# 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
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### 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, ...
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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) ...
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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 ...
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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 ...
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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 ...
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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 ...
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### 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 ...
1answer
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{equation} \begin{array}{c}...
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### 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 ...
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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 ...
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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 ...
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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 ...
1answer
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### How to prove that there is No solution

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)^...
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58 views

### 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 ...