The sums-of-squares tag has no wiki summary.

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536 views

### Would such polynomial identity exist? (related to sum of four squares)

Let $f_1,f_2,f_3,f_4,f_5 \in \mathbb{Q}[x]$ be linear and
coprime and not all constant.
Is it possible $ f_1^2+f_2^2+f_3^2+f_4^2=f_5^2$?
I suppose the answer is negative.
If this is possible, ...

**6**

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**2**answers

439 views

### 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 ...

**7**

votes

**1**answer

377 views

### 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, ...

**5**

votes

**1**answer

323 views

### 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) ...

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vote

**2**answers

214 views

### Impossible Range for Minkowski-Like Sum of Squares

Given coprime positive integers M,N, and a corresponding integer z outside of the range (for all integers x,a,b,c) of $Mx^2-N(a^2+b^2+c^2)$, is there any such z which is "deceptive", meaning that it ...

**2**

votes

**2**answers

246 views

### Four-Square Theorem for Negative Coefficient

What integers are not in the range of $a^2+b^2+c^2-x^2$ (for all integer combinations of a, b, c, and x)? This form is similar to that of Lagrange's Four-Square Theorem, for which the answer would be ...

**4**

votes

**2**answers

412 views

### 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 ...

**3**

votes

**2**answers

500 views

### The four squares theorem from the Gauss-Legendre three squares theorem

Hi,
I've been studying some proofs of The four squares theorem. Some of them are pretty clear. However, I came across a statement that The four squares theorem can be easily derived from ...

**12**

votes

**2**answers

659 views

### Sums of Squares

Every prime $p = 4k + 1$ can be uniquely expressed as sum of two squares, but for which
integers $x$ is $x^2 + y^2 =$ some prime $p$? Stated differently, does the square of
every positive integer ...

**0**

votes

**0**answers

181 views

### Confirm/refute $f(x)$ where $f(x) = $x-th Mersenne prime ($M_p$) where $x$ is [1,2,3,4,5,6,7…]

While tinkering with numbers, I found $n=f(x)$ where $n = \sigma(\sigma(n)-n)$ and $x \neq p$ where $p$ is a Mersenne prime exponent, but I need help input in regard to improving accuracy.
First off, ...

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votes

**3**answers

789 views

### Octonions and the dance of the seven veils

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 ...

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votes

**0**answers

98 views

### maximum of the sum of polynomials

Hi
I have $n$ polynomials, each one is positive over a certain range and the maximum value each can attain is 1. Also each polynomial has atmost one peak(maximum). Is there some way that I can show ...

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vote

**0**answers

82 views

### Information about mutant Leech lattice related to smallest perfect squared square

What happens if we follow the construction of the Leech lattice but replace the relation
$\displaystyle \sum_{n=1}^{24} n^2 = 70^2$
with the smallest perfect squared square? Explicitly, if we set ...

**3**

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**0**answers

141 views

### 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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**2**answers

472 views

### 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 ...

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**0**answers

236 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) ...

**2**

votes

**1**answer

255 views

### Question on Sums of Squares

Is it possible for two different $n$-element sets, each of which consists of $n$ unique positive integers (they can appear in both sets, though) to have the same sum when the squares of their elements ...

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**0**answers

154 views

### Finding a curve of some approximate arc length (with uniform or zero curvature) with a specified distance to a set of points in 3-space

Imagine I define a set of $N$ points in 3-space, $P$, and I would like to define a straight-line or curve, $C$, with uniform or zero curvature, that has some desired distance, $M$, to each of these ...

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**1**answer

506 views

### Representations by positive definite binary quadratic forms

It's known that the number of representations of an integer $k$ by sum of two squares is
$$
4\;\sum_{d|k}\left(\frac{-4}{d}\right)
$$
or
$$
4\sum_{d|k,\; d \textrm{ odd}} (-1)^{\frac{d-1}{2}}= ...

**0**

votes

**2**answers

218 views

### Algorithm for calculating the sum-of-squares distance of a rolling window from a given line function

Given a line function $y = ax + b$, it is easy to calculate the sum-of-squares distance between the line and a window of samples $(1, y_1), (2, y_2), ..., (n, y_n)$ (where $y_1$ is the oldest sample ...

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**1**answer

713 views

### Legendre and sums of three squares

The Three-Squares-Theorem was proved by Gauss in his Disquisitiones, and this proof was studied carefully by various number theorists. Three years before Gauss, Legendre claimed
to have given a proof ...

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votes

**1**answer

409 views

### 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, ...

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votes

**1**answer

537 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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votes

**1**answer

692 views

### On a claim of Ramanujan in his “Lost Notebook”.

As I was flipping through the scanned version of Ramanujan's "Lost Notebook" in our library, I came across a result which caught my attention. And as any excited teenager would do, I immediately ...

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**4**answers

1k views

### Sum of squares modulo a prime

What is the probability that the sum of squares of n randomly chosen numbers from $Z_p$ is a quadratic residue mod p?
That is, let $a_1$,..$a_n$ be chosen at random. Then how often is $\Sigma_i ...

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votes

**1**answer

461 views

### The Hilbert-Waring theorem using the sum-of-squares function.

I've asked the same question at the Math Stack Exchange site, but I didn't have any luck there. So I'm posting the same question here.
Denote by $r_{s,k}(x)$, the number of ways in which $x$ can be ...

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**1**answer

684 views

### 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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1k views

### 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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**1**answer

757 views

### 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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475 views

### 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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**1**answer

318 views

### 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 ...

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2k views

### Euler and the Four-Squares Theorem

There are several questions in the Euler-Goldbach correspondence that
I am unable to answer. Sometimes it does not take very much: in his
letter to Goldbach dated June 9th, 1750, Euler conjectured
...

**5**

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**1**answer

263 views

### 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 ...

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**1**answer

896 views

### Sums of two squares in (certain) integral domains

While giving the first of eight lectures on introductory model theory and its applications yesterday, I stated Hilbert's 17th problem (or rather, Artin's Theorem): if $f \in ...

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3k views

### Ways to prove an inequality

It seems that there are three basic ways to prove an inequality eg $x>0$.
Show that x is a sum of squares.
Use an entropy argument. (Entropy always increases)
Convexity.
Are there other means?
...

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**3**answers

8k views

### Enumerating ways to decompose an integer into the sum of two squares

The well known "Sum of Squares Function" tells you the number of ways you can represent an integer as the sum of two squares. See the link for details, but it is based on counting the factors of the ...

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**1**answer

383 views

### 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 ...

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vote

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837 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 ...

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492 views

### What's the probability that k + n^2 is squarefree, for fixed k?

While playing around with this question (when is the sum of two squares squarefree?), from some experimental computations (and bolstered by the fact that the density of squarefree positive integers is ...