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

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19 views

Sum of squares of series of boolean variables

I am going to simplify the following series: $$\sum^4_{v=1} \left(1 - \sum^4_{i=1} x_{v,i}\right)^2 + \sum^4_{i=1} \left(1 - \sum^4_{v=1} x_{v,i}\right)^2$$ Since $x_{i,j}$ is a boolean variable, ...
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1answer
19 views

Maximum length of a representation of a number as an alternating sum of squares

Define a function $$\mathscr R: \mathbb N \to \mathbb N, \ \ \mathscr R(n) = \lceil \sqrt{n} \rceil ^2 -n.$$ IE. the distance of $n$ to the next prefect square. Sequence A068527 on OEIS. If $\mathscr ...
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4answers
106 views

Represent an integer as a sum of n non-consecutive squares

Is it possible to decompose an integer into a sum of n unique squares ,even though they are not necessarily consecutive. For instance, How would I obtain the sequence 1*1 + 7*7 + 14*14 + 37*37 given ...
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1answer
33 views

Evaluation using Sum of Squares?

So I have a problem that looks like this: Evaluate $\displaystyle\sum\limits_{r=1}^{25} 4r^2-2r+2$ using the Sum of Squares: $\displaystyle\sum\limits_{k=1}^n \frac{n(n+1)(2n+1)}{6}$. I really have ...
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2answers
46 views

Show that an integer of the form $8k + 7$ cannot be written as the sum of three squares.

I have figured out a (long, and tedious) way to do it. But I was wondering if there is some sort of direct correlation or another path that I completely missed. My attempt at the program was as ...
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1answer
35 views

Find $2^{2}+4^{2}….(2n)^{2}$?

I tried subtracting $1^{2}+3^{2}….n^{2}$ from $2^{2}+4^{2}….2n^{2}$, but that didn't work. I know the answer is
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2answers
80 views

Find three numbers given their sum, product and sum of their squares

Given three unknown positive integers. Is it possible to find the three numbers if we are given their Sum->(a+b+c) = X Product-> (abc) = Y Sum of Squares-> (a^2 + b^2 + c^2) = Z
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2answers
19 views

Non negativity condition for quartic polynomials?

Say I have a quartic polynomial $f(x) = ax^4 + bx^3 + cx^2 + d$. I am told that $f(x)$ is nonnegative iff it can be expressed as a sum of squares as follows. $f(x) = \sum_{i=1}^4 q_i(x)^2$. As an ...
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3answers
220 views

If a prime can be expressed as sum of square of two integers, then prove that the representation is unique.

If a prime can be expressed as sum of two squares, then prove that the representation is unique. My attempt: If $a^2+b^2=p$, then it is obvious that $a,b$ of different parity. Now, I assume the ...
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2answers
51 views

Problem Solving Question? Sum of the squares

The sum of the squares of two numbers is 247 and the product of the two numbers is 21. How would I find all possible values for the sum of the two numbers?
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122 views

$2017$ as the sum of two squares

Write the prime $2017$ as the sum of two squares $2017$ can be written as the sum of two squares because it is a prime of the form $p\equiv 1\ ($mod $4)$ Using an appropriate algorithm find the two ...
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1answer
86 views

If $3\mid a,b,c$ and $n=a^2+b^2+c^2$, prove that there exist $x,y,z$ such that $n=x^2+y^2+z^2$, where $3\nmid x,y,z$.

If $3\mid a,b,c$ and $n=a^2+b^2+c^2$, prove that there exist $x,y,z$ such that $n=x^2+y^2+z^2$, where $3\nmid x,y,z$. Here $n\in\mathbb N$, $a,b,c,x,y,z\in\mathbb Z$. This problem is originally ...
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1answer
78 views

Fibonacci numbers expressed as squares of lower Fibonacci numbers

I am no mathematician so my apologies for my ignorance. I notice that every number in the Fibonacci series can be expressed as a previous Fibonacci number squared plus or minus (alternating) another ...
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3answers
84 views

Does this equation have positive integer solutions?

The only solution I can find for $a^2 + b^2 + c^2 = d^2 + e^2 + f^2$ is $a=0$, $b=0$, $c=0$, $d=0$, $e=0$, $f=0$. Are there any positive integer solutions? Any where none of $a,b,c,d,e,f$ are ...
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1answer
39 views

Difference of squares - number of representations

There exists a well-known result concerning a number of representations of $n$ as a sum of two squares. Is there anything similar for a number of representations of $n$ as a difference of two squares? ...
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1answer
61 views

Azuma's inequality: Expected sum of differences

I am looking for an extension of Azuma's inequality which involves the expected sum of squared differences. In particular, recall that Azuma's inequality states \begin{align*} \Pr[X_n-X_0 \geq a] \leq ...
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2answers
87 views

squares which are not the sum of a square and twice a triangular number

I'm trying to determine conditions on integer squares which cannot be written as a square and twice a triangular [all numbers positive], i.e. integers $n \ge 1$ where there are no integers $a,b \ge 1$ ...
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4answers
279 views

sum of 4 squares

Is there any natural number $A$ which cannot be written as: $$A=W^2+X^2+Y^2+Z^2$$ where $W,X,Y,Z \in \mathbb N \cup 0$ I was considering the fact that $a^2+b^2 \not = 1 \mod 4$ and was attempting to ...
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2answers
85 views

Prove that no $n,m, 0<n<m$ exist such that $m^2 +mn+n^2$ is a square number

Prove or disprove the claim that there are integers $n,m, 0<n<m$ such that $m^2 +mn+n^2$ is a perfect square.
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0answers
38 views

A four-digit square is of the form $aabb$. What's $a^2+b^2$? [duplicate]

The 5772 Ulpaniada included the following question: Consider a four digit square number (a number which is the square of a whole number).Its digit notation is $aabb$ (the thousands digit is $a$, ...
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1answer
165 views

Algorithm to find solution to $ax^2 + by^2 = 1$ in a finite field

Let $\mathbb{F}$ be a finite field, and let $a,b \in \mathbb{F}$ be given, subject to $a\ne 0, b \ne 0$. Consider the equation $$ax^2 + by^2 = 1.$$ It is guaranteed that there exists a solution to ...
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0answers
33 views

Showing a Hermitian preserving map is not necessarily positivity preserving?

Say I have a linear map, which is not positivity preserving $$\phi: \mathscr H \to \mathscr H$$where $\mathscr H$ is the set of $n \times n$ Hermitian matrices. Then does there exist a positive ...
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1answer
82 views

Weighted sum of squares, in a finite field

Let $\mathbb{F}$ be a finite field. Let $a_1,\dots,a_n \in \mathbb{F}$ be given. I want to know whether there exists $x_1,\dots,x_n \in \mathbb{F}$ such that $$a_1 x_1^2 + a_2 x_2^2 + \dots + a_n ...
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1answer
25 views

Establishing an Upper Bound for a Curious Function

Suppose I have a sequence of positive real numbers $a_1, \, a_2, \, \dots \,a_n$ such that the following is satisfied: $\sum \limits_{i=1}^{n}a_i = 1$ I am trying to find the smallest value of $L$ ...
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0answers
664 views

Prove that the expectation of residual sum of squares (RSS) is equal to $\sigma^2(n-2)$

The assumed regression model is $E(Y_i|x)=\gamma+\beta(x-\bar{x})$ and $Var(Y|x)=\sigma^2$. So I have: $E(RSS)= E(\sum\limits_{i=1}^n (y_i-\hat{y_i})^2)= E(\sum\limits_{i=1}^n ...
4
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1answer
80 views

Are there identities which show that every odd square is the sum of three squares?

I am looking for algebraic identities of the form $$ (2n+1)^2 = f(n)^2 + g(n)^2 + h(n)^2, $$ where the functions are polynomials in $n$.
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1answer
447 views

Dot product of the column vectors from a matrix and their transposes through matrix multiplication

I have a matrix with data, every dataset is a column vector in my matrix. I want to know the dot product of the transpose of each column vector with the original column vector. If I transpose the ...
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4answers
812 views

Can an integer of the form $4n+3$ written as a sum of two squares?

Let $u$ be an integer of the form $4n+3$, where $n$ is a positive integer. Can we find integers $a$ and $b$ such that $u = a^2 + b^2$? If not, how to establish this for a fact?
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5answers
124 views

Relationships between the elements $(a,b,c,d)$ of a solution to $A^2+B^2+4=C^2+D^2$

I have reduced a certain equation (in positive integers) to the equation $$A^2 + B^2 + 4 = C^2 + D^2. \quad(\star)$$ Assume the positive integers $(a,b,c,d)$ are any solution to $(\star)$. Are there ...
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0answers
23 views

How $\sum_{h=1}^{L}\sum_{h=1}^{L}W_h^2S_h^2=(\sum_{h=1}^{L}W_hS_h)^2$?

Suppose a population is divided into $L$ strata. $N_h=$Total number of units in the $h^{th}$ stratum $N=\sum_{h=1}^{L}N_h$ $W_h=$stratum weight and $W_h=\frac{N_h}{N}$ $S_h^2=$stratum population ...
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1answer
300 views

All elements in $\mathbb{Z}/n\mathbb{Z}$ are representable as sum of a square and a cube?

Is it true that all elements in $\mathbb{Z}/n\mathbb{Z}$ are representable as the sum of a square and a cube? Example: ($n=7$) $0 \equiv 0^2+0^3 \left( \text{mod } 7 \right)$ $1 \equiv 1^2+0^3 ...
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1answer
34 views

Arrangement of the following term

How $$\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{2m}}$$ can be rewritten as $$\sum_{k=1}^{\infty}\frac{(-1)^{2k-2}}{(2k-1)^{2m}}+\sum_{k=1}^{\infty}\frac{(-1)^{2k-1}}{(2k)^{2m}}$$ ???
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1answer
201 views

Rabin and Shallit Algorithm

I want to implement Rabin and Shallit algorithm for representing a positive integer as a sum of three squares. Can anyone give me a rough sketch of the method? I searched through the internet but ...
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2answers
131 views

Representing an Integer as a Sum of at Most $k$ Triangular Numbers

What is the smallest $k$ such that every $n \in \mathbb{N}$ can be represented by a sum of exactly $k$ triangular numbers? For the sake of simplicity, I will assume $0$ is a triangular number. I've ...
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4answers
87 views

Proof that $3 \mid \left( a^2+b^2 \right)$ iff $3 \mid \gcd \left( a,b\right)$

After a lot of messing around today I curiously observed that $a^2+b^2$ is only divisible by 3 when both $a$ and $b$ contain factors of 3. I am trying to prove it without using modular arithmetic ...
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3answers
209 views

$ \exists a, b \in \mathbb{Z} $ such that $ a^2 + b^2 = 5^k $

I saw this problem recently and found an elegant solution to it, and was curious to see if anybody would think of something else. Nice solutions to nice problems are fun to see! Problem: Prove ...
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3answers
123 views

Solving a function for square numbers

Essentially I'm curious; could a perfect square($x$ squared) be less than the sum of all lesser perfect squares by a perfect square, and if so, what would the smallest solution be. Take $36$ for ...
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1answer
143 views

Why does sum of squares appear in so many mathematical applications?

I have some little background in statistics, where in many applications summing of squares is an important calculation. Recently, I came across a mention that summing squares is involved in ...
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2answers
143 views

Pythagorean Quadruples:

Consider the set of integers $x_1, x_2, x_3, x_4$ Such that: $$x_1^2 + x_2^2 + x_3^2 = x_4^2$$ How does one compute all the solutions to this system? I have the following method in place for ...
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1answer
310 views

A prime congruent to 3 modulo 4 & sums of squares

Prove: If $p$ is a prime where $p \equiv 3 \pmod{4}$ then $p$ can't be written as the sum of two numbers squared. I attempted by contradiction, supposing that $p=a^2 + b^2$ where $a,b$ are ...
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0answers
124 views

How to prove Sum of squares of $n$ numbers is unique

I was solving a programming problem where I needed to prove that sum of squares of $n$ numbers is unique. i. e. if calculate the sum of squares of equal number of positive integers then if they are ...
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5answers
4k views

calculate sum of square of first n odd numbers

Is there an analytical expression for the summation $$1^2+3^2+5^2+\cdots+(2n-1)^2$$ and how did you derive it?
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0answers
133 views

Optimization: Minimizing Quadratic Vector Valued Functions

I have a vector valued function $F: \mathbf{R}^{n} \rightarrow \mathbf{R}^{m}$, which consists of quadratic taylor approximations. So one could say that function F consists of stacked approximations ...
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2answers
87 views

Solving for $a,b,c,d$ where $a^2 + b^2 + c^2 + d^2 = 630^2$

How could one solve for $a,b,c,d$ where: $$a^2 + b^2 + c^2 + d^2 = 630^2,\ a>b>c>d$$ $a,b,c,d$ squared is equal to the square of $630$, and $a$ is larger than $b$, and so forth. $a,b,c,d$ ...
3
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2answers
127 views

For any positive integer $n$, is it possible to find a nonzero integer $p$ so that $p^2$ is the sum of $i$ nonzero squares for all $1 \leq i \leq n$?

I need to prove this result for something I am working on: For any positive integer $n$, is it possible to find a nonzero integer $p$ so that $p^2$ is the sum of $i$ nonzero squares for all $1 ...