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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0answers
35 views

Can someone show me a simple example of finding simple squared error in k-means?

I'm working on some homework and one of our assignments asks us to find the final mean of each cluster and the SSE (sum of squared error). I haven't been able to find anything online and was wondering ...
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1answer
38 views

Composition formula for $ax^2 \pm by^2$

It is well known that $(x^2+y^2)(u^2+v^2)=(xu+yv)^2 + (xv-uy)^2$, thus it suffices to characterize the primes $p$ with $x^2+y^2=p$. Are there any similar composition formulae for $ax^2+by^2$ and $ax^2-...
7
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3answers
1k views

Every number is the sum of three squares with signs [duplicate]

The question. Can every $n\in \mathbb N$ can be written: $$n=a^2\pm b^2\pm c^2$$ where $\pm$ are signs of your choice? We know with Lagrange's four-square theorem that every integer can be ...
1
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3answers
235 views

The prime power decomposition of sum of two squares

Show that if an integer is a sum of two squares: $n=x^2+y^2$, then in the prime power decomposition of $n$, all primes $p = 3 \pmod{4}$ appear with even exponents : For $p = 3 \pmod{4}$, if $p^k \...
3
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0answers
80 views

Express Integer as the sum of $k$ squares

I'm looking for a way to efficiently determine the solutions ( i.e. sets of $x_1^2, x_2^2, ..., x_k^2 $ ) that satisfy     $$n^2 = x_1^2 + x_2^2 + x_3^2 + ... + x_k^2 $$ $$x_1, x_2, ...,...
1
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1answer
125 views

Could be this : $7131372917538397234773191167617941438959$ written as $x^{2}+y^{2}$ with $x, y$ integers? [duplicate]

I have constructed the number $7131372917538397234773191167617941438959$, which is prime as shown here, using all primes under $100$ which contains two digits by randomly ordering them as $31,37,29,\...
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3answers
196 views

How can I prove the sum of squares and the sum of cubes with binomial coefficients? [closed]

My main problem is starting. I can't "see" anything that might give me an idea to find a relationship between these two things Thank you :)
3
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1answer
121 views

Prove or disprove a claim involving Pythagorean primes

As we know, every Pythagorean prime $p$ is expressible as $p=x^2+y^2$ such that $x$ is an odd integer and $y$ is an even integer. Prove or disprove: $x$ is a quadratic residue modulo $p$.
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1answer
109 views

Remainder of $\sum_{x=1}^{312} x \times x!$ divided by 2016 [closed]

I have this question: What is the remainder of $$\sum_{x=1}^{312} x \times x!$$ (or just simply) $$(1! \times 1) + (2! \times 2) + (3! \times 3) + \dots + (312! \times312)$$ Divided by ...
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1answer
98 views

Two times the sum of two squares is a sum of two squares

For all $a,b\in\mathbb Z$ there are $c,d\in\mathbb Z$ such that $2(a^2+b^2)=c^2+d^2$. The conjecture is tested for $a^2+b^2<1,000,000$ but I have problems with proving it.
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2answers
4k views

Find the value of: $\frac{\sqrt{45} + \sqrt{18}} {\sqrt{7+2\sqrt{10}}}$ [closed]

Find the value of:$$\frac{\sqrt{45} + \sqrt{18}} {\sqrt{7+2\sqrt{10}}}$$ I need help solving this question. Every reply is appreciated. Thanks!
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2answers
84 views

Find the value of $\frac {1} {1^2 + 1} +\frac {1} {2^2 + 2} +\frac {1} {3^2 + 3} +\frac {1} {4^2 + 4} ++ \dots + \frac {1} {2008^2 + 2008} $

I have this question: Find the value of: $$\frac {1} {1^2 + 1} +\frac {1} {2^2 + 2} +\frac {1} {3^2 + 3} +\frac {1} {4^2 + 4} ++ \dots + \frac {1} {2008^2 + 2008} $$ My attempt: I ...
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0answers
389 views

A simple algorithm for representing a square of the form $4k+1$ as a sum of 3 squares

Let's start with an example, $N=n^2=7^2=49$. $49$ can also be produced by simply writing $49= (6+1)^2 = 36 + 2\cdot6 + 1 = 36 + 13$. So if we can find a way to represent 13 as a sum of two squares, we ...
0
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0answers
60 views

A method to study the factorization of very big numbers

Is there a method to construct very big numbers in a given interval, with rectangular distribution, by selecting prime factors randomly? I want to study factorizations of the kind $a=b\cdot c$, where ...
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3answers
91 views

Is $7m^2-3n^2$ a perfect square?

Is $7m^2-3n^2$ a perfect square for all positive integers $m,n$? I tried using double induction, but failed. Any other approach? By the way, is this related to fermat's theorem on representation of ...
1
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0answers
167 views

Sum of two squares of an integer N, the simplest algorithm?

The algorithm is based on the following observation. An integer $b>a$ can be written as $b=a + k$ with a and k integers. So if we want to find a sum of 2 squares of an integer N, we start by ...
0
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1answer
211 views

About square free semiprimes which are sums of two squares

This conjecture seems to be true: If the product of two different primes is a sum of two squares, the so are the primes. I've tested it for $pq<1000$, but would like to see a proof. Edit: If $p\...
11
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3answers
834 views

Show that $x^2+y^2+z^2=999$ has no integer solutions

The question is asking us to prove that $x^2+y^2+z^2=999$ has no integer solutions. Attempt at a solution: So I've noticed that since 999 is odd, either one of the variables or all three of the ...
3
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0answers
108 views

A surprising link between sum of squares and vector product [closed]

When we look at the two smallest numbers $n$ such that $n$ can not be written as the sum of $3$ squares, we get $7$. And we know that there exists a vector product on $\mathbb R^n$ if, and only if $n\...
2
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1answer
41 views

Find all prime numbers $y$ and $z$ if $x$ is an integer and $x^2-y^2-z^2=2017.$

Find prime numbers $y$ and $z$ if $x$ is an integer and $x^2-y^2-z^2=2017.$ $x^2-y^2-z^2$ is not factorable, so what should I do? My mind is blank here. Any help is appreciated.
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2answers
16k views

Value of: $ \frac {1} {\sqrt {4} + \sqrt {5}} + \frac {1} {\sqrt {5} + \sqrt {6}} + … + \frac {1} {\sqrt {80} + \sqrt {81}}$ [closed]

I have the following question: Find the value of: $$ \frac {1} {\sqrt {4} + \sqrt {5}} + \frac {1} {\sqrt {5} + \sqrt {6}} + ... + \frac {1} {\sqrt {80} + \sqrt {81}}$$ The book ...
1
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2answers
38 views

Summation formula proof

For $k$ and $l$ $\in n$ how can I prove the following : $$(\sum_{k=1}^{k=n}\pi_{k}Y_{k}-\sum_{k=1}^{k=n}Y_{k})^2=\sum_{k=1}^{k=n}\sum_{l=1}^{l=n}Y_{k}Y_{l}(\pi_{k}-1)(\pi_{l}-1)$$ . I have really ...
1
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1answer
285 views

Polynomial equal to sum of squares of polynomials [duplicate]

Given a nonnegative polynomial $p(x)$ on $\mathbb{R}$, does there exist some $k$ such that for some polynomials $f_1,\ldots ,f_k$ we have $p(x)=\sum_{i=1}^k(f_i)^2$? I think yes, because of the ...
4
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2answers
960 views

What Is The Sum of All of The Real Root

I found this question on my test: What is the sum of all of the real root of $x^3-4x^2+x=-6$? A.) $-4 $ A.) $-2$ A.) $-0 $ A.) $2 $ A.) $4 $ My answer: $...
2
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3answers
767 views

Divisibility of the sum of squares of two odd numbers

$A=m^{2}+n^{2}$ with $m$ and $n$ both being odd numbers. We need to find out of this integer is divisible by $2,4$ or $6.$ I don't know how to start solving this. Square of any odd number will have ...
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0answers
50 views

Given A number N conver it to 2 square form

I have a number N i want to subtract square of x from N , where x is any integer ...
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0answers
136 views

Primes as sum of squares in finite field

Is there an algorithm to find integers such that the following holds true : ${a_1}^2 + {a_2}^2 \dots + {a_k}^2 \equiv p $ $(mod $ $n)$, where $p$ is a prime and $n$ is any general integer. By ...
0
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1answer
186 views

Algorithm to Divide the number into 4 square part

I am having a number N i want to divide this number into sum of square contains upto 4 numbers. ...
1
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1answer
69 views

Looking for a tight upper bound of a sum

I try to upper bound tightly the following sum: $$\sum_{i=1}^N \left(\frac{1}{\sqrt{i}} - \frac{1}{\sqrt{i+1}}\right)i$$ I expect something which is asymptotically close to $\sqrt{N}$. Thank you ...
0
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1answer
166 views

Find positive integer solutions of the Diophantine equation $x^4+py^4=z^2$.

I'm Trying to find all integer solutions of the diophantine equation: $$x^4+py^4=z^2,$$ where $p$ is a prime number $p\equiv 13 \quad or\quad 17 \quad (\mod 20)$. I know that $y=0$ is a solution of ...
0
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1answer
60 views

Integer as plus/minus combination of three squares

Some time ago I came across the following statement in a paper: "Every integer $k$ has a representation of the form $k=\pm a^2 \pm b^2 \pm c^2$" Unfortunately I can't remember where I read it, just ...
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0answers
66 views

Density and SOS polynomials

Is the set of sum of square (SOS) polynomials dense (in a sense to precise) in the set of non negative polynomials of degree less than $d$? I don't even know how to ask a well posed question... here ...
2
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3answers
110 views

Alternating binomial sum over even coefficients.

Given a positive integer $n$, I'm looking for a nicer closed form for the expression $$\sum_{\substack{k=0\\2\mid k}}^n(-1)^{\frac k2}2^k\binom{n}{k}.$$ If it helps, it is fine to assume that $n$ is ...
2
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0answers
848 views

On the number of ways of writing an integer as a sum of 3 squares using triangular numbers.

It is well known that integers can be represented as a sum of squares, two, three and more. In what follows will be given a way to represent integers as a sum of 3 squares using triangular numbers. It ...
4
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11answers
10k views

Natural number which can be expressed as sum of two perfect squares in two different ways?

Ramanujan's number is $1729$ which is the least natural number which can be expressed as the sum of two perfect cubes in two different ways. But can we find a number which can be expressed as the sum ...
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2answers
43 views

Squares difference simplifications

I need to simplify this expression $(a+b+c)^4-(a+b)^4-(a+c)^4-(b+c)^4+a^4+b^4+c^4$ Calculating each one will lead to over 30 different items.I observe that the final form should have a,b,c,d at the ...
1
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1answer
64 views

Find all decompositions of 2K² as sum of squares: possible reduction of the problem

I have to deal with a situation where I am trying to decompose numbers written as $2K^2, K\ge0$ into sums of squares, i.e. find: $$S_K=\{(x, y)\ |\ x^2+y^2 = 2K^2\}$$ One of the obvious solutions is $...
1
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2answers
2k views

Proof that sum of squares of error for simple linear regression follows chi-square distribution

I can understand that if Y1~Yn are random samples from N(μ,σ), then the sum of squares of difference between Yi and bar(Y) divided by sigma^2 follows chi-square distribution with n-1 degress of ...
3
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1answer
495 views

Upper bound on sum of square of integers

I have $n$ non-negative integers $x_1, \dotsc, x_n$ which satisfy the constraint $\sum x_i = S$ I want to derive a bound on $\sum x_i^2$. An easy bound can be calculated as: $\sum x_i^2 \le (max_{...
1
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3answers
52 views

Is there a simple way to get the numbers of squares that can fit into an integer N in descending order starting with sqrt(N)?

To make things clear, consider the example of $$N=143= 11^2 + 4^2 + 2^2 + 1^2 + 1^2$$ Is there a way to predict how many squares will add up to N=143 in descending order without finding the squares if ...
0
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1answer
64 views

How to prove that there is No solution

Let $d$ be a square-free positive integer $d>1$. Then there are no integers $x,y,z,t,a,b,c$ with $x\neq \pm z$, and $xt-yz\neq 0$ such that: $$\begin{cases} x^2+y^2=a^2 d \\ z^2+t^2=b^2 d \\ (x-z)^...
2
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1answer
99 views

if m and n are sum of squares then so is $\frac{m}{n}$

prove that if $m$ and $n$are both sum of squares, and $n|m$ then $\frac{m}{n}$ is also the sum of squares. I tried to consider prime divisors of m and n and distinct between $p\equiv 1 \pmod 4$ and $...
10
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4answers
632 views

Find all $x,y,z$ such that $x^2 + y^2 + z^2 = 3^{10}$

By Legendre's 3-squares theorem, a number $n = x^2 + y^2 + z^2$ can be written as the sum of three squares if $n \neq 4^a(8b+7)$. In my case, I am choosing $$n = 3^{10} \equiv (3^2)^5 \equiv 1 \mod 8$...
1
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1answer
76 views

For what sequences of consecutive perfect squares would the sum be a perfect square? [duplicate]

For example, the sum of the sequence of perfect squares from 1^2 to 24^2 = 70^2. What other sequences (of let's say length >= 5) of some starting n^2 and ending m^2 would sum to a perfect square?
2
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8answers
330 views

Finite sum $\sum_{j=0}^{n-1} j^2$

How can I calculate this finite sum? Can someone help me? $$\sum_{j=0}^{n-1} j^2$$
2
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2answers
58 views

Can someone help me with sum of this expression:$ (n-j)+(n-j)^2?$

Can someone help me prove this transition?
1
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3answers
54 views

How can I prove that the numbers whose sum is $ 2n$ and sum of squares is $4n$ are unique?

I know that sum$(a_i)$ = $2n$ and sum($a_i^2)$ = $4n$ and $a_i$ is any natural number > 0. The sums are from 1 to n so they have exactly n terms. I want to prove that any $a_i$ must be 2 in these ...
1
vote
1answer
176 views

Fibonacci primes [closed]

Suppose that $ F_{n} $ is a Fibonacci prime such that $F_{n}+F_{n+1}$ is itself a Fibonacci prime. Is ${F_{n}}^2+{F_{n+1}}^2 $ a third Fibonacci prime?
2
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0answers
61 views

Definite sum of a decreasing series

can we find a closed form for following sum: \begin{equation} \sum_{i=0}^{n} a^{i^2} = a^0 + a^1 + a^4+a^9 +...a^{n^2} \end{equation} where $0<a<1$ , what if $n \rightarrow \infty$ ?
1
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4answers
60 views

A difficult question on representing a square by a sum of 4 (or 5 ) squares

The question above is not about the decomposition of a square $$m^2 = a^2 + b^2 + c^2 + d^2 $$ as a sum of 4 or more squares. We know it is always possible and we also have algorithms to do that. ...