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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Squaring complex number equation with absolute values

I don't understand how you go from the first to the second line in this problem : $$|(a-k)+i(7-2a)|=|(a-2)+i(9-2a)|$$ $$(a-k)^2+(7-2a)^2=(a-2)^2+(9-2a)^2.$$ Firstly, squaring i should make it -1 I ...
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show that there are arbitrarily long sequences of consecutive positive integers which are not sums of two squares

Let $p_1, p_2, . . . , p_k$ be different prime numbers. By the Chinese Remainder theorem, show that for each $k ∈ N$ there exists an integer n such that $p_1 |n+1; p_2 |n+2; ...; p_k |n+k$ but $p^2_k$ ...
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Sum of squares and primes

Let $S$ be the set of the integers that can be represented as the sum of two squares. $S={0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, 32, ...}$. I have made this hypothesis: If an integer ...
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Will iterating the sum of the squares of the digits of any number ultimately give $1$?

I was doing a program where I had to figure out whether the sum of square of digits of a number will ultimately give $1$. For eg: $$68 \;\to\; 6^2+8^2=36+64=100 \;\to\; 1^2+0^2+0^2=1$$ So I wanted to ...
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2 votes
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sum of square of three squares function $r_3(n)$ II

From sum of square of cube function $r_3(n)$, I believed that, as $r_k(n)$ denotes the number of ways that $n$ is the sum of $k$ squares, $$\sum_{n\le X} r_3(n)^2\asymp X^{3/2}\log^{C}X,...(*)$$ but ...
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Find the signature of a bilinear form given by a matrix

I'm trying to complete the bilinear form given by the matrix $$M=\left(\begin{array}{ccc}1 & -1 & 2 \\ -1 & 3 & 1 \\ 2 & 1 & 1\end{array}\right)$$ into squares to find the ...
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convergence of sum of squares over squared sum

If we have $\frac{\sum_{i=1}^{n} a^2_i}{(\sum_{i=1}^{n} a_i)^2}$ where $a_i$ is a positive sequence where each element is finite with probability 1, under which set of conditions does the division ...
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SOSTOOLS in MATLAB: findbound gives wrong output

Here is the MATLAB code that uses SOSTOOLS toolbox: ...
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1 answer
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Finding a tighter bound given sum and sum of squares

Let $x_1, ..., x_{25}>0$ be such that $\sum_{i=1}^{25}{x_i} = 4350$ and $\sum_{i=1}^{25}{x_i^2} = 757770.25$. From the first equality alone, we know that at least one of the $x_i$'s must be less ...
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Finding $(a,b)\in\mathbb{N}^2$ such that $\dfrac{a^2+b^2+1}{a+b} \in \mathbb{N}$.

A pair $(a,b)\in\mathbb{N}^2$ is called good if $a < b$ and $$\frac{a^2+b^2+1}{a+b}\in\mathbb{N}.$$ I think I've shown that there are infinitely many good pairs. However, the family of good pairs ...
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5 votes
3 answers
363 views

Prove that $\frac{a-c}{k}=\frac{d+b}{n}$, where $N=a^2+b^2=c^2+d^2$ and $k=\gcd(a-c,d-b)$, $n=\gcd(a+c, d+b)$

Let $N$ be odd and $N = a^2 + b^2 = c^2 + d^2$, where $a, b, c, d \in \mathbb{N}$ and WLOG let $a, c$ be odd, $b, d$ be even, $a > c$, and $b < d$. Prove that $\frac{a-c}{k}=\frac{d+b}{n}$. I ...
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Is the number of ways to express a number as sum of two coprime squares same as number of solution of $x^2+1\equiv0\pmod n$

The number of representations of $n$ by sum of 2 squares is known as sum of square function $r_2 (n)$. It is known that if prime factorization of $n$ is given as $$2^{a_0}p_1^{a_1}p_2^{a_2}\cdots q_1^{...
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A giant cube is made of $n^3$ small cubes so that its dimensions are $nxnxn$. How many cubes of any size can be made of small cubes in the giant cube?

I am familiar with a similar problem that asks the number of squares that can be made in an $n x n$ chessboard. The answer was $\sum_{k=1}^{n} k^2$. I "feel" that the answer to the cube ...
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Gaussian integral using Euler/Jacobi theta function and $r_2(k)$ (number of representations as sum of 2 squares)

The Euler/Jacobi theta function (using the notation of this question) is $\vartheta_3(\tau) := \sum_{n\in \mathbb Z} q^{n^2}$ where $q = e^{2\pi i\tau}$ is the nome. The square $(\vartheta_3(\tau))^2$ ...
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How would I maximizing the sum of elements given the squared sum?

The question is a more broad approach, but as a concrete example, if I have a point represented by the vector v on a d-dimensional unit sphere, how would I find the elements of the vector v that would ...
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2 votes
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Sum of 3 squares formula using modular forms

I am trying to prove the sums of 3 squares formula in Cohen's 1975 paper 'Sums Involving the Values at Negative Integers of L-Functions of Quadratic Characters'. That is, for $H(N)$ the Hurwitz class ...
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Parameterize matrix with disnict values, same row sums and sum of any 2 values in a row is perfect square.

Problem Suppose I have $3\times3$ matrix $A$ with distinct values: $$ A \in \{B \in \mathbb{N}^{3\times3} | (i,j) \neq (k,l) \implies b_{ij} \neq b_{kl}\} $$ Which also satisfies: $$ (A \times \begin{...
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How do I efficiently calculate the total sum with respect to a changing v in a square root?

How do I efficiently calculate the total sum with respect to a changing $\space v \space\space $in a square root $\sqrt{\dfrac {\hbar{((v)/(6.25E34))}}{4G}}r\space\large{?}$ The sum is based on the ...
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Bound on ratio of sums

I would am interested in upper (or lower) bounds of the following ratio: $$\frac{\sum_{j=1}^k a_j^2}{\sum_{j=1}^k a_j b_j}$$ where $a_j$ are non-negative integers, and $b_j$ are non-negative reals. As ...
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Difference between increasing integer values of $C=x^2+y^2+z^2$

Given are non-negative integer variables $x$, $y$ and $z$. I am trying to deduce the absolute difference between a certain value of $C=x^2+y^2+z^2$ and the very next smallest increase in $C$ possible. ...
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Is it the case that for every $n$, $m$, there is an integer expressible as a sum of $m$ distinct positive integer squares in $n$ distinct ways?

I am not a mathematician, but I learn from various sources that we know that, for every n, there is an integer that can be expressed as a sum of two positive integer squares in n distinct ways, and ...
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3 votes
2 answers
215 views

How to show this inequality $ a^{14}-a^{13}+a^8-a^5+a^2-a+1>0$

Given the real $ a $. Prove that $$a^{14}-a^{13}+a^8-a^5+a^2-a+1>0$$ I tried to factor it as $$(a-1)\Bigl(a^{13}+a^5(a^2+a+1)+a\Bigr)+1$$ I think it should be written as a sum of squares. Any idea ...
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2 votes
1 answer
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Is there a prime $a^2+b^2$ for all $a \in \mathbb N$?

I am wondering whether it is a known result whether for every natural number $a \geq 1$, there is at least one natural number $b$ (of any size) such that $a^2+b^2$ is prime. This seems empirically ...
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For integers $x<y<z$, why are these cases impossible for Mengoli's Six-Square Problem?

I got inspired by this question "Four squares such that the difference of any two is a square?" and rewrote zwim's program that is provided by his answer to the question "Solutions to a ...
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1 vote
1 answer
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Proof that this double summation is nonnegative

I need to prove this equation $$\sum_{i=1}^K {\bigg(x_i\bigg(x_is_i-\sum_{j=1}^K {x_js_is_j}\bigg)\bigg)}\ge0$$ knowing that $K\in\mathbb{N},K\ge2$ $x,s\in\mathbb{R}^K$ $\forall {k\in\{1,\dots,K\}}:...
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1 answer
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Properties of $F(p)=p^2+1$, where $p$ is a prime number

Let $F(p)=p^2+1$, where $p$ is a prime number. For what primes $p_1$, $p_2$ does $p_2$ divide $F(p_1)$ and $p_1$ divide $F(p_2)$? Two examples are $\{p_1,p_2\}=\{5,13\}$, and $\{p_1,p_2\} = \{89,233\}...
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1 answer
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Intuition to faulhabers sum of k-th power of n first integrals

In another post I made, an answer pointed me toward Faulhabers formula for sum of k-th powers of the n first integers. The answer I got of how to reach a formula for $P(n)=\sum_{k=0}^{n}k^2$ looks ...
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1 vote
2 answers
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Cauchy-Schwarz-like Inequality

Let $a_i, b_i >0$ for all $n$ and $0 \le \lambda \le 1$ Is the following result true for all $n$? $$ \sum^n_{i=0} a^\lambda_i b^{1-\lambda}_i \le \left( \sum^n_{i=0} a_i \right) ^\lambda \left( \...
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-1 votes
1 answer
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Write a number as the sum of two squares [closed]

I am trying to write $82900$ as a sum of two squares. I am given a hint that $8290 = 57^2 + 71^2$. How can I use this hint to set up the problem? I have used Fermat's Descent in the past, can I still ...
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2 answers
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please help figure out the error in this sigma sum

the question is "the sum of all possible products of the first n natural numbers taken two at a time is?" and this is how I approached it: first i selected the first number as 1 therefore $\...
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For the mean squared error, prove $E(SSE) = (N-k)\sigma^2$

I am trying to prove that $E(SSE) = (N-k)\sigma^2$ for $k$ treatments and i'm stuck at proving the difference sum between treatment mean squared and total mean squared is zero. Which baffles me as i ...
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Prove existence of certain lattice points in an ellipse

Here is a question that has puzzled me for a long time. I have figured out how to do the question a and b(i). (a) An ellipse, F, is centred on the origin. There are two distinct points inside F with ...
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1 answer
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Is there any special relation between squared root of sum of squares and sum of the values themselves?

I mean, is there any relation between $a_1 + a_2 + a_3 + ... + a_n$ and $\sqrt{a_1^2 + a_2^2 + a_3^2 + ... + a_n^2}$ ? This relation can be of any kind or any use. Thank you all in advance.
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2 votes
5 answers
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Characterizing integer solutions to $a^2 \mid \bigl((b^2)^2 + (b^2+1)^2\bigr)$

I’m looking to characterize all integers $a$ and $b$ satisfying $$a^2 \mid \bigl((b^2)^2 + (b^2+1)^2\bigr), \qquad a > b \ge 1.$$ Brute force searches have so far turned up the two solutions $(a,b) ...
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2 answers
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eucledian geometry

from facebook A square. Calculate the ratio of shaded area to the area of square but one solution appear in the second fiqure used ratios could any one explained it geometrical thank you i try ...
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4 votes
0 answers
113 views

How many ways can an integer be written as a sum of three integer squares

I'm working on a fairly simple Physics problem (particle in an infinite potential cube), and I'm asked: "Are any of the energy eigenvalues degenerate? If so, what is the degeneracy?". The ...
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2 votes
1 answer
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Products of sums of $2n$ squares

It is well-known that the product of two squares is a square, and the product of sums of two squares is a sum of two squares (Brahmagupta-Fibonacci identity): $$(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2$$...
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1 answer
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Properties of ordered pairs that satisfy $x^2 +y^2 =a$

I want to know number of ordered pairs $(x,y)$ satisfying $x^2 +y^2 =a$. Let $n(a)$ be the number of ordered pairs $(x,y)$ satisfying $x^2 +y^2 =a$. If $x^2 +y^2 =a$ and $z^2 +w^2 =b$, then $(xz+yw)^2 ...
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  • 518
2 votes
3 answers
153 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 ${6d^2+30d+55 \ne n^2 \Rightarrow 6(d^2 + 5d ...
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$\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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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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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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7 votes
2 answers
381 views

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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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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3 votes
2 answers
79 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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2 answers
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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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6 votes
2 answers
101 views

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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2 answers
41 views

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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2 answers
84 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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-2 votes
1 answer
54 views

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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