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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4
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2answers
136 views

Minimum over Probability Measures

Let $f$ and $g$ be polynomials in $\mathbf x \in \mathbb R^n$. Let $X$ be a compact subset of $\mathbb R^n$. Finally, Let $\mathcal M(X)$ be the set of probability measures over $X$. Can the ...
1
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1answer
38 views

Quadratic fit using sum of least squares without matrices

I've been hunting around for examples using the sum of least squares + partial derivative method to fit a polynomial to a set of points but am completely stuck. All the examples I've found involve ...
4
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3answers
83 views

$\forall n \in \mathbb{N}_{>5}\implies\exists (a,b)\in \mathbb Z^+:a^2+b^2\notin\mathbb P\;\wedge\;n=a+b$

Conjecture: $\forall n \in \mathbb{N}_{>5} \implies\exists (a,b)\in \mathbb Z^+:a^2+b^2\notin\mathbb P\;\wedge\;n=a+b$ Tested $\forall n\leq 100,000$. Small exceptions: {1,2,3,5}. I would like to ...
1
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1answer
78 views

Solving a Quadratic Ternary Form with Large Coefficients

I encountered this expression $$1215696x^2+566544y^2-103776z^2=0$$ which I've understood is called a "quadratic ternary form" and have been trying to find solutions $x,y,z\in\mathbb{Z}$ One can ...
16
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4answers
1k views

Value of $(\alpha^2+1)(\beta^2+1)(\gamma^2+1)(\delta^2+1)$ if $z^4-2z^3+z^2+z-7=0$ for $z=\alpha$, $\beta$, $\gamma$, $\delta$

Let $\alpha$, $\beta$, $\gamma$, $\delta$ be the roots of $$z^4-2z^3+z^2+z-7=0$$ then find value of $$(\alpha^2+1)(\beta^2+1)(\gamma^2+1)(\delta^2+1)$$ Are Vieta's formulas appropriate?
3
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3answers
128 views

How to prove that $441 \mid a^2 + b^2$

How to prove that $441 \mid a^2 + b^2$ if it is known that $21 \mid a^2 + b^2$. I've tried to present $441$ as $21 \cdot 21$, but it is not sufficient.
2
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0answers
56 views

Arithmetic progression of four square number with the same common difference [duplicate]

I want to know, if is possible found the arithmetic progression of four square number, with the same common difference. \begin{align} \ & a^2 +r = b^2 \\ & b^2 +r = c^2 \\ & c^2 +r =...
5
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3answers
273 views

Solve $32x^2 -y^2 = 448$

I am trying to find all integer solutions to the following equation: $$32x^2 - y^2 = 448$$ This is what I have tried so far: The equation describes a hyperbola, and so I try the usual trick of ...
12
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3answers
1k views

Can we prove a stronger claim?

Here it is asked whether for every $k>1$, there is a prime $p$ and $w_1,w_2,\cdots ,w_k>1$, such that $p$ divides $w_1^2+w_2^2+\cdots +w_k^2$, but none of the summands. Can we prove there is ...
5
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2answers
143 views

Primes dividing sums of squares but not dividing any of summands

I am inexperienced when it comes to number theory (even elementary one) so do not have an idea at this moment on how to solve this one. Let us go through some examples. For $k=2$ we can have prime ...
3
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0answers
67 views

Proof algorithm to find representations as sum of two squares

I saw in a book the following algorithm to find, given a prime $p\equiv 1 \pmod 4$, integers $a $ and $b $ such that $p=a^2+b^2$. Step 1 : Find an integer $0 <m <p$ such that $m^2\equiv -1\...
0
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1answer
160 views

Difference of squares - number of representations

There exists a well-known result concerning a number of representations of $n$ as a sum of two squares. Is there anything similar for a number of representations of $n$ as a difference of two squares? ...
1
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0answers
88 views

The sum of the squares of $5$ consecutive primes is again prime. True for infinite many quintuples?

Let $(p,q,r,s,t)$ be a quintuple of consecutive primes. Are there infinite many such tuples such that $$p^2+q^2+r^2+s^2+t^2$$ is prime again ? Motivation : $p^2$ is never prime $p^2+q^2$ is ...
0
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1answer
52 views

A simple explanation to this asymmetry?

Let $S_k^\delta=\{a+b\le k\mid(a,b)\in\mathbb N_+^2\wedge a^2+b^2\in\mathbb P\wedge a^2+b^2+\delta\in\mathbb P\}$, where $\mathbb P$ is the set of primes. If $\delta=2$ then the condition on $a^2+b^...
1
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5answers
65 views

$\sum_{n=1}^{2017}\left(\left((n+2)^4\bmod{(n+1)^4}\right)\bmod{4}\right)$

What's $$\sum_{n=1}^{2017}\left(\left((n+2)^4\bmod{(n+1)^4}\right)\bmod{4}\right)$$ What have I tried? $$(n+2)^4=n^4+8n^3+24n^2+32n+16$$ $$(n+1)^4=n^4+4n^3+6n^2+4n+1$$ Remainder: $$4n^3+18n^...
0
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1answer
40 views

Explain the following derivation

Can you explain the derivation in the given image? Which steps lead to the incorrect conclusion?
0
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1answer
43 views

Bivariate Form Sum of Squares

Let $F \in \mathbb{R}[x,y]$ be nonnegative and homogeneous of degree $2n$. Then it can be written as a sum of two squares.
1
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1answer
257 views

Sum of four squares

I was looking for numbers who can be expressed as sum of exactly four squares and not less. And I think I have found them. They are all the integers of the form $$4^{n}\,(7+8k);\;k,\,n\in\mathbb{N}...
0
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0answers
110 views

Reduce $(a+b)^2(c+d)^2-16abcd$ to sum of squares

Is it possible to reduce $$(a+b)^2(c+d)^2-16abcd$$ to sum of squares? This expression is used in proof of AM-GM inequality. It is known that $$\dfrac{a+b}{2}\ge\sqrt{ab}\tag{1}$$ So, it can be proved ...
7
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2answers
100 views

$\{a+b|a,b\in\mathbb N^+\wedge ma^2+nb^2\in\mathbb P\}=\{k>1|\gcd(k,m+n)=1\}$

Conjecture: $\{a+b|a,b\in\mathbb N^+\wedge ma^2+nb^2\in\mathbb P^{>2}\}=\{k>2|\gcd(k,m+n)=1\}$ if $m,n\in \mathbb N^+$ and $\gcd(m,n)=1$. This is a generalization of Any odd number is of form $...
4
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0answers
132 views

Even numbers has the form $a+b$ where $\frac{a^2+b^2}{2}$ is prime

Conjecture: All even numbers $n>2$ can be written $n=a+b$ where $a,b\in\mathbb N^+$ and $\frac{a^2+b^2}{2}\in\mathbb P$. Is tested for $n<10,000,000$. This conjecture is related to and maybe ...
2
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4answers
201 views

Find all solutions the diophantine equation $x^2+y^2=z^2+w^2+1$

Let $x,y,z,w$ be postive integers,find the diophantine equation all solution $$x^2+y^2=z^2+w^2+1$$ However, I'm looking for a identity or something like that. Very important is that must be a ...
8
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2answers
219 views

Most even numbers is a sum $a+b+c+d$ where $a^2+b^2+c^2=d^2$

I have no idea how hard this conjecture is to prove: Any even number $n\ge 36$ can be written as $n=a+b+c+d$ where $a^2+b^2+c^2=d^2$ and $a,b,c,d\in\mathbb Z^+$. Small exceptions are $n=2, 4, 6, 10, ...
2
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2answers
69 views

Any integer is of the form $n=a_1^2+a_2^2+a_3^2-a_4^2-a_5^2$

I've struggled with this conjecture, which probably can be proved: Any natural number $n$ can be written as $n=a_1^2+a_2^2+a_3^2-a_4^2-a_5^2$ for some $a_1,a_2,a_3,a_4,a_5\in\mathbb Z^+$. I guess ...
2
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0answers
85 views

Hard for-all-integer-problems proveable for very great integer limits

Computational experiments suggests the conjecture: All big enough odd numbers $N$ is of the form $N=\sum_{k=1}^n m_k$, where $\sum_{k=1}^n m_k^2$ is prime and all $m_k\in\mathbb Z^+$. Since the ...
7
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0answers
351 views

Yet an other conjecture about odd numbers $n=a+b$ such that $a^2+b^2$ is prime

This question is related to A conjecture about an unlimited path and Any odd number is of form $a+b$ where $a^2+b^2$ is prime but I present it on its own if anyone would like to help finding ...
1
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1answer
194 views

ANOVA: What are the values SSW, SSB and the test statistics?

In a test of the ability of a certain polymer to remove toxic wastes from water, experiments where conducted at three different temperatures. The data below give the percentages of impurities that ...
0
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0answers
64 views

Why $a^4 + b^4$ ($a, b$ are positive integers) is not a perfect square?

I've realized that there are maybe no positive integers $(a, b)$ that $a^4 + b^4$ is a perfect square, because I tested for $a, b \le 10000$ and cannot find any solution. I think that's weird, and ...
47
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3answers
1k views

Any odd number is of form $a+b$ where $a^2+b^2$ is prime

This conjecture is tested for all odd natural numbers less than $10^8$: If $n>1$ is an odd natural number, then there are natural numbers $a,b$ such that $n=a+b$ and $a^2+b^2\in\mathbb P$. $\...
1
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0answers
60 views

Formula or way for the number expressed as sum of two squares in two different ways? [duplicate]

Number which can be expressed as sum of two squares in two different ways - e.g - a). $85 = {2^2} + {9^2} = {6^2} + {7^2}$ b). $125 = {2^2} + {11^2} = {5^2} + {10^2}$ I do not need more example. I ...
2
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1answer
171 views

Does the interval $[x, x + 10 x^{1/4})$ contain a sum of two squares?

Is it true that there exist an integer $N \in \mathbb N$ , such that $\forall x > N$ , the interval $[x, x+10x^{1/4} )$ always contains a sum of two squares ? I know that $n \in \mathbb N$ is a ...
1
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2answers
210 views

Are there 4 consecutive numbers that are each the sum of 2 squares?

Some numbers can be expressed as the sum of two squares (ex. $10=3^2+1^2$) such as: $$0,1,2,4,5,8,9,10,13,16,17,18,20,25,...$$ Other numbers are not the sum of any two squares of integers: $$3,6,7,11,...
0
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1answer
61 views

Which is the correct divisor when calculating an average of summed up squares

If I have the following: $$(1 * 1) + (2 * 2) + (3 * 3) + (4 * 4) + (5 * 5) + (6 * 6) = 91$$ Is the average $\frac{91}{6} = 15.16$ because I have $6$ distinct numbers that are being squared and then ...
3
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2answers
224 views

If N is a sum of two squares, then so is the odd part of N

Given two natural numbers $m,n$, where $m$ is odd. Please prove that if $m\cdot2^n$ is a sum of two squares, then $m$ is a sum of two squares. Tested for all $m\cdot2^n<100,000$. I love the ...
3
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1answer
143 views

Expressing the hypergeometric function as an infinite sum of squares

The hypergeometric function evaluated on paramaters of a specific form, can be expressed as an infinite sum of squares. I'm wondering if this gives it any interesting closed form expressions. So in ...
9
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1answer
136 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 \...
0
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1answer
354 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 + ...
0
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3answers
76 views

looking for some shortcut or faster method for solving a question involving square root

If $x =\dfrac{ \sqrt 3 - \sqrt 2 }{ \sqrt 3 + \sqrt 2 } $ and $ y = \dfrac{\sqrt 3 + \sqrt 2 }{ \sqrt 3 - \sqrt 2}$, find the value of $\dfrac{x^2 + xy + y^2}{x^2 - xy + y^2}.$ I obviously know the ...
4
votes
1answer
74 views

Is there a theory of “sums-of-squares residues”?

The theory of quadratic residues is long- and well-studied. Recall that, [somewhat simplified] if $x,a,b$ are integers, with $0 \le a < b$, such that $$x \equiv a^2\!\!\!\pmod{b},$$ then we say ...
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0answers
58 views

Finding the pdf of sum of squared weighted gaussian variables

I have 3 sets of weighted gaussians that are part of 3 different Gaussian Mixture Models, phi1,phi2 and phi3. phi1 has n1 gaussian components with weights wj, mean mu_j and variance eta_j_squared, j ...
3
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2answers
214 views

Express $355$ as a sum of three squares.

PROBLEM Find all the ways to express $355$ as a sum of three squares. MY ATTEMPT By Legendre's three-square theorem, since $355$ is not of the form $n = {4^a}(8b+7)$ (for $a, b \in \mathbb{Z}$), ...
0
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2answers
85 views

Can two the sum of two primitive perfect squares be equal? $a^2+b^2 = c^2 + d^2$?

Given a primitive perfect square, $n^2=a^2+b^2$ where $gcd(a, b) = 1$ and $m^2=c^2+d^2$ where $gcd(c,d)=1$ Can $n=m$? a, b, c, d, n, and m are positive integers. And I forgot to mention: EDIT 1: Two ...
2
votes
3answers
2k views

Supposed $a,b \in \mathbb{Z}$. If $ab$ is odd, then $a^{2} + b^{2}$ is even.

Supposed $a,b \in \mathbb{Z}$. If $ab$ is odd, then $a^{2} + b^{2}$ is even. I'm stuck on the best way to get this started. My thinking is that I could use cases. i.e. Case 1: a is even and b is ...
4
votes
3answers
79 views

What is the maximum number of distinct positive integer's square that sums up to $2002$?

What is the maximum number of distinct positive integer's square that sums up to $2002$ ? My tries: $$\frac{n(n+1)(2n+1)}{6} = 2002$$ $$\implies n\approx 17 $$ but am clueless as to how to proceed ...
2
votes
1answer
341 views

Can 2017 be written uniquely as a sum of 2 squares? [duplicate]

I know that $2017 = 9^2+44^2$, but how can I prove that this is the only possible way you can write 2017 as a sum of 2 squares?
1
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2answers
147 views

Which numbers less than 5 billion have the most representations as the sums of two squares?

1842675848 has four representations as the sum of two squares: 1842675848 = 3518^2 + 42782^2 = 5458^2 + 42578^2 = 11338^2 + 41402^2 = 19702^2 + 38138^2. It got me to wonder which numbers in this ...
0
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1answer
272 views

Are the 4 square representations of a sum of 4 squares of an integer equivalent?

Every integer $N$ can be represented by a sum of 4 squares $N=a^2+b^2+c^2+d^2$. We usually have more than one representation for a given integer N. For example $7*13=91$ has the following ...
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0answers
105 views

Proof verification: Alternative proof of Lagrange's four square theorem.

We assume Fermat's two-square theorem to begin with. Note that, in our proof, the terms numbers and natural numbers are everywhere taken to mean nonnegative integers. Lemma 1: The product of two ...
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1answer
62 views

How can i find number of ways presenting number as sum of no more than 4 squares

I am given a number and I have to find number of ways to present that number as sum of no more than 4 squares.For example $25$ can be presented as $1^2+2^2+2^2+4^2$, $3^2+4^2$ and $5^2$.
1
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1answer
61 views

help: sum of squares summation simplification

this is my first question, so please don't be too harsh on me. In order to simplify a standard deviation i have managed to isolate some terms. I'm however unsure of how i can rewrite or simply ...