The sums-of-squares tag has no usage guidance.

**13**

votes

**1**answer

372 views

### Realization of numbers as a sum of three squares via right-angled tetrahedra

De Gua's theorem
is a $3$-dimensional analog of the Pythagorean theorem:
The square of the area of the diagonal face of a right-angled tetrahedron
is the sum of the squares of the areas of the other ...

**6**

votes

**0**answers

104 views

### 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*}
...

**11**

votes

**2**answers

361 views

### “Pythagoras number” for integral matrices

It is classically known that every positive integer is a sum of at most four squares of integers, i.e. every sum of squares of integers is a sum of four squares of integers. Now consider a symmetric ...

**11**

votes

**0**answers

317 views

### Sums of squares via semidefinite programming for the complex free group algebra

In the algebra of real noncommutative polynomials (the “free monoid algebra” over the real field) it is possible to reduce the question of whether an element is a sum of hermitian squares and ...

**1**

vote

**2**answers

174 views

### Sharply Estimating Pythagorean Triples [closed]

Given $m,n\in\Bbb N$ with $m<n$, how many pythagorean triples $p^2+r^2=q^2$ satisfy $$m\leq p<r\leq n?$$
Is there a way to give a sharp estimate?

**6**

votes

**1**answer

200 views

### Lattice points on the boundary of an ellipse

How many points of the integer lattice ${\mathbb Z}^2$ can an axis-parallel ellipse of radius $r$ contain on its boundary? (that is, we consider ${\mathbb Z}^2$ as lying in ${\mathbb R}^2$). ...

**3**

votes

**2**answers

248 views

### The number of solution of $x_1^2 + \cdots + x_k^2 \equiv \lambda \bmod q$

I'm playing with exponential sums...
If $q$ is an odd prime and $a$ an integer such that $q \nmid a$, then the following formula for the Gaussian sum is known
$$\sum_{x=0}^{q-1} e_q(ax^2) = ...

**10**

votes

**2**answers

582 views

### Many representations as a sum of three squares

Let $r_3(n) = \left|\{(a,b,c)\in {\mathbb Z}^3 :\, a^2+b^2+c^2=n \}\right|$. I am looking for the maximum asymptotic size of $r_3(n)$. That is, the maximum number of representations that a number can ...

**0**

votes

**0**answers

121 views

### Representation as sum of squares

Given $N\in\Bbb N$ such that $\prod_{i=1}^mp_i=N$ with $p_i$ being similar sized primes such that $p_i\neq p_j$ if $i\neq j$ where $m\in[1,\log\log N]$, consider ...

**6**

votes

**4**answers

257 views

### 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 ...

**2**

votes

**1**answer

147 views

### How many finite subsets in $\mathbb{Z}^d$ have a given sum of squares?

Let $|\cdot|$ denote the usual norm in $\mathbb{Z}^d$. Given a finite subset $S \subset \mathbb{Z}^d$, let $\varphi(S) = \sum_{z \in S}|z|^2$. Given $m \in \mathbb{N}$, what is the size of ...

**1**

vote

**1**answer

277 views

### 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**

vote

**0**answers

91 views

### The hypertriangular function of $n$

I'm looking for papers or recent results on the hypertriangular function of $n$:
$$H_t(n)= \displaystyle\sum\limits_{k=1}^{n} k^k$$
This is A001923 in the OEIS.
I don't have much experience with ...

**3**

votes

**0**answers

271 views

### Generating Function of distinct way of partitioned square sums of positive integers

Let's define a function $p_2(n)$ that it is total distinct way to write $n$ positive integer as sum of square of positive integers. For example, $9$ can be partitioned as sum of squares in 4 distinct ...

**3**

votes

**1**answer

428 views

### 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), ...

**6**

votes

**1**answer

143 views

### 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 ...

**0**

votes

**1**answer

189 views

### Longest sequence of sum of distinct squares [closed]

I want to find longest sequence of distinct squares that $\alpha_{_1}$ + ... + $\alpha_{_n}$ is given number.
In particular I want to find largest square in that sequence.
I've tried use Lagrange's ...

**6**

votes

**2**answers

294 views

### 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 ...

**24**

votes

**1**answer

509 views

### Distribution of $a^2+\alpha b^2$

It is well known that size of the set of positive integers up to $n$ that can be written as $a^2+b^2$ is asymptotic to $C \frac{n}{\sqrt{\log n}}$. Here I'm interested mostly in the weaker fact that ...

**2**

votes

**0**answers

180 views

### Sum of Squares and Harmonic Functions

Let $a_0, a_1, \dots a_k$ be non-negative reals. Any homogeneous polynomial $p$ of degree $2k$ in $\mathbb{R}^{d}$ can be decomposed as
$$
p(x)=\sum_{i=0}^{k}c_i|x|^{2(k-i)}p_{2i}(x)
$$
for some ...

**3**

votes

**1**answer

130 views

### Sum of Squares Length of a Product

Let $n \geq 2$. Let $g_1, \ldots , g_{n-1} \in \mathbb{R}[x_1,\ldots,x_n]$ such that $q=g_1^2+\ldots +g_{n-1}^2$ is not divisible by $p=x_1^2+\ldots +x_n^2$. Let $m \geq 1$ be the smallest integer ...

**6**

votes

**2**answers

645 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**

votes

**2**answers

536 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 ...

**6**

votes

**1**answer

543 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, ...

**6**

votes

**1**answer

418 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) ...

**1**

vote

**2**answers

227 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

272 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

449 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

777 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 ...

**11**

votes

**2**answers

772 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

200 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, ...

**6**

votes

**3**answers

1k views

### 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 ...

**0**

votes

**0**answers

102 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 ...

**1**

vote

**0**answers

87 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 ...

**6**

votes

**1**answer

380 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 ...

**3**

votes

**2**answers

684 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 ...

**4**

votes

**0**answers

248 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

297 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 ...

**9**

votes

**1**answer

670 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

597 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 ...

**13**

votes

**1**answer

1k 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 ...

**5**

votes

**1**answer

439 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, ...

**4**

votes

**1**answer

623 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 ...

**2**

votes

**1**answer

725 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 ...

**10**

votes

**4**answers

2k 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 ...

**0**

votes

**1**answer

520 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 ...

**3**

votes

**1**answer

926 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 $$\lambda(n) \sim \frac{n}{\sqrt{\ln(n)}}$$
...

**6**

votes

**0**answers

2k 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 ...

**5**

votes

**1**answer

1k 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 ...

**5**

votes

**1**answer

491 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 ...