The sums-of-squares tag has no wiki summary.
5
votes
2answers
537 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
2answers
440 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
1answer
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
1answer
324 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
2answers
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
2answers
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
2answers
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
2answers
501 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
2answers
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
0answers
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, ...
5
votes
3answers
792 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 ...
0
votes
0answers
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 ...
1
vote
0answers
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
votes
0answers
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 ...
2
votes
2answers
475 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
0answers
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
1answer
256 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 ...
1
vote
0answers
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 ...
7
votes
1answer
508 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
2answers
220 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
1answer
716 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
1answer
410 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
1answer
538 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
1answer
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 ...
10
votes
4answers
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 ...
0
votes
1answer
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 ...
3
votes
1answer
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)}}$$
...
6
votes
0answers
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 ...
5
votes
1answer
759 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
1answer
476 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 ...
7
votes
1answer
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 ...
31
votes
2answers
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
votes
1answer
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 ...
7
votes
1answer
897 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 ...
7
votes
9answers
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?
...
6
votes
3answers
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 ...
4
votes
1answer
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 ...
1
vote
3answers
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 ...
14
votes
1answer
493 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 ...
