# Questions tagged [sums-of-squares]

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

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### 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 ...
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### An integer is chosen at random. The probability that the sum of the digits of its square is 39, is [closed]

I need to find the probability that the integer I have chosen follows the rules. Its square's digit sum must add up to 39. I also observed that we can make infinite numbers. How to prove conclusively ...
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### How to compute SSR with just residuals and Xi?

How do we calculate SSR? I know SSE is the square of residuals all added together, but SSR is a subtraction between prediction for each observation and the population mean. Not sure how calculate SSR. ...
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### Prove that the square of a prime of form $4k+1$ is the sum of two squares [duplicate]

Suppose that $p$ is a prime of the form $4k + 1$. I'm trying to prove that there exist nonzero integers $a$ and $b$ such that $p^2 = a^2 + b^2$. I begin by noting that $p$ itself can be written as the ...
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### Prove there $\exists n$ such that $\left|\left\{\{x, y, z\}: x, y, z \in \mathbb{Z}^{+}, x<y<z, x^{2}+y^{2}+z^{2}=n\right\}\right| \geq 2021$

The Problem is that: Show that there is a positive integer $n$ such that $$\left|\left\{\{x, y, z\}: x, y, z \in \mathbb{Z}^{+}, x<y<z, x^{2}+y^{2}+z^{2}=n\right\}\right| \geq 2021$$ I tried ...
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### Clarification of sums of squares formula

I’m trying to find all the ordered sums of squares: $n=x^2+y^2$, $0\leq x\leq y$ for particular $n$. I’m confused about the sum of squares formula presented on Wolfram Alpha “ To find in how many ways ...
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### Can you multiply if all you know is squaring?

I am interested in this problem as a novelty; in the direction of constructing known things (which are usually taken to be somewhat fundamental) in terms of other, less-orthodox-to-call 'fundamental' ...
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### On the properties of the expansion into a sum of $4$ squares of the difference of squares of primes and composites

This question is about the properties of the expansion of $N= a^2-b^2 = (x_1^2+x_2^2+x_3^2+x_4^2)-(y_1^2+y_2^2+y_3^2+y_4^2)$ for primes and composite integers. The following examples will provide more ...
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### Prove subset sum of squares errors are less than sum of squares error of the original set

As part of a proof I'd like to show if $R_1 \cup R_2 \subseteq R_3$, then $SSE(R_1) + SSE(R_2) \leq SSE(R_3)$ where SSE is the sum of squared deviations from the mean of each group. I'm assuming the ...
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### Fermat's 2 Square-Like Results from Minkowski Lattice Proofs

Minkowski's Convex Body Theorem for lattices in the plane: Suppose $\mathfrak{L}$ is a lattice in $\mathbf{R}^2$ defined as $\mathfrak{L}=\{m\vec{v_1}+n\vec{v_2}:m,n\in\mathbf{Z}\}$, where $\vec{v_1}$ ...
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### Find all solutions: $x^2 + 2y^2 = z^2$

I'm use to finding the solutions of linear Diophantine equations, but what are you suppose to do when you have quadratic terms? For example consider the following problem: Find all solutions in ...
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Espress $2017$ as sum of two squares. attempt: by Fermat's Theorem on sums of squares, the prime $p = 2017$ is the sum of two squares $2017 = a^2 + b^2$ , $a,b \in \mathbb{Z}$, if and only if $p \... 1answer 111 views ### Finding squares that add to a certain sum I have been looking for an algebraic way to solve for 2 squares when given a sum, but I got nothing so far. For example: $$y^2 + x^2 = 9797$$ I thought that the solution would have to do something ... 1answer 45 views ### Expected value calculations I am reading Montgomery book for DOE. this is part of his proofs. I could figure out what he does until this last line, how did he get the second line from the first line. I know E(x) formulas for ... 0answers 18 views ### Decomposing Simpson's Diversity Index/Herfindahl Index When working with indices it can be instructive to decompose changes, holding variation components constant. I find this tough when working with indices such as Herfindahl or Simpson's Diversity Index.... 1answer 70 views ### Finding$a^2 + b^2 =n$. How do I find the possible sets of two numbers for any positive integer if the sum of squares of the two number is equal to that integer. I mean if$n=a^2+b^2(n,a,b \in N)$. How can I find the ... 3answers 77 views ### Can two distinct sets of N numbers between -1 and 1 have the same sum and sum of squares? Is is possible to find two different sets of numbers$\{ a_1, a_2, \dots, a_N\}$and$\{ b_1, b_2, \dots, b_N\}$with$a_i,b_i\in[-1,1]$such that$\sum a_i = \sum b_i$and$\sum a_i^2 = \sum b_i^2$... 3answers 1k views ### Numbers that are the sum of the squares of their prime factors A number which is equal to the sum of the squares of its prime factors with multiplicity:$16=2^2+2^2+2^2+2^227=3^2+3^2+3^2$Are these the only two such numbers to exist? There has to be an easy ... 1answer 61 views ### How to prove$2a^4+2b^4 \geq c^2(2a^2+2b^2-c^2)$for any positive$a$,$b$and$c$? [closed] How to prove$2a^4+2b^4 \geq c^2(2a^2+2b^2-c^2)$for$a,b,c>0$? 1answer 170 views ### When a number$N$is expressible as a sum of two squares in EXACTLY three ways Does there exist an$N$such that$$N=x_1^2+y_1^2=x_2^2+y_2^2=x_3^2+y_3^2\neq x_4^2+y_4^2\qquad \land \qquad \begin{cases}\gcd(x_1,y_1)&=1 \\\gcd(x_2,y_2)&=1\\\gcd(x_3,y_3)&=1 \end{cases}$...
I am working on the problem: I need to quickly check if positive number $n$ can be expressed as $n^2=a^2+b^2$. I found this theorem: https://en.wikipedia.org/wiki/Sum_of_two_squares_theorem But it ...