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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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Gauss' Proof of Lagrange Four-Square Theorem?

I read recently that Gauss provided a proof of Lagrange's Four-Square Theorem using his ideas about equivalence classes of quadratic forms (i.e. linear substitution of variables) somehow applied to $w^...
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1answer
34 views

Constructing a sum of squares in $\mathbb R[x]$ with given complex valuation

Fix a polynomial $g(x)\in\mathbb R[x]$ and a complex number $u\in\mathbb C\setminus\mathbb R$. My main question is How can we construct a polynomial $s(x)\in\mathbb R[x]$ such that $s(x)$ is a sum ...
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1answer
629 views

How much of an infinite board can a N-mover reach?

This question is inspired by the question on codegolf.SE: N-movers: How much of the infinite board can I reach? A N-mover is a knight-like piece that can move to any square that has a Euclidean ...
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ternary quadratic form as a sum two squares of linear forms

Let $\phi(x,y)=ax^2+2bxy+cy^2$ be a binary quadratic form with integer coefficients. Pall proved in 1951 that $\phi(x,y)$ can be written as the sum of two squares of two linear forms if and only if $...
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18 views

What is a closed form partial sum formula for the q-digamma function?

It is known that the partial sum of the digamma function can be expressed in closed form, but what of the q-digamma function? $$\sum_{x=1}^n\psi_q{\small(x)}\ =\ ?$$ $$q\in\mathbb N$$
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2answers
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Show that if $a,b,c\in \mathbb{N}$ and ${a^2+b^2+c^2}\over{abc+1}$ is an integer it is the sum of two nonzero squares

I was reading about the fascinating problem in the IMO ($1988 $ #$6$) that asks: Let $a$ and $b$ be positive integers such that $(ab+1) | (a^2+b^2)$. Show that ${a^2+b^2}\over{ab+1}$ is a perfect ...
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3answers
92 views

Let a, b, c, d be any four distinct integers such that $a>b>c>d>1$. Show that if $ad=bc$, then $a^2+b^2+c^2+d^2$ is composite. [closed]

Let $a, b, c, d$ be any four distinct integers such that $a>b>c>d>1$. Show that if $ad=bc$, then $a^2+b^2+c^2+d^2$ is composite.
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117 views

Can factoring with the sum of 4 squares be made more efficient?

We have seen that it was possible to use the sum of two squares to factor numbers (see Can the sum of two squares be used to factor large numbers? ) The main drawback is the fact that the method ...
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2answers
30 views

Adding sequence of square roots [closed]

How to add sequence of square roots from square root 2 till square root 99 and how to add the sequence of their reciprocal here is the original problem
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1answer
54 views

Hypergeometric function 3F2 with unit argument

Recently I obtained the following expression $${}_3F_2(-n,a - b ,1-b-n; b + 1, 1-a-n; 1), $$ with $b>a>0$ and $n\in\mathbb{N}$. My question is: If someone knows a closed form solution to the ...
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3answers
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Understanding some proofs-without-words for sums of consecutive numbers, consecutive squares, consecutive odd numbers, and consecutive cubes

I understand how to derive the formulas for sum of squares, consecutive squares, consecutive cubes, and sum of consecutive odd numbers but I don't understand the visual proofs for them. For the ...
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78 views

Numbers that are the sum of 2 distinct nonzero squares in exactly 1 way

Is there a formula that says if a number is the sum of 2 distinct nonzero squares ? This is the sequence I'm trying to emulate: 5, 10, 13, 17, etc.. http://oeis.org/A004431 (Invalid it contains 85, ...
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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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134 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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22 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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277 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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187 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
96 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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$\exists\infty$ 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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29 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
46 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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4answers
45 views

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
30 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
46 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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113 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
67 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
39 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
29 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
208 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
90 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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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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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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1answer
24 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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58 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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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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33 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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221 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
79 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
268 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
585 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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55 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
124 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
86 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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0answers
155 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
51 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
36 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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37 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 ...