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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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616 views

Diophantine equation $a^2+b^2+c^2=a^2b^2$

I am trying to find all non trivial integers for which $a^2+b^2+c^2=a^2b^2$. As suggested I have tried working (mod 4). This is what I've gotten so far: Squares can have a remainder of 0 or 1 (mod 4)....
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60 views

How many numbers up to $x$ are sums of two squares?

I know that the primes which are representable as a sum of two squares are a specific type of prime, that is, $p=4k+1$, where $k$ is a positive integer. and from this, I could deduce which integers ...
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1answer
198 views

Matrix notation for weighted sum of squares

While going through page 1 of Lecture 24: Weighted and Generalized Least Squares [PDF], I got the following questions. Weighted sum of squares is defined as below: $$ \sum_{i = 0}^{n}{w_i(Y_i - X_ib)...
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1answer
87 views

When is $n^2-1$ a sum of two squares?

I am trying to work out when $n^2-1$ is a sum of two squares. Is there a formula for such $n$? I have found $n=1$, $n=3$ and $n=9$ so far but am struggling to find a pattern that will generalise. If ...
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156 views

Which primes $\mathfrak{p}$ can be written as sums of two squares in $\mathbb{Q}(\sqrt[3]{2})$?

which primes $\mathfrak{p}$ can be written as sums of two squares in $\mathbb{Q}(\sqrt[3]{2})$? I am asking to solve an equation: $$ \mathfrak{p} = \big(a_1+a_2\sqrt[3]{2}+a_3\sqrt[3]{4}\big)^2 + \...
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1answer
54 views

A Peculiar Sum Of Squares [duplicate]

I have been trying to solve a question that arose in my mind a couple of days ago. What is the sum of: $1+4+9+16+25+\cdots +n^2$? It is the sum of squares of each numbers starting from $1$ to $n$....
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1answer
49 views

New Figurate Number Relation? [closed]

Has anyone seen the relation $$ nP_{2}(n)=P_{3}(n-1)+\sum_{i=1}^ni^2 $$ where $P_2(n)$ is the $n$th triangular number and $P_3(n)$ is the $n$th tetrahedral number? I know the straightforward algebra ...
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0answers
40 views

Sum of squares for truncated normal distribution

I need to find a distribution of sum of squares of N variables, when N-1 variable has a normal distribution and one variable has a truncated normal distribution. Should be similar to chi-square ...
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0answers
38 views

why sum of squares equals to assign the variable evenly in linear programming

I have a quesiton about linear programming. My objective is trying to make the utilization evenly. Example $U_{1}, U_{2},...,U_{k} $ $\sum_{i=1..k} U_{i} = C $ $C$ is some constant. $U_{i}$ is ...
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2answers
131 views

Product of $(4k-1)$ primes can't be sum of 2 squares

I am trying to prove, Product of primes of the form $(4k-1)$ can't be sum of 2 squares. My approach is- Let the product is $M=m_1m_2...m_n$ where $ m_1, m_2, ...m_n$ are primes. Assume, $M$ can ...
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2answers
111 views

Integer as sum of 6 squares.

Can every integer be written as sum of exactly 6 squares? I am also curious to know when an integer can be written as sum of exactly 8 squares. I know the problem is related to Waring-Hilbert ...
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1answer
36 views

How can be proved that every square number can be expressed as the sum of another square number and some semiprime number? [closed]

Any help to focus the problem would be welcomed! Something to do with quadratic residues?
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1answer
24 views

What the difference is between TypeI/II/III SS in ANOVA?

My background is not mathematics and I do not really understand what this mathematical symbols means: Type I SS: SS(A) SS(B|A) SS (AB|A, B) Type II SS: SS(A|B) SS(B|A) SS (AB|A, B) Type III SS: SS (A|...
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2answers
63 views

Does $\exists n : 6 + \sum_{i=2}^n p_i = x^6+y^6$ for $p_i$ the $i^{\text{th}}$ prime and $x,y\in\mathbb{Z}$?

I noticed something about the prime numbers: Pick the number $2$. Then, add the first odd prime, namely $3$. The result is $5 = 1^2+2^2$. Notice that the exponents are also the number we picked. ...
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5answers
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How do I find the sum of the series -1^2-2^2+3^2+4^2-5^2… upto 4n terms? [duplicate]

I tried by giving $$ S = \sum_{k=0}^{n-1} \left((4k+3)^2+(4k+4)^2-(4k+1)^2-(4k+2)^2\right) $$ but I am stuck here. I have no idea what to do next. The answer in my book says 4n(n+1). How can I get ...
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1answer
114 views

Does every sum-of-squares equation have a plane geometric interpretation?

The 2.1.2 sum-of-squares [SOS] equation $$a^2=b^2+c^2$$ can be thought of in the [regular XY] plane as a right-angled triangle. Question: Is there an analogous interpretation for every SOS equation?...
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1answer
47 views

Confusion about sum of squares.

First, I want to declare that this problem have been solved by myself, I just want to share my work to everyone. It has known that a prime has the form $4k+3$ cannot be sum of two squares, and also a ...
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2answers
320 views

Sum of two squares equal to $2018^{2019}+2018$ [closed]

$$x^2+y^2 = 2018^{2019}+2018$$ is expressed as sum of two perfect squares. Any pair of perfect squares can satisfy?
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1answer
76 views

Find all triples $(x,y,z)$ of positive integers such that $2018^x=y^2+z^2+1$ [closed]

Find all triples $(x,y,z)$ of positive integers such that $$2018^x=y^2+z^2+1$$
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1answer
264 views

For any integer n, $78n^2+72n+2018$ is expressed as sum of four perfect cubes. [closed]

For any integer n, $$78n^2+72n+2018$$ is expressed as sum of four perfect cubes. Is this possible that any integer can fit into this expression with 4 perfect cubes?
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1answer
211 views

Is it possible to use partitions of an odd integer to generate primes in a given interval?

We start with the partition of $N=5$. $$5$$ $$4+1$$ $$3+2$$ $$3+1+1$$ $$2+2+1$$ $$2+1+1+1$$ $$1+1+1+1+1$$ Then we form the sum of squares (no limit on the number of elements) to get: $$4^2+1^2=17$$ ...
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1answer
81 views

Find sum of squares of elements of QB

Given that Q is an orthogonal nxn matrix and B is an mxn matrix, how can we find the sum of squares of all elements of QB in terms of the sum of squares of all elements of B? I know that the sum of ...
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2answers
121 views

Proving $\left ( x+ y- z \right )\left ( \frac{3}{x+ y}- \frac{1}{y+ z}- \frac{1}{z+ x} \right )\leqslant \frac{1}{2}$ [closed]

Give $x$, $y$ and $z$ be nonnegative numbers. Prove that $$\left ( x+ y- z \right )\left ( \frac{3}{x+ y}- \frac{1}{y+ z}- \frac{1}{z+ x} \right )\leqslant \frac{1}{2}$$ First solution $$LHS- RHS= \...
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1answer
139 views

Mandelbrot and Julia fractals for $z_{n+2} = z_{n+1}^2 + z_n^2 + c$

The Mandelbrot and Julia type fractals are very Well known. But such fractals follow from $$z_n = f(z_{n-1},c)$$ In other words a recursion that only depends on the previous value and a constant. (...
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2answers
358 views

Perfect square as sum of two perfect squares

I refer to this question. Given a perfect square, can you prove that it is a sum of two perfect squares? I recently saw this: Let $p,q$ be primes. $p_i \equiv 1 \pmod 4$ and $q_i \equiv 3 \pmod 4$...
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3answers
110 views

Is $a^2 + b^2 = c^2 + d^2 = e^2 + f^2$ possible, where a+e=d and b+c=f

Here's my problem: Is $a^2 + b^2 = c^2 + d^2 = e^2 + f^2$ possible where $a$, $b$, $c$, $d$, $e$, and $f$ are all positive integers, and $(a, b)$, $(c, d)$, $(e, f)$ are all distinct pairs ($(3, 4)$ ...
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79 views

Can Gauss-Newton algorithm give better optimization performances than Newton algorithm?

I am currently facing the problem of a robotic manipulator calibration: the goal is to find the best correction that must be applied to a set of kinematic parameters describing the robot model, in ...
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0answers
73 views

Sums of descending squares

I am interested in integers that can be expressed as a sum of squares. Specifically I am interested in integers that can be expressed as follows: $n=6*Sum (k^2+(k-a)^2+(k-2a)^2.....1^2)$ These ...
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1answer
65 views

Unique Representation of Primes as a Quadratic Form [closed]

I have been wondering how to solve certain problem types, but specifically this one: "If $p$ is a prime that can be written in the form $5x^2+6y^2$, where $x$ and $y$ are positive integers, prove ...
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2answers
47 views

Prove that any power of $10$ can be written as sum of two squares

I do know various techniques to solve this problem, but I need an elementary solution which can be explained to a fifth grader (that is, with as little algebra as possible, no modulo arithmetic).
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1answer
89 views

Closed form for the sum $\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}{\left(n^2+m^2\right)^{-{p}}}$

Is it possible to express the following sum in closed form? $$S(p)=\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}{\left(n^2+m^2\right)^{\large-{p}}}$$ where the point $(n,m)=(0,0)$ is not taken ...
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1answer
71 views

On near-Pythagorean triples $(n^5-2n^3+2n)^2 + (2n^4-2n^2+1)^2 = n^{10} + 1$

We have, $$\begin{aligned} (n^3-2n)^2 + (2n^2-1)^2 &= n^6 + 1\\ (n^5-2n^3+2n)^2 + (2n^4-2n^2+1)^2 &= n^{10} + 1\\ (n^7-2n^5+2n^3-2n)^2 + (2n^6-2n^4+2n^2-1)^2 &= n^{14} + 1\end{aligned}$$ ...
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2answers
329 views

How many sub-square matrices does a square matrix have and is there a simple formula for it?

Consider an $n \times n$ matrix $M$. I want to find the determinant for ALL sub-square matrices of $M$. There may be a better way but my method is to find all sub-square matrices and check them ...
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2answers
45 views

Sum of the Squares of the First n square Numbers is not a perfect square number

In other words, $x(x+1)(2x+1)=6y^2$ has no nontrivial integral solutions. I thought this is a well-known result, but surprisingly could not find a recorded (easy) proof. Can someone provide a proof ...
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2answers
53 views

consecutive integers that are not the sum of 2 squares. [closed]

Are there any $n\in\mathbb{N}$ such that no element $k\in\{n,n+1,n+2,...,n+2017\}$ can be expressed as $a^2+b^2$ for some $a,b\in\mathbb{Z}$?
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0answers
147 views

How to interpret this visual proof for Archimedes' derivation of Sum of Squares?

I'm interested in knowing how was the first closed form solution of the sum of squares derived (for historical context/curiosity). I stumbled across an MAA Page that provides a visual proof, without ...
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3answers
120 views

If $x^2-dy^2 = -1$ has a solution in $\mathbb{Z^2}$, then $d$ is the sum of two coprime squares.

I want to prove: Let $d\in\mathbb{N}$ be square-free. If $x^2-dy^2 = -1$ has a solution in $\mathbb{Z^2}$, then $d$ is the sum of two coprime squares. I've already shown, that the equation has no ...
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1answer
63 views

How many variations of new primitive Pythagorean triples are there when the hypotenuse is multiplied by a prime?

When I was looking at all the Pythagorean triples from 1 to 1000, I noticed a certain trend which I cannot seem to explain. Given $a, b, c$ are integers where $c$ is the hypotenuse, and $a < b$, ...
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2answers
66 views

How to find the sum of the Series $\sum_{n \in \Bbb{Z}} \vert c_n \vert^2$ which terms are the special integral representations

For $n \in \Bbb{Z}$, define $$ c_n=\frac{1}{\sqrt{2\pi}} \int_{-\pi}^{\pi} e^{i(n-i)x} dx$$ where $i^2=-1$. Then $\displaystyle\sum_{n \in \Bbb{Z}} \vert c_n \vert^2$ equals........? a) $\cosh(\...
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1answer
41 views

Confirmation of Proof: $\forall n \in \mathbb{N}^*, \ p = 4261 + 2308n : \text{prime} \Rightarrow p = \{a^2 + b^2 \mid \gcd(a, b) = 1\}$

I developed a conjecture which I would like to confirm whether or not it holds truth for all members $n$ of the set of $\mathbb{N}^* = \mathbb{N} \ \cup \ \{0\}$. $$\forall n \in \mathbb{N}^*, ...
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1answer
19 views

Is there a pattern defining the existence of root integer distances in an isometric grid?

In a standard square grid pattern the distances to integer root locations is simply the sum of two squares. We find that these distances have $\sqrt1, \sqrt2, \sqrt4, \sqrt5, \sqrt8, \sqrt9, \sqrt10, \...
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1answer
146 views

Can this be continued forever?

We have: $$1=1^2$$ and $$25=5^2=3^2+4^2$$ and $$441=21^2=20^2+4^2+5^2$$ So, for $k=1,2,3$ we have a $k$-digit number that is a perfect square and a sum of $k$ different non-zero perfect squares. ...
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2answers
87 views

Is there an infinite number of numbers like $1600$?

My reputation is at this moment at $1600$. I did some experimenting with $1600$ and obtained the following: Evidently, it is a perfect square $1600=40^2$ Also, it is a hypothenuse of a Pythagorean ...
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1answer
38 views

Calculating the Sum of Squares between in ANOVA table

I can't figure out where I have gone wrong here. There are 3 independent samples all of which have 12 observations, which give values: $\Sigma x_a = 4913, \Sigma x_b = \Sigma x_c =5177, \Sigma x_a^2 ...
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2answers
102 views

sum of square derivative

What is the partial derivative of the following expression with respect to $U_i,V_j$ and M, respectively: $$L=\sum_{i}^m \sum_{j}^n(P_{ij} - g(U_i^T M V_j))^2 $$ where $$ U \in R^{d*m} , V \in R^{d*...
4
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0answers
151 views

Iterations of $x^2 + y^2$

We construct a sequence $S$ of distinct positive integers as follows 1) the sequence $S$ starts as $1,2,3$ 2) If $x,y$ are in the sequence , then $x^2 + y^2 $ is also in the sequence. 3) the ...
2
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1answer
116 views

More of this type $T_n^2+T_{n+1}^2+T_{n+2}^2=X^2?$

Given triangular numbers, $$T_n:= {n(n+1)\over 2} = 1,3,6,10,15,...$$ Let $T_n, T_{n+1}$ and $T_{n+2}$ be three consecutive triangular numbers. My question is: Are there more of $T_n^2+T_{n+1}^2+...
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4answers
82 views

Integer solutions to $a^{2}+b^{2}+c^{2} = d^{2}+e^{2}+f^{2}$, where $a, b, c, d, e, f \neq 0$

How does one approach something like this? Is there an equivalent Legendre's three-square theorem for the sum of three squares in two different ways? It seems like the only way to approach it would ...
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0answers
33 views

Is there reference sheets or techniques to writing numbers as sums of squares?

For example if I wanted to write $x$ as the sum of $n$ squares in $m$ different ways, is there something I can just look up? Also I have other constraints with what I am working with, if I knew $x$ I ...
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1answer
37 views

Show that the number of solutions of $x^2+y^2 ≡ 1$ (mod $p$) where $0<x<p$, $0<y<p$, $p$ is odd prime is even iff $p ≡ 3, -3$ (mod $8$)

Show that the number of solutions of $x^2+y^2 ≡ 1$ (mod $p$) where $0<x<p$, $0<y<p$, $p$ is odd prime is even iff $p ≡ 3, -3$ (mod $8$). I learned about quadratic residue and sums of ...