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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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How to calculate $\sum_{i=0}^N\lfloor ai\rfloor^2$ for $0<a<1$ and $N\approx10^{16}$?

I have an irrational number $0<a<1$ for which I need to calculate the sum $$\sum_{i=0}^N\lfloor ai\rfloor^2\mbox{ mod }m.$$ Here $m$ is an $8$-digit number and $N$ is a $15$-digit number. The ...
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66 views

Primality test using the sum of 2 squares representation of a prime $p^2= a^2+b^2$

We have seen that the sum of two squares (2 sq rep) of an integer $N^2$ can be used to factor $N$. Can the sum of two squares be used to factor large numbers? Here we use the same method to show ...
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21 views

Find the next least number N that is N+1 = X^2 and (N/2)+1 = y^2?

To go with this following task to create a proper equation ? Many numbers, especially from smaller ones, can rightly claim that they are of particular interest to scientists. In the kingdom of ...
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204 views

Can the sum of two squares be used to factor large numbers?

The method described below is different from Euler sum of two squares factorization method. The method below does not require 2 sum of two squares representations to factor a number. This post was ...
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128 views

Can the sum of two squares be used to determine if a number is square free?

Mathword (http://mathworld.wolfram.com/Squarefree.html) stated that "There is no known polynomial time algorithm for recognizing squarefree integers or for computing the squarefree part of an integer."...
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5answers
90 views

If $xy$ divides $x^2 + y^2$ show that $x=\pm y$ [duplicate]

Problem statement : Let $x,y$ be integers, show that if $xy$ divides $x^2 + y^2$ then $x=\pm y.$ What I have tried: I can reduce this to the case where $\gcd(x,y)=1$, since if $x$ and $y$ have a ...
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33 views

$\exists\infty$ many pairs of consecutive squares s.t. their sum is also a square

First of all, the term "pairs" is two of them, I assume (question's formulation is rather difficult to understand for me). So I guess this is the statement: $$\exists\infty\text{ many pairs of ...
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27 views

Does bounded version of Lagrange's 4-square theorem follow from this result?

Let $S_4(n)$ be the number of ways of representing $n$ as a sum of 4 integer squares and suppose I can give you a proof of the result that $\sum_{i=1}^nS_4(i) \sim \frac{\pi^2}{2}n^2$. Is this result ...
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1answer
45 views

$\lim_{n\to\infty}\frac{\sum_{i=1}^ni^{4\lambda}}{\Big(\sum_{i=1}^ni^{2\lambda}\Big)^2}=0$

Is it true that $\forall \lambda>0$ $$\lim_{n\to\infty}\frac{\sum_{i=1}^ni^{4\lambda}}{\Big(\sum_{i=1}^ni^{2\lambda}\Big)^2}=0$$ I cannot find a way to prove it, nor can I find a counterexample. ...
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Diophantine equation: solving $a^2+4n=b^2$

I found myself working with diophantine equations but I have no experience at all with them. Given an integer $n$, can I find two integers, $a$ and $b$, such that $$a^2+4n=b^2$$ How would you guys ...
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1answer
27 views

Inequality regarding sum of squared probabilities

I'm working on a problem set for a course on Machine Learning and one the problems asks me to prove a given inequality. As an aid for that, the problem gives me the hint to use the following result, ...
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2answers
44 views

How to express a number as a sum of $k$ squares?

My question is the following: Show that for each integer $k \ge 5$, there is an integer $N(k)$ such that every integer $n \ge N(k)$ can be written as a sum of $k$ nonzero squares. What is the process ...
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110 views

Odd of the form $a^2+b^2+c^2+ab+ac+bc$.

The computation below shows that (for $a,b,c \in \mathbb{N}$) the form $$a^2+b^2+c^2+ab+ac+bc$$ covers every odd integer less than $10^5$ except those in $$I= \{ 5, 15, 23, 29, 41, 53, 59, 65, ...
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1answer
65 views

Sum of four Squares relation

Cheers, As is well known due to Lagrange the number of ways to write $n$ as a sum of four squares is given by $$ r_4(n)=8\sum_{d|n} d $$ if $n$ is odd. Now define $$ \tilde{r}_4(n)=\#\left\{1 + \...
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1answer
34 views

Upper bound on Sum of square roots

Let $k_1,k_2\ldots k_t$ be integers and $\sum_{i=1}^{t}{k_i}=k$ where $k$ is fixed. What is the maximum value of $\sum_{i=1}^{t}\sqrt{k_i}$.
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1answer
27 views

Quasiconvexity of a specific quartic bivariate polynomial family

I have a quartic polynomial of shape $$(a\cdot x+b+c\cdot xy-d\cdot y+y^2\big)^2$$ where $a,b,c,d\in\mathbb Z_{>0}$ are coefficients. The polynomial is not convex. However is it quasiconvex for ...
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2answers
84 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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2answers
52 views

Product of two numbers which are expressable as the sum of two perfect squares

Problem: "Let m and n be integers such that each can be expressed as the sum of two perfect squares. Show that mn has this property as well" Proof: Let $m =a^2+b^2$ , $n=c^2+d^2$ ,where $a, b , c , d$...
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23 views

What is the error propagation for $r = \sqrt{a^2+b^2}$ if the error in $\delta a$ and $\delta b$ are known?

Suppose I have some value $r$ which is defined as $$r = \sqrt{a^2+b^2}$$ where $a$ and $b$ have errors of $\delta a $ and $\delta b$, respectively. What is $\delta r$? Using "standard" error ...
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38 views

How to find how similar two vectors are, while giving weight to their lengths?

I asked a question about this yesterday and got a really good response! Apparently I should use the euclidian (sum of squares) distance between the two vectors. This works well, but I'm having a bit ...
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66 views

Sums of powers of algebraic numbers that are rational

$\newcommand\Q{\mathbb Q}$I have a series of questions related to sum of $k$-powers of algebraic numbers (it is actually only one question that I try to weaken/strengthen the conditions). Suppose we ...
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23 views

How to show that sum of squares is n times of mean?

I gave $X_i\sim N(\mu=8, \sigma^2=1)$ for $i=1,...91$ with observed $\bar{x}=7.319$ and I calculate $f(\bar{x}=7.319|\mu_0=8)$ and I stuck in one step of calculation, actually the very first: $(\...
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45 views

Can a positive polynomial on sphere be represented as the sum of squares of spherical harmonics

Let $p\in {\mathbb{R}}[x_1,\ldots, x_d]$ be a homogenous polynomial degree $2n$. We know that if $p$ is positive on $[-\pi,\pi]^d$, $p$ is sum of squares polynomial, i.e. $p$ can be witten as sum of ...
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2answers
463 views

Exploiting a Diophantine approximation of $\pi^4$ into giving a series of rationals for $\pi^4$

A note about this question: The original question asked seems likely impossible so I am really asking if we can exploit the technique below into giving us a 'nice' form for $\pi^4$. By nice form I ...
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31 views

Optimization of sum of squares over permutations

Suppose I have fixed, positive values $n_1, \cdots, n_\ell$ and $T$. I'm looking for an algorithm to optimize \begin{align*} f(\boldsymbol{n}) = T\left(\sum_{j=1}^{\ell}\left(\sum_{i=1}^{j}n_i\right)^...
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1answer
77 views

Positive integers have the form $a^2+b^2+c^2+2^d$

For any positive integer $n$ there seems to be non-negative integers $a,b,c,d$ such that $$n=a^2+b^2+c^2+2^d.$$ Due to Legendre's three-square theorem a natural number can be represented as the ...
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139 views

Sum of a Sequence of Odd Numbers that are Squared [duplicate]

What is the sum of all the numbers in the sequence $1^2 + 3^2 + 5^2 + 7^2 + 9^2 + \ldots + k^2$. Note that all the numbers being squared in the sequence are all odd numbers. This is what I have done ...
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1answer
78 views

Pythagoras theorem quick calculation [closed]

How can I easily calculate this equation $x^2+y^2=76149513$ when $x$ and $y$ are whole numbers?
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1answer
198 views

How to prove that all primes of the form $4k+1$ can be represented by the sum of two squares in only one way regardless of the order?

I am reading a book about Number Theory as a new learner. The book has proved that all primes of the form $4k+1$ can be represented by the sum of two squares. This question is given as exercise and ...
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2answers
541 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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49 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
73 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
84 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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154 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
50 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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28 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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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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106 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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95 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
34 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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54 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
146 views

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
109 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
43 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
256 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
65 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
215 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
198 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: $$4^2+1^2=17$$ ...