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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83 views

$\sigma(x,y,z)=xy+yz+zx$ and its connection to sums of squares

I derived a theorem regarding cyclic sums of three variables. Define $\sigma:\mathbb R^3\to\mathbb R$ by $$\sigma(x,y,z)=xy+yz+zx.$$ Given $\alpha,\beta,\gamma\in\mathbb R$ such that $\alpha+\beta+\...
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119 views

Show that the sum of the squares of 3,4,5 and 6 consecutive numbers can not be a square

$\textbf{Edit:}$ Thank you so far for the answers. I still do not understand how to prove that the sum of 6 consecutive squares is not a square. I've tried ${6d^2+30d+55 \ne n^2 \Rightarrow 6(d^2 + 5d ...
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1answer
84 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 ...
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2answers
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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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1answer
2k views

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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3answers
88 views

Distinct positive integers $x_1,x_2,x_3,x_4$ that satisfy $4x_1^2=x_2^2+2x_3^2+x_4^2.$ [closed]

So I want to find a solution to the Diophantine equation $4x_1^2=x_2^2+2x_3^2+x_4^2,$ such that $x_1,x_2,x_3$ and $x_4$ are distinct positive integers which also satisfy the inequalities $3x_1^2-x_3^2-...
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27 views

Equality on sum of squares constraint

I've been working on an optimization problem that is almost a semi-definite program (SDP) barring a sum of squares constraint (expressed as under) which is making it a non-convex problem: \begin{...
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How to find the sum of squares of $1\ldots N-1$ that add to a squared number $N^2$?

So here's a question my friend recently gave me and ever since I've been trying to solve it without much success: There's a number $N$, and out of the set $U = \{1,2,3,\ldots,(N-1)\}$ we have to find ...
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38 views

For simple linear regression, show that $\mathbb{E}(\text{SSE})=(n-2)\sigma^2$

In the textbook, $$ S^2 = \frac{SSE}{n-2}$$ $$ E(S^2) = \frac{E(SSE)}{n-2} = \frac{(n-2) \sigma^2}{n-2} = \sigma^2$$ I don't understand the reason $\mathbb{E}(\text{SSE})=(n-2)\sigma^2$ My thought is ...
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Is there a way to simplify $ \sum_{i=0}^k \cos^2\left(\frac{\pi}{2}+\left\lfloor\frac{n}{b^i}\right\rfloor \frac{2 \pi}{b}\right) $?

Sorry for my poor English language, but I really need help. I would like to simplify the sum down in terms of: $n$, $k$, and $b$. $$ \sum_{i=0}^k \cos^2\left(\frac{\pi}{2}+\left\lfloor\frac{n}{b^i}\...
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Number of integer solutions of $x^2 + y^2 = k$

I'm looking for some help disproving an answer provided on a StackOverflow question I posted about computing the number of double square combinations for a given integer. The original question is ...
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2answers
49 views

Variance of the Square

Suppose $X_1, \cdots, X_n$ are a sample of independent variables taken from a normally distributed population with mean $\mu$ and variance $\sigma^2$. I would like to determine the variance of the ...
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How to work out sum of Square Number from given numbers?

Which of the following cannot be written as the sum of two distinct square numbers? A.106 B. 109 C. 112 D. 117 What would be the correct answer here and can someone explain in detail please.
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57 views

Is there another formula which generates Pythagoras' triples such that the largest $2$ of the triple differ by $3$?

I was thinking about Pythagoras' triples recently, and I wondered if I could find a formula that generated Pythagorean triples such that the largest of $2$ numbers of the triple differ by $1$, and I ...
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Sums of Squares algorithm

I want to express a large number as the sum of two squares, given that it is possible and given its prime factors. Let's say the number is $273097$. It's prime factors are $11^2, 37$ and $61$. Here ...
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1answer
957 views

Does the equation $a^2 + b^2 + c^2 = d^2$ have solutions in integers if $(a, b, c, d) > 0$?

I tried working out a solution to satisfy this equation and I got that this has no solution, however: $$1^2 + 2^2 + 2^2 = 3^2$$ so it does have a solution. I started off with the equation: $$a^2 + ...
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Proving a Sum Identity for Variance Estimator

I'm trying to prove the following identity: for a set of numbers: $$ x_1, ... ,x_n $$ $$\frac{1}{n(n-1)} \sum_{i=1}^n (x_i- \bar x)^2 = \bar x^2 - \frac{1}{n(n-1)} \sum_{i \neq j} x_ix_j $$ but I cant ...
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45 views

Upper bound to sum of squares using given sum

If $\sum_{i = 1}^n a_i = x$ for $a_i \geq 0$, then is it possible to find upper bound to $\sum_{i=1}^n a_i^2$? I know that the lower bound can be easily determined using Cauchy- Schwarz Inequality. ...
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103 views

Solve for integers $x, y$ and $z$: $x^2 + y^2 = z^3.$

Solve for integers $x, y$ and $z$: $x^2 + y^2 = z^3.$ I tried manipulating by adding and subtracting $2xy$ , but it didn't give me any other information, except the fact that $z^3 - 2xy$ and $z^3+2xy$...
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proving that an equation has infinitely many solutions

I have the equation $(4n+3)^2-48m^2=1$ that I changed into $X^2-48m^2=1$ so that it would be a Pell's equation, that has infinitely many solutions, that I found being $\left\{\begin{align} x_k = \...
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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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1answer
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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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How to show this series grows like $\sqrt{\log{x}}$?

Let $b_n$ denote the function such that $$b_n=\begin{cases}1 & n \text{ is the sum of two squares}\\0&\text{otherwise}\end{cases}$$ How do I show the sum $$\sum_{n \leq x} \frac{b_n}{n} = O(\...
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Can a square number be expressed as sum of squares of two other members.? [closed]

Is there any theorem to tell if square of a number can be expressed as sum of squares of two other distinct numbers. I have one such set. ${5, 4, 3}$ $5^2 = 4^2 + 3^2$ Given a number $n$ how to find ...
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192 views

A Conjecture on the Existence of Integer Solutions for $(f_1)^2 + (f_2)^2 +( f_3)^2… = I^2$

Using fractions($f_n$) where integer $n$ is the number of fractions, prove that the sum of the squares $f_1$ to $f_n$ has no integer($I$) solutions when $n >= 1$ given each fraction has a distinct ...
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108 views

For two odd primes $p<q$, can we deduce positive integers $a,b$ solving $a^2+b^4=pq$ without trial & error (brute force)?

Let us fix two primes $p,q$ with $2<p<q$. How can we find positive integers $a,b$ which solve the equation $a^2+b^4=pq$ without brute force? Interestingly there exist sometimes two solutions: $...
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201 views

Prove that there exists at least a form $g\left ( a, b, c \right )\pm f\left ( a, b, c \right )\left ( a- b \right )\left ( b- c \right )$ of $H$

With nonnegative cyclic polynomial $H$ as form $m\left ( a, b, c \right )\left ( c- 1 \right )\left ( a- 1 \right )+ n\left ( a, b, c \right )$ for $$m\left ( a, b, c \right ), n\left ( a, b, c \right ...
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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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1answer
39 views

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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185 views

$ 7\mid x \text{ and } 7\mid y \Longleftrightarrow 7\mid x^2+y^2 $

Show that $$ 7\mid x \text{ and } 7\mid y \Longleftrightarrow 7\mid x^2+y^2 $$ Indeed, First let's show $7\mid x \text{ and } 7\mid y \Longrightarrow 7\mid x^2+y^2 $ we've $7\mid x \...
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Solving multivariate quadratic equations over the integers

I am looking for a method (if it exists) to solve over the integers the following sum of squares equation: $$ x_1^2 + x_2^2+x_3^2 + \cdots + x_n^2 = m,$$ with $m \in \mathbb{N}.$ Someone has any idea ...
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Numbers which are not the sum of distinct squares

We are defining square factorization as representation a positive natural number as sum of squares of different positive, integer numbers. For example $5 = 1^2 +2^2$ and $5$ has no more ...
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Let a and b be natural numbers such that $2a - b, a - 2b$ and $a + b$ are all distinct squares. What is the smallest possible value of $b$?

Let a and b be natural numbers such that $2a - b, a - 2b$ and $a + b$ are all distinct squares. What is the smallest possible value of $b$? Let, $2a-b=k^2, a-2b=p^2, a+b=q^2$. $k^2=p^2+q^2$ after ...
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2answers
47 views

n unique sums of squares for 2p^(2n) for prime p (1 mod 4) and n natural number >= 1?

After looking at many sums of squares, this is something I am hypothesizing. I'd love to know if there is any work related to this, if it is easy to solve or even if there exist counterexamples. ...
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3answers
2k views

How do the normal equations *always* have a solution?

My professor says that "the normal equations always have a solution", even when $A$ is not full rank. HOwever, this does not make sense to me. The normal equations are $$A^\dagger=(A^TA)^{-1}A^T$$ ...
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4answers
81 views

Find a general method to find particular solutions where the sum of the squares of two consecutive integers is equal to the square of another integer

Question: If the sum of the squares of two consecutive integers is equal to the square of another integer, then find a general method to find particular solutions. E.g., $27304196^2+27304197^2=...
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express prime as sum of squares, $p = a^2 + b^2$

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 \...
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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 ...
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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 ...
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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....
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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 ...
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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 $\sum a_i^2 = \sum b_i^2$ ...
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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: $16=2^2+2^2+2^2+2^2$ $27=3^2+3^2+3^2$ Are these the only two such numbers to exist? There has to be an easy ...
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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 $a,b,c>0$?
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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}$...
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3answers
64 views

Sum of two squares theorem

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 ...

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