# Questions tagged [sums-of-squares]

For questions concerning various representation of integers as sums of squares, which are studied in number theory.

108 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
55 views

### Is this fraction a natural number only in case ${n_1}^2+{n_2}^2+{n_3}^2={m_1}^2+{m_2}^2+{m_3}^2$?

Suppose that $m_1,m_2,m_3,n_1,n_2,n_3 \in \mathbb N$ and $m_1<m_2<m_3$ and $n_1<n_2<n_3$. If $\dfrac {{n_1}^2+{n_2}^2+{n_3}^2-3}{{m_1}^2+{m_2}^2+{m_3}^2-3}$ is a natural number, ...
51 views

### Linear Regression: Fitting a cubic polynomial model in R and comparison with quadratic fit

Let $Y_i=\beta_0 +\beta_1 x_i+\beta_2 x_i^2+\beta_3 x_i^3+\epsilon^2$ I need to plot the model: $E(Y)=\gamma_0+\gamma_1(x)+\gamma_2(x^2-4)+\gamma_3(x^3-7x)$ (Orthogonal polynomials) ...
90 views

### How is distributed the sum of square of correlated z-score?

I have a $1\times k$ matrix representing $z$-scores and each element is correlated to each other according to a covariance matrix $\Sigma$. I would like to compute their sum of square and to know the ...
847 views

### Sum of Residuals equal to Zero

It is clear that running a regression, the sum of Residuals should equal Zero. I also understand that when running a weighted regression the sum of weighted Residuals should equal Zero. My model is ...
116 views

### Generalizing Landau-Ramanujan Theorem for sum of three squares

Let $S_{3}(x)$ denote the number of positive integers not exceeding x which can be expressed as a sum of three squares. Can we find an asymptotic formula for $S_{3}(x)$, maybe using Landau-Ramanujan ...
29 views

### From 4 squares to 2 squares

Let $r_k(n)$ denote the number of representations of $n$ as a sum of $k$ squares of integers. Suppose I know that $r_4(n) = 8\sum_{4 \nmid d \mid n} d$ (Jacobi's Theorem). Is there a way to deduce ...
83 views

476 views

### Optimization: Minimizing Quadratic Vector Valued Functions

I have a vector valued function $F: \mathbf{R}^{n} \rightarrow \mathbf{R}^{m}$, which consists of quadratic taylor approximations. So one could say that function F consists of stacked approximations ...
206 views

### SOS relaxations for polynomial optimization

I do not understand how SOS (Sum-Of-Squares) relaxation for polynomial optimization works in some cases. For instance, consider the polynomial optimization problem: \begin{array}{c}...
46 views

29 views

389 views

### A simple algorithm for representing a square of the form $4k+1$ as a sum of 3 squares

Let's start with an example, $N=n^2=7^2=49$. $49$ can also be produced by simply writing $49= (6+1)^2 = 36 + 2\cdot6 + 1 = 36 + 13$. So if we can find a way to represent 13 as a sum of two squares, we ...
59 views

### A method to study the factorization of very big numbers

Is there a method to construct very big numbers in a given interval, with rectangular distribution, by selecting prime factors randomly? I want to study factorizations of the kind $a=b\cdot c$, where ...
50 views

### Given A number N conver it to 2 square form

I have a number N i want to subtract square of x from N , where x is any integer ...
65 views

### Density and SOS polynomials

Is the set of sum of square (SOS) polynomials dense (in a sense to precise) in the set of non negative polynomials of degree less than $d$? I don't even know how to ask a well posed question... here ...
Let $d$ be a square-free positive integer $d>1$. Then there are no integers $x,y,z,t,a,b,c$ with $x\neq \pm z$, and $xt-yz\neq 0$ such that: \begin{cases} x^2+y^2=a^2 d \\ z^2+t^2=b^2 d \\ (x-z)^...
### How can I prove $Q(x,y,z)=x^{2}+y^{2}+z^{2}$ represents integer mod p?
p is an odd prime. Clearly Q represents all squares modulo p and all integers divisible by p, so the problem is really just the nonsquares modulo p. I started by letting $t$ be the smallest ...