same genus that represent the same numbers
There are three genera of positive, integral, ternary quadratic forms in which both forms (classes...) are regular, so the paired forms represent the same numbers. These pairs (complete genera) are:
…
- 25.2k
7
votes
2
answers
630
views
Verifying an example in the Geometry of Numbers and Quadratic Forms
In answer to Pete L. Clark's question Must a ring which admits a Euclidean quadratic form be Euclidean? on Euclidean quadratic forms, I gave an example in seven variables, repeated below. Pete's Eucl …
- 25.2k
6
votes
1
answer
353
views
Erdos Kac for imaginary class number
In answer to A coverage question Cam mentions an article by SOUND. I have been running a computer program for THIS and would like to know if there are a reasonable average and standard deviation for …
- 25.2k