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

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1answer
12 views

Showing Residual Sum of Squares for Multiple Linear Regression is 0

Problem: I have the linear regression model: $y_i=\beta_0+\sum_{k=1}^p \beta_kx_{ik}+\epsilon_i$ where $\epsilon_i\sim N(0,\sigma^2)$, for $i = 1,2,\ldots ,n$. I want to prove that the residual sum ...
8
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3answers
267 views

Find the smallest positive integer which can be written as the sum of the squares of two positive integers in two different ways

Find the smallest positive integer which can be written as the sum of the squares of two positive integers in two different ways. I took extremely long to solve this I got $50= 7^2 + 1^2 $ $50= ...
2
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1answer
38 views

Algorithm for finding the representation of an integer as a sum of two squares

We know that an integer $n$ is the sum of two squares if and only if all its prime divisors $p$ of the form $p \equiv 3 \pmod4$ have an even exponent in the prime factor decomposition of $n$. My ...
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1answer
97 views

showing SSE of simple regression model is larger than or equal to SSE of multiple regression model

Lets say we have 2 linear regression models: $y_i = B_0 + B_1x_{i1} + \epsilon_i,$ where $\epsilon_i$ follows $N(0,σ_1^2)$ $y_i = B_0 + B_1x_{i1} + B_2x_{i2} + \lambda_i,$ where $ \lambda_i$ follows ...
2
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1answer
53 views

Proof of Lagrange's four square theorem using Cauchy-Davenport Theorem

Cauchy used the Cauchy-Davenport theorem to prove that $ax^2 + by^2 + c \equiv 0 \pmod p$ has solutions provided that $abc \neq 0$. Lagrange used this result to establish his four squares theorem. I ...
2
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1answer
77 views

Is there any formula to calculate the number of different Pythagorean triangle with a hypotenuse length $n$, using its prime decomposition?

Lets define $N(n)$ to be the number of different Pythagorean triangles with hypotenuse length equal to $n$. One would see that for prime number $p$, where $p=2$ or $p\equiv 3 \pmod 4$, $N(p)=0$ also $...
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2answers
66 views

prove if n - natural number divide number $34x^2-42xy+13y^2$ then n is sum of two square number

prove if n - natural number divide number $34x^2-42xy+13y^2$ where x,y are relatively prime then n is sum of two square number. I don't know what is going on in this exercise. I will be grateful ...
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0answers
40 views

Additive basis of order n: Sets which allow every integer to be expressed as the sum of at most n members of that set. [closed]

Every integer can be expressed as the sum of at most 3 triangular numbers. That is, the set of triangular numbers is an additive basis of order 3. The sum of the inverse triangular numbers is 2. (1/1 +...
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3answers
63 views

Show that if $n\equiv 3, 6 \pmod9 $ then $n$ is not a sum of two squares

Show that if $n\equiv 3, 6 \pmod9 $ then $n$ is not a sum of two squares. I started by: Assume $n=a^2+b^2$ a sum of two squares. Then $a^2,b^2\equiv 0,1,4,7 \pmod9$, and no combination these numbers ...
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2answers
53 views

Student test statistic and self normalizing sum.

I study the asymptotic distribution of self normalizing sums which are defined as $S_n/V_n$ where $S_n=\sum_{i=1}^n X_i$ and $V_n^2 = \sum_{i=1}^n X_i^2$ for some i.i.d RV's $X_i$. Motivation ...
3
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0answers
18 views

Is there a theory of “sums-of-squares residues”?

The theory of quadratic residues is long- and well-studied. Recall that, [somewhat simplified] if $x,a,b$ are integers, with $0 \le a < b$, such that $$x \equiv a^2\!\!\!\pmod{b},$$ then we say ...
4
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0answers
91 views

Probability of having at least $j$ collisions when tossing $m$ balls into $n$ bins

Suppose that we throw $m$ balls into $n$ bins uniformly and independantly at random. We consider collisions as distinct unordered pairs, e.g., if 3 balls are tossed in one bin, we count 3 collisions. ...
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3answers
46 views

Solving $rX_1^2+sY_1^2+tZ_1^2=rX_2^2+sY_2^2+tZ_2^2$ completely in integers

Given pairwise relatively prime integers $r,s,t$, I’m looking for a complete solution (i.e., integer parameterization or similar) for the Diophantine equation $$ rX_1^2+sY_1^2+tZ_1^2=rX_2^2+sY_2^2+...
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2answers
32 views

Pattern in digits of sums of consecutive squares

I am interested in patterns in square numbers as well as the reasons behind them and I can't seem to figure out (also how to prove) why do the sums of two consecutive squares only end in digits 1, 3 ...
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1answer
60 views

Prove that $4n+2=x^2+y^2+z^2$ for some odd $x,y$ and even $z$

Show that for all $n\in \mathbb{N}$, exists $x,y,z \in \mathbb{N}$, such that $x,y$ are odd and $z$ is even, such that $4n+2=x^2+y^2+z^2$. I started by using the fact that every natural number has a ...
3
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0answers
70 views

$m^2+n^2$ and $m^2-n^2$ cannot both be squares [duplicate]

I need to show that there aren't any $m$ and $n$ such that $m^2+n^2$ and $m^2-n^2$ are both squares. First of all, assume without loss of generality that $m$ and $n$ are co-prime, since otherwise we ...
2
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3answers
85 views

Show that $\forall n\in\mathbb{N}$, $14^n$ can be represented as a sum of three perfect squares.

Show that $\forall n\in\mathbb{N}$, $14^n$ can be represented as a sum of three perfect squares. I checked $(\mod 7)$ and deduced that the three squares can be $1,4,2(\mod 7) $ or all divisible by $7$...
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3answers
34 views

What does the algorithm s = s + k * k do?

I just finished an exam in my math class and I did well except for one question that I just can't get out of my head, it seems simple but I just can't figure it out: PROBLEM: ...
4
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2answers
282 views

Sum of squares of integers divisible by 3

Suppose that $n$ is a sum of squares of three integers divisible by $3$. Prove that it is also a sum of squares of three integers not divisible by $3$. From the condition, $n=(3a)^2+(3b)^2+(3c)^2=9(a^...
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0answers
29 views

Algorithm Identification

Background I'm currently working with a system that has a 4-dimensional function. Currently, an algorithm is used to speed up calculation of the final value via interpolation, and two of the ...
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3answers
138 views

My formula for sum of consecutive squares series?

I stumbled upon a specific series, who's Sum of squares of consecutive integers equals the sum of squares of the continuation of that consecutive integers. For exmaple, this first number in the ...
4
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2answers
65 views

Why is a polynomial $f(x)$ sum of squares if $f(x)>0 $ for all real values of $x$?

If a polynomial $f(x)>0 $ for all real values of $x$, then $f(x)$ is sum of squares. Why is this true ? I understand that the roots of this $f(x)$ will be complex and hence will exist as ...
4
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1answer
125 views

For which polynomials $f$ is the subset {$f(x):x∈ℤ$} of $ℤ$ closed under multiplication?

You surely know about the Brahmagupta–Fibonacci identity, $$(a_1^2 + b_1^2)(a_2^2 + b_2^2) = (a_1a_2 \pm b_1b_2)^2 + (a_1b_2 \mp a_2b_1)^2$$ which tells us that the product of two numbers, each of ...
2
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1answer
51 views

variant of Lagrange's four square theorem using a restricted set of squares

The well-known four square theorem states that any positive integer is the sum of at most four squares. Suppose that, instead of allowing all squares, I only consider the following set of squares: $$ ...
2
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4answers
76 views

Triangular Numbers and Sum of Two Squares

"If n is a triangular number, show that each of the three consecutive integers, $8n^2, 8n^2+1, 8n^2+2$ can be written as a sum of two squares." I have spend hours working on this problem and cannot ...
1
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1answer
24 views

Maximize sum of squares

Lets say that I know that $n$ values $x_i$ sums up to $\mu$: $$ \mu=\sum_{i=1}^n x_i $$ I also now that $0\leq x_i\leq 1$ for all $i=1\cdots n$. I want to find an upper bound as tight as possible ...
0
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1answer
33 views

Upper bound for sum of squares given mean of elements

I need upper and lower bounds as tight as possible for the following expression of the elements of a $n$ x $n$ matrix: $$ \sum_{i,j}\rho_{ij}^2-\frac{2}{n}\sum_i(\sum_j\rho_{ij})^2+\frac{(\sum\rho_{ij}...
4
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2answers
97 views

Show that $x^2 + y^2 + z^2 = x^3 + y^3 + z^3$ has infinitely many integer solutions.

Show that $x^2 + y^2 + z^2 = x^3 + y^3 + z^3$ has infinitely many integer solutions. I am not able to find an idea on how to proceed with the above questions. I have found only the obvious solution $...
0
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3answers
84 views

If $n = a^2 + b^2 + c^2$ for positive integers $a$, $b$,$c$, show that there exist positive integers $x$, $y$, $z$ such that $n^2 = x^2 + y^2 + z^2$.

If $n = a^2 + b^2 + c^2$ for positive integers $a$, $b$,$c$, show that there exist positive integers $x$, $y$, $z$ such that $n^2 = x^2 + y^2 + z^2$. I feel that the problem basically uses algebraic ...
0
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0answers
37 views

If $p=x^2+y^2$ is a prime number, then $\left( \frac{x+y}{p} \right) = \left( \frac{2}{x+y} \right) $

Let $p=x^2+y^2$ be a prime number. How to prove that $\left( \dfrac{x+y}{p} \right) = \left( \dfrac{2}{x+y} \right) $ (where $\left(\frac ab\right)$ denotes the Jacobi symbol)?
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1answer
34 views

Relatively Prime Cases of the Brahmagupta-Fibonacci Identity

The Brahmagupta-Fibonacci identity states that the set of the sum of two squares is closed under multiplication. $$(u^2 + v^2)(A^2 + B^2) = (uA \pm vb)^2 + (vA \mp uB)^2$$ This is easy to verify as \...
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2answers
39 views

Sums of squares minus square of sums

I have the following equation in a statistics textbook and cannot see how the right side comes into being. $$\frac{1}{n} \sum_{i=1}^n x^2_i - \left(\frac{1}{n} \sum_{i=1}^n x_i\right)^2 = \frac{1}{n} ...
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5answers
72 views

Why is the square root of a sum not equal to the square root of each its addends?

Example: Let's presume one was attempting to isolate m below: A common mistake would be: $k^2 = m^2 + n^2 \to k = m +n$ Even though: $k^2 = m^2 + n^2 \to k \neq m +n$ If you apply a square root to ...
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1answer
42 views

What is the relation between the square root of the sum of squares and the sum of the absolute values?

I want to prove that $\sqrt{\sum a_{i}^{2}} \geq \sum \left | a_{i} \right |$, is it possible ?
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1answer
46 views

Prove that there are arbitrarily long sequences of consecutive integers, none of which can be written as the sum of two perfect squares.

Prove that there are arbitrarily long sequences of consecutive integers, none of which can be written as the sum of two perfect squares. First few numbers are $3,6,7,11,12,14,15,19,21,22,23,24,27,28,...
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1answer
50 views

Let $n$ be a positive integer. Show that $n$ and $2n$ have the same number of representations as a sum of two squares of non-negative integers. [closed]

Let $n$ be a positive integer. Show that $n$ and $2n$ have the same number of representations as a sum of two squares of non-negative integers. Hint: $(a^2+b^2)(c^2+d^2)=(ac+bd)^2+(ad-bc)^2$
3
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1answer
64 views

Sum of Consecutive Perfect Squares

If $n\in\mathbb{Z}$ and $2n+1$ is a perfect square, then is it true that $n+1$ is a sum of two consecutive perfect squares?
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3answers
93 views

Write $x_n=22..244…45$ as sum of $2$ squares

I've recently came across this problem and, although I've spent time looking for a solution, I don't have any interesting ideas. Let the numbers $$x_1=25$$ $$x_2=2245$$ $$x_3=222445$$ and ...
0
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1answer
62 views

Prove that for any prime $p$ there exist natural numbers $a,b$ for which $ p$ divides $a^2+b^2+1$ [closed]

Prove that for each prime $p$ there exist natural numbers $a,b$ for which $p$ divides $a^2+b^2+1$
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1answer
20 views

Hurwitz's matrix equations

I have a question about the proof of Hurwitz's 1-2-4-8 theorem about the sum of squares. I have consulted Chapter 1 of Rajwade's "Squares" book, notes by Keith Conrad, and notes by Daniel Shapiro. ...
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1answer
39 views

Can a number of the form $n = {2^r}{b^2}$ always be represented as a sum of two squares, if $r \geq 1$ and $b$ is an odd composite?

Can a number of the form $n = {2^r}{b^2}$ always be represented as a sum of two squares, if $r \geq 1$ and $b$ is an odd composite? More generally: Can a number of the form $n = {2^r}{b^2}$ ...
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2answers
67 views

Fermat's eleventh $F_{11}$ represented as sums of two squares?

There's an unanswered quest in another forum, that makes me sleepless: Fermat's Elevnth Let $a,b \in \mathbb{N^{+}}$ and $a>b$ then, there are $k$-representations for $F_{11}=2^{2^{11}}+1$ as sum ...
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2answers
33 views

Non negativity condition for polynomials?

Let $p(x)=3ax^2+2bx+(1-a-b), \,\,\,\,0<x<1$ What are the conditions on a and b to $p(x)>0$ Could someone give me a hint ?
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4answers
62 views

If a prime $p>2$ is expressible as $p=a^2+b^2$, then $4\mid (p-1)$

Show that if $p\in\mathbb{P}, p>2$ is a sum of squares, i.e $p=a^2+b^2$, then $p\equiv 1\pmod{4}$. Have established that the remainder can't be either of $0,2$ for obvious reasons. So we must ...
1
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1answer
28 views

Pythagoras number of $\mathbb{F}_p$

For a commutative ring $A$ and $a \in A$, define the length of $a$ as $$ l(a) = \inf \lbrace n \in \mathbb{N} \mid \exists a_1, \ldots, a_n \in A : a = \sum_{i=1}^n a_i^2 \rbrace . $$ Let $\Sigma A^2$ ...
0
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1answer
20 views

$a =\prod_{i = 1}^{r} p_{i}^{ai}$ with $a_{i} > 0$ for each $i$ is the canonical representation of $a$…

$a =\prod_{i = 1}^{r} p_{i}^{ai}$ with $a_{i} > 0$ for each $i$ is the canonical representation of $a$, deduce a formula for the sum of the squares of the positive divisors of $a$. -The section we ...
7
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2answers
197 views

Chinese New Year Equation 2016

In the spirit of Chinese New Year, here's a problem to commemorate the year. $\color{black}{\text{Solve the following equation for positive integers $a$ and $b$:}}$ $$\color{red}{a^2+b^2+(a+8)^...
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1answer
35 views

Manipulating a sum and showing it has an upper bound

Let $x_1, \ldots ,x_n$ and $y_1, \ldots , y_n$ for $n \in \mathbb{N}$ be complex numbers such that $$\left| \sum_{k=1}^n x_k y_k \right| \leq 1. $$ Further I know that $$ \sum_{k = 1}^n |x_k|^2 \leq ...
0
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0answers
29 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? ...