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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3
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0answers
82 views
+50

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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1answer
23 views

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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1answer
43 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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2answers
42 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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0answers
14 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....
3
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3answers
70 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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1answer
57 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
35 views

Number theory problem with sets of elements - $x = (1 + 3 + \dots + m)p + (m + 2) + (m + 4) + \dots + n$

Given $p \in \mathbb{N}$, $p > 1$ and the set: $$A_p = \{x \ | \ (\exists) \ m ,n \in 2\mathbb{N} + 1, x = (1 + 3 + \dots + m)p + (m + 2) + (m + 4) + \dots + n\}$$ Prove that, if $x, y, \in A_p$, ...
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1answer
160 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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1answer
75 views

If n and m are sums of two squares so is $\frac{n}{m}$

So I found the following post regarding this problem, but I don't find the solution to be helpful at all, so I decided to ask the question again. Furthermore, my question is a bit different than the ...
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1answer
62 views

Prove that the prime factors $p$ of $n=x^2+y^2$ of the form $p=4m+3$ can only have even exponents. [duplicate]

I'm looking for an elementary proof of the statement in the title. Actually I have already proved that if $-\bar{1}$ is a square then $p\equiv 1 \pmod 4$ or $p=2$. I try to deduce from this statement ...
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40 views

p congruent to 1 mod 4 is a sum of square

I am studying the proof that if $p \equiv 1 \pmod 4$ then $p$ can be written as a sum of squares. That is one implication of Fermat's two square theorem. I have stumbled upon this text: https://people....
5
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1answer
77 views

How many zeros does $x^2 + y^2 \pmod d$ have on $[0, d-1]^2$?

I have been doing some work on Pythagoras's Theorem with my Year 8 Maths class (7th Grade in US speak). I had them investigating what values were unobtainable for the square on the hypotenuse of right-...
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3answers
56 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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infinte sums, inequalities, optimal rate.

Suppose we have a sequence $\{a_k\}_{k=1}^\infty$ such that $$ \sum_{k=1}^\infty k^{2s}a_k^2<1\,\,\,\text{for some s > 1 }. $$ I need to show for all $\gamma>1$ that $$ \frac{\sum^n_{l=1}l^{\...
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1answer
40 views

Expression of sum of squares as a sum of a specific form

This question is a follow-up of this one. Let $x,y \in \mathbb{Z}$, and suppose that $x^2+y^2 \ge 4$, and that $x,y$ are not both odd. Do there exist $a,b,c,d \in \mathbb{Z}$ such that $ (a+d)^2+(b-c)^...
5
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1answer
80 views

Can any of sum of squares be realized as a sum of this specific form?

Let $x,y \in \mathbb{Z}$, and suppose that $x^2+y^2 \ge 4$. Do there exist $a,b,c,d \in \mathbb{Z}$ such that $ (a+d)^2+(b-c)^2=x^2+y^2 $ and $ad-bc=1$? This question is motivated by an attempt to ...
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50 views

How to prove the value o $\zeta(2)$ by Functional Analysis? [duplicate]

I have a question about $\sum_{n=1}^{\infty} 1/n^2$ = $\pi^2/6$ I know it can be proven with standard 1 variable analysis (working on Taylor series of $\arcsin$ or something like that) or basic ...
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1answer
266 views

Elementary Inequality regarding Sum of Squares

\begin{align} &\mbox{If}\quad A = \sum_{i = 1}^{m}a_{i}\,x_{i}^{2} \quad\mbox{and}\quad B = \sum_{j = 1}^{n}b_{j}\,y_{j}^{2} \\[2mm] &\ \mbox{where}\quad x_{i}, y_{j}, a_{i}, b_{j} \geq 1, \...
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53 views

If a number can be expressed as sum of $2$ squares then every factor it can be expressed as sum of two squares

Lemma : If a number which can be written as a sum of two squares is divisible by a number which is not a sum of two squares, then the quotient has a factor which is not a sum of two squares. (This is ...
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1answer
266 views

An Explicit Solution to $a^2+b^2=p$

$\def \op {\operatorname*} \def \C#1#2 {\begin{pmatrix} #1\\#2 \end{pmatrix}}$ I got the following theory from the internet and I seek a proof: Let $p=4k+1 (k \in \Bbb{Z}^+)$ be a given prime. Assume ...
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1answer
53 views

Number of $3\times3$-matrices with integer coefficients satisfying $\operatorname{Tr}(A^{\top}A)=6$.

We are asked to find the number of $3$x$3$ matrices $A$ with all integer entries satisfying $\operatorname{Tr}(A^{\top}A)=6$. My approach Assuming $A=\begin{bmatrix} a_{11}&a_{12} & a_{13}\\ ...
8
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1answer
77 views

Is it possible to put all of the $\frac{1}{n}$ side squares into the unit square? [duplicate]

It's well known that $\displaystyle\underset{n\geq1}\sum\frac{1}{n^2}=\frac{\pi^2}{6},$ which is less than $2$, hence $\displaystyle\underset{n\geq2}\sum\frac{1}{n^2}<1$. However, this inequality ...
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1answer
21 views

Minimum value of RSS: Why does the the coefficient of $d^2$ being positive tell us that this value is a minimum?

I am currently studying the textbook Statistical Inference by Casella and Berger. Chapter 11.3.1 Least Squares: A Mathematical Solution says the following: For any line $y = c + dx$, the residual sum ...
4
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0answers
131 views

Proving $\sum_{\text{cyc}}\, (23a-5b-c)(a-b)^2(a+b-3c)^2 \geqslant 0$

Problem (KaiRain's problem). For $a,b,c\geqslant 0.$ Prove $$\displaystyle \sum_{\text{cyc}}\, (23a-5b-c)(a-b)^2(a+b-3c)^2 \geqslant 0$$ I only found a proof by $pqr.$ (Note that from pqr's proof we ...
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1answer
72 views

Non-Existence of Three-Dimensional Numbers

I have recently begun my study on Naive Lie Theory by John Stillwell. I am trying to understand his historical argument on the early warning signs that 3-dimensional numbers cannot exist. This is an ...
2
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4answers
99 views

Proving $\frac{a^3+b^3+c^3}{3}-abc\ge \frac{3}{4}\sqrt{(a-b)^2(b-c)^2(c-a)^2}$

For $a,b,c\ge 0$ Prove that $$\frac{a^3+b^3+c^3}{3}-abc\ge \frac{3}{4}\sqrt{(a-b)^2(b-c)^2(c-a)^2}$$ My Attempt WLOG $b=\text{mid} \{a,b,c\},$ $$\left(\dfrac{a^3+b^3+c^3}{3}-abc\right)^2-\dfrac{9}{16}(...
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4answers
126 views

Revisiting $(W^2 + X^2 + Y^2 + Z^2) = (A^2 + B^2)$

This query presents a narrow question that was broached by some of the responses to the following query: Can a sum of $n$ squares be expressed as the sum of $n/2$ squares? I'm unsure whether the ...
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3answers
92 views

Fermat's theorem on sums of two squares (every prime $p$ s.t. $p \not\equiv 3 \pmod 4$ is a sum of two squares)

I'm reflecting the following proof (see below). My question is where it uses the given fact ($p \not\equiv 3 \pmod 4$)? I'm not sure it uses this fact, and it kind of makes me think that something is ...
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4answers
233 views

Can a sum of $n$ squares be expressed as the sum of $n/2$ squares?

The answer for the special case when the squares are Pythagorean triple is yes. The Pythagorean triples are the case of the lowest $n$, namely $2$. Two Pythagorean triples can be combined to form a ...
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1answer
33 views

estimate for Möbius function of order k

Let $\mu_{k}(n)$ be the MÖBIUS function of order $k$, defined by $$ \mu_{k}(n)= \begin{cases} 1 &\text{if }\: n=1,\\ 0 &\text{if }\: p^{k+1}\mid n, \\ (-1)^r & \text{if }\: n = p^k_1· ...
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3answers
91 views

Proving $\displaystyle \sum_{cyc}\frac{(a^2+b^2)}{a+b}\leqslant \frac{3(a^2+b^2+c^2)}{a+b+c}$ [duplicate]

For $a,b,c>0$, prove $\displaystyle \sum_{cyc}\frac{(a^2+b^2)}{a+b}\leqslant \frac{3(a^2+b^2+c^2)}{a+b+c}$ I've simplified the inequality by multiplying both sides with $(a+b+c).$ So the inequality ...
3
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2answers
160 views

Prove $( xy+2x+2y+1 ) ^{2} ( x+y+2 ) ^{2}\geqslant \frac14 [xy(x+y)+2(x^2+y^2)-18xy+5(x+y)+2 ] ^{2}+144xy ( {x}^{2}+{y}^{2}+2) $

For $x,y \geqslant 0.$ Prove$:$ $$ \left( xy+2\,x+2\,y+1 \right) ^{2} \left( x+y+2 \right) ^{2}\geqslant \frac14 \left[xy(x+y)+2({x}^{2}+{y}^{2})-18\,xy+5(x+y)+2 \right] ^{2}+144\,xy \left( {x}^{2}+{y}...
2
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1answer
107 views

How many squares in a three-dimensional $n \times n \times n$ cartesian grid?

This brings the classical question to three dimensions. Given a three-dimensional Cartesian grid of $n \times n \times n$ points (that is $(n-1) \times (n-1) \times (n-1)$ unit cubes), how many ...
5
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2answers
139 views

If the largest positive integer is n such that $\sqrt{n - 100} + \sqrt{n + 100}$ is a rational no. , find the value of $\sqrt{n - 1}$ .

So here is the Problem :- If the largest positive integer is n such that $\sqrt{n - 100} + \sqrt{n + 100}$ is a rational no. , find the value of $\sqrt{n - 1}$ . What I tried :- I think that for $\...
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3answers
83 views

What is the proof that $(a+b)^2 >a^2 + b^2$? [closed]

I would like to know if there is a theorem that proves that $$(a+b)^2>a^2+ b^2$$ where $ab>0$ I am also wondering whether there is a name associated with this inequality.
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49 views

Is it an open problem? (Sums of squares) [duplicate]

Has it been proven that if n can be represented as a sum of squares of two rational numbers, then it can also be represented as a sum of squares of two integers? This is a relatively simple statement, ...
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1answer
33 views

Optimization problem that obviously should have solution leads to gradient that can't be $0$?

Let's say I have an image which is a matrix of pixel values. The intensity at pixel position $(x,y)$ is given by $g(x,y)$. Somewhere in the image is hidden a circle or an arc of a circle (pixel values ...
3
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4answers
75 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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0answers
32 views

Upon integers that sum up to perfect squares

I have some issues to find three positive integers that sum up to a perfect square... The complete problem wants you to find a triplet of integers $a,b,c>0$ such that $a+b$, $b+c$, $c+a$ are ...
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2answers
212 views

Solution to Lagrange's Four Square Theorem with Fewest Terms

Lagrange's four square theorem states that any natural number $n$ can be written as the sum of the square of 4 other integers. For most values of $n$, there are multiple square combinations that work. ...
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0answers
39 views

Lower bound for the square root of the sum of variables including squares

I already got a nice answer for a similar question (https://math.stackexchange.com/q/3734505) while now i am looking for a lower bound for the square root of the sum of variables including squares and ...
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2answers
90 views

Prove: $\sqrt{\frac{bc}{a(3b+a)}} + \sqrt{\frac{ac}{b(3c+b)}} + \sqrt{\frac{ab}{c(3a+c)}} \ge \frac{3}{2}$.

Prove: $\sqrt{\dfrac{bc}{a(3b+a)}} + \sqrt{\dfrac{ac}{b(3c+b)}} + \sqrt{\dfrac{ab}{c(3a+c)}} \ge \dfrac{3}{2}$ with $a, b, c$ are positive real numbers. Let $a \le b \le c$: \begin{align*} \sqrt{\...
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1answer
80 views

Prove $\frac{a^2}{(a+b)^2} \geqslant \frac{4a^2-b^2-bc+7ca}{4(a+b+c)^2}$

Let $a,\,b,\,c$ are positive numbers. Prove that $$\frac{a^2}{(a+b)^2} \geqslant \frac{4a^2-b^2-bc+7ca}{4(a+b+c)^2}. \quad (*)$$ Note. My proof is use sos. Form $(*)$ we get know problem $$\frac{a^2}{(...
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2answers
50 views

Solution for $\beta$ in ridge regression

The RSS of the ridge regression in matrix form is: $$RSS(\lambda) = (y−X\beta)^T(y−X\beta) +λ\beta^T\beta$$ the ridge regression solutions are easily seen to be $$β_{ridge}= (X^TX+λI)^{−1}X^Ty$$ See ...
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1answer
25 views

Help eliminating a term inside a square root

I have the somewhat ugly expression: $((V_iV_j\lambda_2 + X_iX_j\lambda_1)^2 + \lambda_1\lambda_2(V_iX_j - X_iV_j)^2)^{\frac{1}{2}}$ Every single term here is a scalar over the reals. The goal is to ...
8
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2answers
509 views

Conjecture: All but 21 non-square integers are the sum of a square and a prime

Update on 6/19/2020. This discussion led to deeper and deeper results on the topic. The last findings are described in my new post (including my two answers), here. I came up with the following ...
5
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1answer
110 views

Proving ${ \left\{\sum \left( ab+{b}^{2}+{c}^{2}+ac \right)\right\} }^{4}\geq 27\,{ \sum} ( ab+{b}^{2}+{c}^{2}+ac ) ^{3} ( c+a) ( a+b) $

For $a,b,c>0.$ Prove$:$ $$ \left\{ \sum\limits_{cyc} \left( ab+{b}^{2}+{c}^{2}+ac \right) \right\}^{4}\geq 27\,{ \sum\limits_{cyc}} \left( ab+{b}^{2}+{c}^{2}+ac \right) ^{3} \left( c+a \right) \...
3
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2answers
119 views

Proving $\frac{a(b+c)}{a^2+bc}+\frac{b(a+c)}{b^2+ac}+\frac{c(b+a)}{c^2+ba}\geqq 1+\frac{16abc}{(a+b)(b+c)(c+a)} $

For $a,b,c \in (0,\infty).$ Prove$:$ $$\frac{a(b+c)}{a^2+bc}+\frac{b(a+c)}{b^2+ac}+\frac{c(b+a)}{c^2+ba}\geqq 1+\frac{16abc}{(a+b)(b+c)(c+a)} $$ My proof by SOS$:$ $$ \left( {a}^{2}+bc \right) \left(...
2
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0answers
99 views

On sums of two squares

The original question was: In my recent article (see section 2.3 here) I claim that the number of solutions to $x^2 + y^2 \leq k$ with unknown $(x, y)$ being integers (positive or negative) is ...

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