For questions concerning various representation of integers as sums of squares, which are studied in number theory.

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16 views

What is the distribution of sum of the squares of k dependent standard normal random variables?

It is known that the sum of the squares of k independent standard normal random variables is chi-squared distributed, what happens if we look at the sum of the squares of k dependent normal variables? ...
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0answers
20 views

Generalizing Landau-Ramanujan Theorem for sum of three squares

Let $S_{3}(x)$ denote the number of positive integers not exceeding x which can be expressed as a sum of three squares. Can we find an asymptotic formula for $S_{3}(x)$, maybe using Landau-Ramanujan ...
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1answer
21 views

Sum implies null product. True? [closed]

Let $x, y, z \in \mathbb{C}$. Assuming that $x+y+z=x^2+y^2+z^2=0$, is it true that $x^2y^2+y^2z^2+z^2x^2=0$?
2
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1answer
34 views

The sum of the squares of two factors of a number is a perfect square

Is there any way to get two factors of a number whose sum of the squares is a perfect square. As an example $19354423920$ is a number. which has $4262$, $4541160$ as factors ($19354423920 = 4262 * ...
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0answers
51 views

When is sum of squares a perfect square? [duplicate]

Recall that $$\sum_{j=1}^nj^2=\frac{n(n+1)(2n+1)}{6}.$$ When is this quantity a perfect square? It appears that the only solutions are $n=0,1,24.$ By setting $x=12n+6$, the problem reduces to finding ...
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2answers
45 views

Given a number N how many pairs of numbers have square sum less than or equal to N?

Let's define $F(N)$ as the number of pairs of distinct positive integers $(A, B)$ such that $A^2 + B^2 \leq N$ If $N=5$ the only possible such pair is $(1, 2)$, for $N=10$ the pairs are two: $(1,2)$ ...
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4answers
49 views

What is 100,000 as a sum of some number of distinct squares? [closed]

I cannot find the solution to this problem. It is part of a larger homework question but I can't go on until I solve this question.
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2answers
45 views

Dividing the squares $1^2,2^2,\ldots,54^2$ into three equal groups with the same total sum

Is it possible to divide the squares $1^2,2^2,\ldots,54^2$ into three groups, each of which contains $18$ squares, such that the sum of squares within each group is the same for all three groups?
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0answers
9 views

Lattice points with next-largest norm

In a 2D integer grid, the points in increasing distance from the origin are: $(0,0)$ $(\pm1,0)$ and $(0,\pm1)$ $(\pm1,\pm1)$ etc By symmetry we need only consider one-eighth of the lattice, $x\ge0$ ...
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2answers
56 views

Diophantine equation $x^2+y^2=z^2+t^2$?

I would like to find some source books or articles which discuss the Diophantine equation $$ x^2+y^2=z^2+t^2,\qquad |y-z|=1 $$ for which $x,z$ are odd positive and $y,t$ are even positive integers. ...
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3answers
46 views

$Dm^2 - n^2D^2$ is a perfect square then $D$ is the sum of two squares

How do I show that if $$Dm^2 - n^2D^2$$ is a perfect square for some integers $m$ and $n$ ($n \neq 0$), $D$ is the sum of two (non-zero) perfect squares? I tried solving for $D$ but that only gives me ...
2
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1answer
67 views

Sum of eight squares over a finite field.

Consider the split-octonions $\mathbb{O}$ with coefficients in $\mathbb{F}_q$. Suppose $a \in \mathbb{F}_q$ and $b \in \mathbb{F}_q^*$. I want to find the amount of elements $x \in \mathbb{O}$ such ...
2
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2answers
73 views

Finding $\sqrt{(14+6\sqrt 5)^3}+\sqrt{(14-6\sqrt 5)^3}$

Find $$\sqrt{(14+6\sqrt{5})^3}+ \sqrt{(14-6\sqrt{5})^3}$$ A.$72$ B.$144$ C.$64\sqrt{5}$ D.$32\sqrt{5}$ How to cancel out the square root?
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2answers
28 views

Is this generalized factorization of a difference of powers correct?

Is this factorization true? $$(x^n - y^n) = (x+y)^{n-1}(x-y)^{n-1}$$ I am trying to use it in my computation of the determinant of a Vandermonde matrix. Thanks,
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3answers
123 views

Fermat's infinite descent for finding the squares that sum to a prime

Fermat's theorem on sum of two squares states that an odd prime $p = x^2 + y^2 \iff p \equiv 1 \pmod 4$ Applying the descent procedure I can get to $a^2 + b^2 = pc$ where $c \in \mathbb{Z} \gt 1$ I ...
4
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1answer
45 views

Reference for a Post about Gauss formula for primes written as sum of two squares?

In a post Efficiently finding two squares which sum to a prime I read "In 1825 Gauss gave the following construction for writing a prime congruent to $1 \pmod{4}$ as a sum of two squares: Let ...
3
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1answer
53 views

express prime as sum of squares, $p = a^2 + b^2$

Espress $2017$ as sum of two squares. attempt: by Fermat's Theorem on sums of squares, the prime $p = 2017$ is the sum of two squares $2017 = a^2 + b^2$ , $a,b \in \mathbb{Z}$, if and only if $p ...
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2answers
55 views

Number writable as sum of cubes in $9$ “consecutive” ways

Let's say that a given $n\in\mathbb{N}$ is writable as sum of cubes in $k$ consecutive ways if it can be written as sum of $j,j+1,\ldots, j+(k-1)$ nonzero cubes, for some $j\geqslant 1$. For ...
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1answer
35 views

How to open square brackets of the sum? [closed]

For example Σ(y[i]-mean(y[i]))^2 = ? Please describe the way how it opens.
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0answers
32 views

Determining the uncertainty from a reduced chi-square test

The questions wants me to generate 4 random numbers from a standard normal distribution (mean=0 and $\sigma$=1 ) and then calculate the sum $$x_1^2+x_2^2+x_3^2+x_4^2$$ If I do this 1000 times, how ...
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2answers
64 views

Writing an integer as a sum of two square in many ways, with consecutive arguments

Let $n\in{\mathbb N}$. I call $n=x_1^2+y_1^2=x_2^2+y_2^2=\ldots x_r^2+y_r^2$ (where $(x_1,y_1),(x_2,y_2),\ldots,(x_r,y_r)$ are distinct uples in ${\mathbb N}^2$) a multi-decomposition of $n$, of ...
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1answer
17 views

Sup-multiplicative function related to sum of two squares

Say that a representation $n=a^2+b^2$ (where $n,a,b$ are integers) is normalized if $0\leq a \leq b$. Among those normalized representations, there is a unique one minimizing $a$ (or equivalently ...
4
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2answers
49 views

Four square Lagrange Theorem.

Today teacher tell us that every natural number is the sum of four positive squares. and let us this work: write $747004$ as the sum of $4$ squares. I'm at high school and don't understand the ...
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0answers
37 views

Algebra 2 Factoring Questions

Prove that the "sum of squares" is always a prime polynomial. Prove that all trinomials generated by the sum of cubes formula have no real roots. All the proofs online use concepts we haven't ...
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2answers
74 views

Can a square number be expressed as sum of squares of two other members.? [closed]

Is there any theorem to tell if square of a number can be expressed as sum of squares of two other distinct numbers. I have one such set. {5 4 3} 5^2 = 4^2 + 3^2 Given a number n how to find if ...
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2answers
43 views

Strange induction proof

I'm trying to solve an induction proof exercise but this time I can't even understand how to proceed. I must prove that for every given $n\in \mathbb{N}$ with $n\geq2$ there exist $a,a_1,a_2,...,a_n$ ...
0
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1answer
21 views

Lagrange Method of Quadratic Form the a Billinear Form

In the following question I have to present the bilinear form as sum of squares with Lagrange method. $$q(x_1,x_2,x_3,x_4)=2x_1x_4-6x_2x_3$$ However I don't know how I can do it here since none of ...
2
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3answers
62 views

Determine if a positive integer is a sum of two positive squares

Given a positive integer $N$, what is the best/most efficient way to determine if $N=x^2+y^2$ for some positive integers $x<y$? Only the decision is needed, not explicitly $x,y$.
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2answers
50 views

Sum of two squares representation

I found the following condition: A positive integer $n$ is re-presentable as the sum of two squares, $n=x^2+y^2$ if and only if every prime divisor $p≡3$ mod $4$ of $n$ occurs with even exponent. And ...
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0answers
24 views

Representations of integer in the form f(x) - f(y)?

Let $f(x) \in \mathbb{Z}[x]$ be a polynomial. I would like to have an estimate for the number of representations $R(n)$ of $n \in \mathbb{Z}$ in the form $$ f(x) - f(y) = n, \qquad x,y \in \mathbb{N}. ...
3
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0answers
113 views

Gauss' “Eureka” theorem

Gauss proved that every integer is expressible as the sum of three triangular numbers. I was wondering if the proof is anywhere to be found?
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1answer
53 views

If $p\equiv 3\pmod{4}$ and $p\mid x^2+y^2$, prove $p\mid x,y$.

I have to prove that if $p$ is a prime number of the form $p = 4n - 1$, $n\in N$ and $x^2+y^2\equiv 0\pmod{p}$, then $x\equiv 0\pmod{p}$ and $y\equiv 0\pmod{p}$. I have gone about this as follows and ...
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1answer
85 views

Sum of squares of random variables that satisfy $x_1+\dots+x_n=1$

I am trying to solve the following problem and would very much appreciate some help. Let $x_1,\dots,x_n$ identically distributed continuous random variables that take values in $(0,1)$ and satisfy ...
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0answers
19 views

Proof related to ranks and sums of squares

Let $\mathbf{Y}$ be a $r \times c$ matrix with mean-centered columns, $r \lt c$, $\mathsf{rank}(\mathbf{Y})=k$. Let $\mathbf{P}$ be a projection matrix that maps from $\mathbb{R}^r$ to a subspace in ...
3
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1answer
68 views

Can two distinct sets of $N$ non-negative numbers have the same sum and sum of squares?

Suppose I have a set of $N$ non-negative numbers that sum to $A$. The sum of squares of these $N$ non-negative numbers sum to $B$. Here's the question: can there be a different set of $N$ ...
3
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2answers
94 views

Integer solutions to $x^2 + dy^2 = c$

I'm trying to find all integer solutions of an equation $x^2 + dy^2 = c$ with $d,c \in \mathbb{Z}_{>0}$. I'm well aware of the methods that exists when $d \in \mathbb{Z}_{<0}$ or when $c$ is a ...
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0answers
24 views

From 4 squares to 2 squares

Let $r_k(n)$ denote the number of representations of $n$ as a sum of $k$ squares of integers. Suppose I know that $r_4(n) = 8\sum_{4 \nmid d \mid n} d$ (Jacobi's Theorem). Is there a way to deduce ...
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0answers
61 views

Integer $2n^2+2$ as the sum of 2,3,4, and 5 squares

If $n-1$ and $n+1$ are both primes, establish that the integer $2n^2+2$ can be represented as the sum of 2, 3, 4, and 5 squares. I managed to solve 2 and 4 squares, since: $$2n^2+2 = ...
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0answers
49 views

On Catalan's complete solution to the equation $T^2=U^2+V^2+W^2$

Catalan proved the following: If $t,u,v,w$ are coprime integers such that \begin{equation*} t^2 = u^2 + v^2 + w^2, \end{equation*} then there exist integers $\alpha,\beta,\gamma,\delta$ such that ...
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0answers
41 views

Square matrix whose sum of squared elements equals 1.

I'm doing some applied work where I've come across examples that involve real valued square matrices that hold the following property, which expressed using tensor notation is $$A_{ij}A_{ij} = 1$$ ...
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4answers
70 views

Express 2104 as the sum of four squares

How to write 2104 as the sum of four squares. I know the general equation for factoring a number into the sum of four squares but I don't know how to go about this when some of the prime factors are ...
3
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2answers
76 views

How to show that that the following three consecutive numbers $3^{2^{10}} − 1$, $3^{2^{10}}$,$3^{2^{10}}+ 1$ are the sum of two squares?

Show that the following three consecutive numbers: $$ 3^{2^{10}} − 1, 3^{2^{10}} , 3^{2^{10}} + 1 $$ can be represented as sums of two integer squares.
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5answers
82 views

$5^m$, where m is any natural, can be expressed as the sum of two perfect squares? [closed]

Prove that for all natural $m$, $5^m$ can be expressed as the sum of two perfect squares. Also, prove that $5^m + 2$ can be expressed as the sum of three perfect squares.
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2answers
121 views

How to derive the formula for the sum of the first $n$ perfect squares? [duplicate]

How do you derive the formula for the sum of the series when $S_n = \sum_{j=1}^n j^2$? The relationship $(n, S_n)$ can be determined by a polynomial in $n$. You are supposed to use finite differences ...
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1answer
93 views

Combining sums and/or differences of squares

I'd like to combine a sum of as many squares as possible into a sum of as few squares as possible. The signs of the squares doesn't matter. For example, the Brahmagupta-Fibonacci Identity combines a ...
0
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1answer
69 views

Find all natural solutions to $x^2+2y^2 = z^2$ [duplicate]

I need to find all natural solutions to $x^2 + 2y^2 = z^2$ What I tried: I did $\pmod 2$ to the equation receiving $z^2 - x^2 \equiv 0 \pmod 2$. Then there are two possibilities: $x^2 \equiv 0 ...
2
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6answers
74 views

find all natural solution that satisfy $x^2+y^2 = 3z^2$

I need to find all natural $x,y,z$ that satisfy the following $x^2+y^2 = 3z^2$ $(0,0,0)$ is an answer of course. What I tried: I tried solving with congruences. I know that every square number ...
5
votes
3answers
74 views

Is it possible to have $a^2 + b^2 = c^2 + 1$ for $a$, $b$, $c$ being coprime integers?

As stated above. I'm working on a possible proof. It appears that $$(b+1)(b-1)=(c+a)(c-a)$$ That's where I'm stuck. Any help please? A clear, simple proof desired, thanks!
1
vote
1answer
47 views

Show that there are infinitely many primes $p$ of the form $p=a^2+b^2+c^2+1$

I know that any prime can be written as the sum of four squares. But I don't know how to know one of these squares is $0$.
5
votes
1answer
113 views

Is it known whether any positive integer can be written as the sum of $n$ different squares?

Is it known whether any sufficiently large positive integer can be written as the sum of four different squares? I know that every positive integer can be written as the sum of four not necessarily ...