112
votes
Zagier's one-sentence proof of a theorem of Fermat
Let me answer your question "where do these involution come from" with an elementary geometric explanation. You can skip ahead to the pictures, which are somewhat self-explanatory, I hope.
The ...
27
votes
Accepted
Positive quadratic polynomial
No. The main idea is to find a $q$ on $S$ that is the sum of squares, but where the expressions being squared are not linear combinations of the coordinates in ${\bf R}^n$.
Consider the space curve
\...
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 ...
22
votes
The statement that $A \ge B$ implies $A^{-1} \le B^{-1}$ is still true for matrices?
This is a well-known fact. A simple proof : setting $y=B^{1/2}x$, we have $\|y\|^2\le y^TB^{-1/2}AB^{-1/2}y$, that is $I_n\le B^{-1/2}AB^{-1/2}$. The eigenvalues of the latter symmetric matrix are ...
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 ...
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
$$r_Q(n)\ll_{k,\...
20
votes
Accepted
A diophantine equation in $\mathbb{N}$
This is an elaboration of Emil Jeřábek's important comment, and contains no original contribution. The OP's problem was examined in depth by Borwein-Choi (1999), and their article is available for ...
20
votes
Accepted
Positive 4-form
(6 October 2023) I'll leave the original argument below because it seems that many people liked it, but, in fact, it wanders around and introduces a lot of unnecessary information, which obscures the ...
19
votes
Accepted
Upper bound on answer for Pell equation
Let $d$ be a positive fundamental discriminant, $\epsilon_d$ denote the fundamental unit, $h(d)$ the class number, and $\chi_d$ the primitive character associated to the discriminant $d$. The class ...
19
votes
A criterion for real-rooted polynomials with nonnegative coefficients
Also simply for $P(x)=x^4-1$ we have
$$P\cdot P''+\left(\frac{1}{4} -1 \right)\left(P'\right)^2=-12x^2\leq 0$$
and $P$ has imaginary roots.
The stronger version is also false. Take $P(x)=x^5-x$ ...
16
votes
Accepted
Do almost all systems of quadratic equations have solutions?
Yes. The magic words are "elimination theory" and "resultant". In essence, the system has a solution unless some determinant (the iterated resultant) vanishes.
14
votes
Accepted
Set of quadratic forms that represents all primes
Every prime $p$ is represented by at least one of the following quadratic forms: $x^2+y^2$, $x^2+3y^2$, $3x^2-y^2$:
if $p=2$ or $p\equiv 1\pmod{4}$, then $p$ is represented by $x^2+y^2$;
if $p=3$ or $...
14
votes
Accepted
How to describe all integer solutions to $x^2+y^2=3z^2+1$?
Ok, I now was able to solve the equation myself.
If $(x,y,z)$ is any solution, then $(y,x,z)$, $(x,-y,z)$, $(x,y,-z)$, and $(x,3z-2y,2z-y)$ are also solutions. To check the last one, observe that
$$
...
13
votes
When does $axy+byz+czx$ represent all integers?
Just so you know, one of Dickson's students (A. Oppenheim) finished classifying (indefinite) universal ternaries; the final family is $xy - M z^2.$ Page 161 in Modern Elementary Theory of Numbers. ...
13
votes
Accepted
Simple conjecture about rational orthogonal matrices and lattices
Proof
Let $R$ be any matrix. We have the obvious exact sequence
$$ 0 \longrightarrow\mathbb{R}^N \xrightarrow[\left(\begin{matrix} I \\ R \end{matrix}\right)]{} \mathbb{R}^N \oplus \mathbb{R}^N \...
12
votes
Upper bound on answer for Pell equation
Let $u_0=x_0+y_0\sqrt{p}$ be the smallest solution with $x_0,y_0>0$. (I assume you're talking about an upper bound for the smallest solution, since obviously there are solutions that are ...
12
votes
A criterion for real-rooted polynomials with nonnegative coefficients
It seems that inverse fails in general.
In terms of roots $z_1,\dots,z_n$ your inequality $P\cdot P'' + (\frac{1}{n}-1)P'^2 \leqslant 0$ may be rewritten as $\sum_{1\leqslant j<k\leqslant n} (\...
12
votes
Which quaternary quadratic form represents $n$ the greatest number of times?
GH from MO gave in his answer a bound due to himself and Valentin Blomer of $O(\sigma(n))$. I thought it would be interesting to compute the $O$. Here I'm looking for an effective bound of the form $\...
12
votes
Accepted
Intersection of two quadrics that have a common inscribed sphere
This is a nice observation about quadrics, I haven't seen it stated anywhere. It can be proved with the help of pencils of quadrics.
1) The curve of tangency is a circle.
Consider the pencil of ...
12
votes
Accepted
how to prove an equation involving sums of Kronecker symbol
The identity can be rewritten as
$$\sum_{\substack{|x|<p\\ 2|x}}\sum_{r|p^2-x^2}\left(\frac{-3}{r}\right)=p+2,$$
because for $x=0$ the inner sum is $1-1+1=1$. Writing $x=2c$, the identity becomes
$$...
11
votes
Accepted
Asymptotic formula for sums of four squares?
The function you are asking for is $r_4(n)$, the number of ways to write $n$ as a sum of four squares. The exact formula was discovered by Jacobi, and is given as
$$\displaystyle r_4(n) = 8\sum_{\...
11
votes
Accepted
Are lattice points in thin spherical shells uniformly distributed?
Yes, they are equidistributed as long as $\delta<11/16$ and $r=R^{-\delta}$ and $R\to\infty$. Without loss of generality, we shall assume that $\delta>-1$ (i.e. $r<R$).
To see this, let $\...
11
votes
Accepted
Integer positive definite quadratic form as a sum of squares
This is a well-known problem, called the Waring's problem of integral quadratic forms. Every semi-positive definite quadratic form in $n \leq 5$ variables is a sum of $n + 3$ squares of linear forms. ...
10
votes
Accepted
Reference request: correspondence between central simple algebras and quadratic forms
Everything is in Lam's book Introduction to Quadratic Forms over Fields. Theorem III 5.1 says:
All central simple algebra $A$ of dimension $4$ is quaternion. That is $A \cong \left(\frac{a,b}{k}\...
10
votes
Reference request: correspondence between central simple algebras and quadratic forms
In the formulation, presumably on the right side what is intended are 3-dimensional non-degenerate quadratic spaces (up to isomorphism), with discriminant 1 (same as $4^3$ mod squares as John Ma notes)...
10
votes
Asymptotic formula for sums of four squares?
Note that
$$L(x) = \sum_{n\leq x} r_4(n)$$ is the number of lattice points in the ball of radius $\sqrt{x}.$
It is known that $$L(x) = \frac{\pi^2}2 x^2 + O(x \log(x)).$$ (the error term can be ...
10
votes
Solutions to the Diophantine equation $x^2+3y^2+3z^2=n$
The theta function associated to your quadratic form (here $\theta(\tau)\theta(3\tau)^2$) belongs to the modular form space $M_{3/2}(\Gamma_0(12))$,
and you are in luck because there are no cusp forms,...
9
votes
Zagier's one-sentence proof of a theorem of Fermat
A recent article gives another very elegant proof of the two-square theorem in the Zagier style, but is easier. See
Stan Dolan, A very simple proof of the two-squares theorem,
The Mathematical ...
9
votes
Equivalence of binary quadratic forms over $\operatorname{GL}_2(\mathbb{Q}_p)$ or $\operatorname{GL}_2(\mathbb{Z}_p)$
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 ...
9
votes
Quadrics in the Grothendieck ring
Edit. Following Remy van Dobben de Bruyn's excellent suggestion, I clarified the use of "irreducible quadrics of dimension $0$."
Daniel Loughran's observation about Chevalley-Warning is the key to ...
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