Questions tagged [sums-of-squares]
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163 questions
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Is this Negativstellensatz with uniform denominators known?
A theorem of Reznick states that if $f>0$ is a real homogeneous polynomial in several polynomials that is positive away from the origin of ${\mathbb{R}}^n$, then for large $N$, the form $(\sum x_i^...
user2529
7
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2
answers
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Sums of squares of primes [closed]
Question: What is the least number that is a sum of three squares of primes in exactly six ways?
... I know it is not research mathematics. Happy new year!
EDIT: Now that it is answered I should ...
- 25.5k
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3
answers
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Realizing proper pure octonions as conjugates
Let us take the octonions as having all integer coefficients and the multiplication table at BAEZ
We have a standard conjugation operator with $\bar{1} = 1$ and $\bar{e_i}= - e_i,$ extend by ...
- 25.7k
7
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2
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426
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Divisor sums over values of binary forms of primes
Let $\tau$ be the divisor function, that is
$$
\tau(n)=\sharp\{d \in \mathbb{N}, d|n\}.
$$
I was wondering if anyone has ever proved an asymptotic estimate
for the sum
$$S(x):=\sum_{p,q\leq x}\tau(p^...
- 3,062
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0
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Is there another simple formula for the sum-of-squares function?
The sum-of-squares function (denoted $r_{2}(n)$) gives the number of ways in which a given number $n$ is expressible as the sum of two squares. The following is from the article on this function from ...
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4
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550
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Application and usage of representation of integers as sum of powers?
We know that there are many articles and manuscripts from the ancient to date talking about representation of integers as sum of squares, cubes etc. I would like to know what is it the usage and ...
- 841
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2
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Hurwitz integers represented as sums of two squares of Hurwitz integers
I wonder if there exists a characterisation of Hurwitz integers which are represented as sums of two squares of Hurwitz integers, up to multiplication by a unit. And if so, could you please point to a ...
6
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1
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Using the decomposition $641 = 5^4 + 2^4$ to factor $F_5$
The question in the title arises from a problem in Stewart's "Galois Theory, Third Edition" (and possibly elsewhere) which has been bugging me for a few days since reading it:
Problem 19.5 (p. 224) ...
- 5,191
6
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1
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361
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A simple way to bound the density of sums of two odd squares
Define
$$S(x) ~=~ \# \left\{ n^2+m^2\leq x : n,m\in\mathbb{N}\right\}$$
Landau (1908) proved that with
$$ B(x) ~=~ K\,\frac{x}{ \sqrt{\log x}} ~~\text{ one has}~~~ \lim \limits_{x\to \infty} \frac{S(...
- 1,676
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On permuted sum of squares of primes in a list
We want to pick a set of distinct primes (if not possible, then just positive numbers) $p_1,p_2,\dots,p_k$ such that there exists $t$ permutations, $\sigma_1(\cdot)$,$\sigma_2(\cdot),\dots,\sigma_t(\...
- 13.9k
6
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1
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425
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Spherical Bessel functions. Sum of squares
In (1) there is a property of spherical Bessel functions, which's derivation I can not find in the literature.
${\mathsf{j}_{n}^{2}}\left(z\right)+{\mathsf{y}_{n}^{2}}\left(z\right)=\sum_{k=%
0}^{n}\...
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6
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1
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542
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An S-lemma for polynomials of degree 4 in three variables
Might the following be true:
Let $p,q\in\mathbb{R}[x,y,z]$ be homogeneous polynomials, with $\deg(p)\leq 4$ and $\deg(q)= 2$. Suppose $q(x,y,z)>0$ for some $x,y,z\in\mathbb{R}$. Then the ...
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Representing a symmetric polynomial as a conical sum of squares
This question in inspired by the recent solution to another question.
The following inequality for monomial symmetric polynomials in 4 positive variables $x_1,x_2,x_3,x_4$:
$$m_{(4, 3, 2, 1)} + m_{(4, ...
- 34.3k
6
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1
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For a Sum-of-Squares cost functions J(x) is it true that J(x)-j* is also SOS?
For polynomial optimization problems the sum-of-squares theory and Lasserre relaxation hierarchy provides a theoretically handy way of getting the solution. There are also results saying that finite ...
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Denominators in the solution to Hilbert's XVII
Hilbert's seventeenth problem asks to prove that every positive semidefinite form can be written as the sum of squares of rational functions. Currently we don't seem to have a good understanding of ...
- 85.5k
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Legendre's three-square theorem and squared norm of integer matrices
Let $\mathbb{N}$ be the set of non-negative integers. Let $E_n$ be the set of integers which are the sum of $n$ squares. Let $F_n$ be the set of integers of the form $\Vert A \Vert^2$ with $A \in M_n(...
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535
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When is $ \sigma(n!-1) $ a perfect square?
I am looking for pairs of positive integers $(m,n)$ such that $ \sigma(n!-1) =m^2$, where $\sigma$ is the sum of divisors function. Examples occur with $(m,n)=(12,5),(1,2)$.
Question: Are there ...
user99666
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0
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About the properities of sum of powers of items in a polynomial
Given a polynomial $f_1=a_1x+a_2x^2+\cdots+a_{p-1}x^{p-1}\in\mathbb{Z}[x]/(x^{p}-1)$, with prime $p$, we may generate the other $p-1$ polynomials:
\begin{eqnarray*}
f_2&=&a_1^2x^2+\cdots+a_{p-...
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1
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Is there a theorem which provides conditions under which a power series satisfies the reciprocal root sum law?
Kalman - Six ways to sum a series discusses Euler's original proof for the Basel problem $\sum\limits_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6} $:
$$\frac{\sin(\sqrt x)}{\sqrt x} = 1- \frac{x}{3!}+ \...
- 541
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1
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Is there an upper bound on the number of representations as a sum of squares?
I am interested in finding upper bounds for the Sum of squares function defined as $r_k(n) = |\{(a_1, a_2, \ldots, a_k) \in \mathbb{Z}^k \ : \ n = a_1^2 + a_2^2 + \cdots + a_k^2\}|$ whenever the ...
- 313
5
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1
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How often is the value of a quadratic polynomial equal to a sum of two integer squares?
Let $b:\mathbb N\to \{0,1\}$ be the indicator function of integers that are a sum of two non-zero integer squares. Let $f(t)\in \mathbb Z[t]$ be an irreducible polynomial of degree $2$ with positive ...
- 3,062
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1
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549
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A 'generalized Four Squares Theorem'?
The $4$-dimensional lattice $\mathbb{Z}^{4}$ has vectors of length $\sqrt{n}$ for any positive integer $n$ by the Four Squares Theorem, but this need not be true for higher-dimensional integral, ...
- 3,645
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265
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polynomial maps from reducible plane curves to conics
It is classically known that every smooth plane quartic curve $C$ can be represented by an equation $q_1 q_3 = q_2^2,$ with $q_j\in\mathbb{C}[X,Y,Z]$, $1\leq j\leq 3$ quadratic forms, and the same is ...
- 14k
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538
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Least sum squares given constraints on subcomponents
Hi all,
I recently encounter a difficult problem.
I wish to minimize in $ \mathbf{x} $ the sum $\min \sum_{i=1..n} (\mathbf{x}^T \mathbf{A}_i \mathbf{x})^2$ given the constraints on the norms of ...
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Sum of two squares and implication of Bunyakovsky conjecture
Bunyakovsky's conjecture states that a polynomial with integer
coefficients takes infinitely many prime values at integers,
unless this is impossible for trivial reasons.
Let $a_1(x), a_2(x), a_3(x), ...
- 25.4k
5
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0
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284
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On $w^4+x^4+y^2+z^2$ over a number field
In 1921 Siegel confirmed a conjecture of Hilbert by proving that for any number field $K$ each element of
$$K_{\geq0}=\{a\in K:\ \sigma(a)\geq0\ \mbox{for all}\ \sigma\in\mathrm{Gal}(K/\mathbb Q)\}$$ ...
- 15.6k
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Can we write each positive integer as $x^2+y^2+\varphi(z^2)$?
As odd squares are congruent to $1$ modulo $8$, any integer of the form $4^k(8m+7)$ with $k,m\in\mathbb N=\{0,1,2,\ldots\}$ cannot be written as the sum of three squares.
To avoid such congruence ...
- 15.6k
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237
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Translation of Hilbert's paper on sums of squares
Does anyone know if there is a French or English translation of Hilbert's paper on sums of squares: Ueber die Darstellung definiter Formen als Summe von Formenquadraten?
- 151
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3
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Infinite sum of Laguerre polynomials: is $\sum_{k=0}^{\infty}\frac{n!}{(k+n)!}x^kL_n^k(x)^2=e^x+P(x)$, with $P$ a polynomial of degree $2n-1$?
I have the following expression:
$$
\sum_{k=0}^{\infty}\frac{n!}{(k+n)!}x^k(L_n^k(x))^2,
$$
where
$$
L_n^k(x)=\sum_{j=0}^n(-1)^j\binom{n+k}{n-j}\frac{x^j}{j!}
$$
is the usual associated Laguerre ...
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4
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Convergence of the sum of squares of averages of a sequence whose sum of squares is convergent
Can we find a sequence $u_n$ of positive real numbers such that
$\sum_{n=1}^\infty u_n^2$ is finite, yet $\sum_{n=1}^\infty ({u_1+u_2+...+u_n\over n})^2$ is infinite ?
After several attempts, I ...
- 18.7k
4
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1
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Efficient computation of $\sum_{i=1}^{\sqrt{n}} i^2\cdot\left\lfloor{\frac n{i^2}}\right\rfloor$
I need to compute efficiently the sum
$$
\sum_{i=1}^{\sqrt{n}} i^2\cdot\left\lfloor{\frac n{i^2}}\right\rfloor.
$$
We can do this in $O({\sqrt{n}})$ but I need a faster algorithm: for example, it ...
- 43
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votes
1
answer
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Exact formula for the number of integers in an interval which are the sum of two squares.
Denote by $\lambda(n)$, the number of numbers between $0$ and $n$ which are the sum of two squares. Landau, and Ramanujan have proven independently, that $$\lambda(n) \sim \frac{n}{\sqrt{\ln(n)}}$$
...
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Sum of squares of determinants of principal minors
I am interested in computing the sum of squares of determinants of principal minors. Let $A$ be an $n\times n$ positive semidefinite matrix and $A_S$ be a principal minor of $A$ indexed by the set $S ...
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Which integers can be expressed as $P(t)^2 + Q(t)^2 + R(t)^5$?
Inspired by this article and that one, I have two questions:
(1) Is the question of whether every integer can be expressed in the form $x^2 + y^2 + z^5$ ($x$, $y$, $z$ in $\mathbb{Z}$) an open problem?...
- 655
4
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1
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Specific quaternary quartic that is positive semi-definite but not sum of squares
Does there exist a quaternary quartic $f$ (a form in $\mathbb{R}[x_1,x_2,x_3,x_4]$ of degree $4$), which is positive semi-definite ($f \geq 0$ on $\mathbb{R}^4$) but not a sum of squares, such that ...
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what is this sum of squares of algebraic functions?
This question is inspired by the MO query here, although it has no direct implications.
Define the family of polynomial functions
$$f_n(x)=n^2x^{n-1}-\frac{d}{dx}\left(\frac{x^n-1}{x-1}\right),$$
and ...
- 43.1k
4
votes
1
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406
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Ensuring that a sum of squares is non-zero (in a finite field)
The following problem bears some similarity to the Additive Basis Conjecture [ALM91,JLPT92] saying (in characteristic $3$) that there is an absolute constant $N$ such that for any positive integer $m$,...
- 23k
4
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The power of chi-square test
Under the null hypothesis, if we have
$$\sqrt{n} \vec{x} \, \rightarrow_d \, N(0, I_p),$$
the test statistic can be construct as:
$$\hat{\Psi} = n \vec{x}^{\top} \vec{x} \, \rightarrow_d \,\chi^2_p.$$
...
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2
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Numerical coincidence?
(Nobody's answered this one on stackexchange after several days.)
My brother built a garage whose horizontal cross-section is a rectangle that measures $45$ feet by $30$ feet. To make sure the right ...
4
votes
1
answer
830
views
quartic diagonal as a sum of squares of quadratic forms
I would appreciate if someone can point out to the literature related to characterizing the set of all different ways to write real quartic diagonal $\sum \limits_{k=1}^n x_k^4, x \in \mathbb{R^n}$ as ...
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1
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Witt ring of a field with Pythagoras number $2$
I am currently looking at a few simple properties of the Witt ring of a field $K$ (by which I mean the ring of Witt classes of quadratic forms, not the ring of Witt vectors), which are clearly true ...
- 272
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0
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507
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Four-square Conjecture
Lagrange's four-square theorem states that every nonnegative integer
can be written as the sum of four squares. My following conjecture is much stronger than this classical theorem.
Four-square ...
- 15.6k
4
votes
0
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276
views
Product on representations of an integer by a quadratic form?
Define the quadratic form
$$Q(z_1,z_2,z_3,z_4) = 13 + \sum_{i=1}^4 (10+i)z_i +5 \sum_{1 \le i \le j \le 4} z_iz_j.$$
Then, $r_Q(n) := \left|\{(z_1,z_2,z_3,z_4) \in \mathbb{Z}^4 : Q(z_1,z_2,z_3,z_4) ...
3
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1
answer
547
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$x^2+7y^2=2^n$ and sums of four squares
Lagrange's four square theorem states that each $m\in\mathbb N=\{0,1,2,\ldots\}$ can be written as a sum of four squares.
Recently, I found that the diophantine equation $x^2+7y^2=2^n$ has certain ...
- 15.6k
3
votes
3
answers
958
views
solutions to equation mod a prime
I know that characterizing the solutions to an equation in a finite field is generally difficult, but I was wondering if anyone had anything to say about the equation
(ab)^2 + a^2 + b^2 = 0 mod p
I ...
- 39
3
votes
2
answers
2k
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Integer partition and sum of squares
Hello,
The question below might be well known, and using different words (I made these up, I'm not a number theorist or specialist in combinatorics)
For all integers $n\geq 2$ denote by $\mathcal{P}...
- 2,829
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2
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677
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Sum of two squares - Number of steps in Fermat descent
If a prime $p$ can be written as the sum of two squares, then one can construct this representation via Fermat descent if we know an $x$ such that $x^2 \equiv −1 \mod p$. Is there a possibility to say ...
- 1,101
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1
answer
389
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How are natural numbers that cannot be written as a sum of exactly four squares of naturals characterized? [closed]
This is most probably asked here already in some other form (as it seems to be a very basic question) but since I have't found some question very similar to this one I feel obliged to ask (also ...
user153451
3
votes
2
answers
659
views
Number of ways to write an integer as a sum of squares modulo $k$
Given a natural number $n$ and an element $k \in \mathbb{Z}_n$, how many solutions are there in $\mathbb{Z}_n$ to the equation $x^2+y^2 =k$? That is, I'm wondering whether there is a mod-$n$ version ...
- 143
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2
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386
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Sieve bound for the sum of two squares
Let $$S(n) = \sum_{p \le n} b(n-p),$$ where
$b(a)=1$ is $a$ is a sum of two squares of positive integers and $b(a)=0$ otherwise.
Trivially by PNT we have
$$S(n) \le \sum_{p \le n}1 \ll \frac{n}{\log n}...
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