# Questions tagged [sums-of-squares]

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

721 questions
Filter by
Sorted by
Tagged with
80 views

24 views

1 vote
57 views

138 views

### How can I generate solutions to the Diophantine equation: $a^2 + b^2 + 2c^2 = d^2$ [closed]

Is there a way to reduce or quickly find integer solutions to the equation $a^2 + b^2 + 2c^2 = d^2$ (where $a, b, c$ and $d$ are distinct natural numbers) ? Sorry I’m really bad at Diophantine ...
390 views

### Parametrization of integer solutions of the equation $a^2+b^2=c^2+d^2=2x^2$

I need the general form of integer solutions to this equation $$a^2+b^2=c^2+d^2=2x^2$$ Here is my partial solution:- The parametrization of the integer solutions of the equation $$p^2+q^2=2y^2$$ is ... 146 views

### Proving that an integer $n = 2^{e_2} \cdot 3^{e_3} \cdots$ can be expressed as the sum of two squares iff $e_p$ is even whenever $p \equiv 3 \pmod 4$

This problem is tripping me up big time. So far, this is what I've got (it's not much): First, prove that if $e_p$ is even whenever $p \equiv 3 \pmod 4$, then $n = 2^{e_2} \cdot 3^{e_3} \cdots$ can be ...
70 views

### What numbers are sums of 2 squares in $\mathbb Z/n\mathbb Z$ (modulo $n$)?

The negative case of the Fermat Christmas Theorem (i.e. that all numbers, in particular primes, that are $3 \pmod 4$ can't be expressed as the sum of two squares) is very quickly proven by seeing that ...
1 vote
52 views

### How to simplify a Summation within a (nested) summation: $\sum_{a=0}^{\ T/2 -1}$ $\sum_{b=2a}^{\ T-1} b*b$

How Would you Simplify a summation with-in a summation, like so $\sum_{a=0}^{\ T/2 -1}$ $\sum_{b=2a}^{\ T-1} b*b$ I honestly have tried numerous approaches to simply the inner part first, but I get ...
55 views

### Does for every $n>1$ exist a sum of $n$ squares of consecutive natural numbers, that equals to the sum of squares of next $n-1$ consecutive numbers?

Can we, for every natural $n>1$, find such a natural number $k$ that sum of $n$ squares of consecutive natural numbers starting with $k$, that is $k^2+(k+1)^2+...+(k-1+n)^2$, will be equal to the ...
205 views

### All natural number solutions to the equation $a^2+b^2=c^2+d^2=2x^2$

Yesterday, I posted this question, and got that if $a$, $b$ and $c$ are in the form $$a=k(m^2-n^2+2mn)$$ $$b=k(n^2-m^2+2mn)$$ $$c=k(m^2+n^2)$$ where $m$ and $n$ are natural numbers, $a$, $b$ and $c$ ... 203 views

### All natural number solutions for the equation $a^2+b^2=2c^2$

$a$, $b$ and $c$ of all Pythagorean triplets can be written in the form $$\begin{split} a &= 2mn\\ b &= m^2-n^2 \\ c &= m^2+n^2 \end{split}$$ where $m$ and $n$ are natural numbers. For ... 151 views

118 views

### For what integers $n$, do we have $30n+11=6x^2+5y^2$ for some integers $x,y$?

I am trying to find all integers $m$ such that $m$ is relatively prime to 30, and $m=6x^2+5y^2$ for some integers $x,y$. Note that we must have: $y$ is odd, $(y,3)=1=(x,5)$. Using these conditions, I ...
165 views

### Why do I get this value?

Can somebody explain this? Why does this happen? Yesterday I was on a popular chat bot and I asked it to make me a code to generate a sequence of numbers. What I wanted, was a script that given a ...
1 vote
70 views

### show (or disprove) that numbers with at most two prime factors can be expressed as a sum of two squares in at most 2 ways

Show (or disprove) that numbers with at most two prime factors can be expressed as a sum of two squares in at most 2 ways. This post is inspired by an answer to this other post, say post A. I found ...
19 views

101 views

### Proof about the properties of the centroid in a finite set?

The whole text was too long to fit in the title. I found this statement without much of a citation or proof: "The sum of the squared distances from every point to the centroid is equal to sum of ...
84 views

### There is a set of numbers whose sum is equal to the sum of the elements squared. What's bigger: the sum of the cubes or the sum of the fourth powers?

Question: There is a set of numbers whose sum is equal to the sum of the elements squared. What's bigger: the sum of the cubes or the sum of the fourth powers? This is a question taken from a set of ...
89 views

### The polynomial $1 + x_1^2 + \dots + x_n^2$ cannot be written as a sum of $n$ rational squares of polynomials.

Artin (1927) settled Hilbert's 17th problem– any nonnegative polynomial can be written as a sum of squares of rational polynomials. Cassels (1964) proved that a if a polynomial admits an SOS ...
I have three timeseries $x_{t1}$,$x_{t2}$ and $x_{t3}$ that occur at different times and are independent. I bootstrap the timeseries and I end up with ca. 2000 distributions per timeseries. As a ...
### Expressing $\sum_{k=1}^3(A_k\cos(x-\alpha_k)-C_k)^2$ in the form $A\cos(x-\alpha)+B$
I have the following sum $$f(x) = \sum_{k=1}^3 (A_k \cos(x - \alpha_k) - C_k)^2$$ which I want to bring into the form $$f(x) = A\cos(x - \alpha) + B$$ I know I can rewrite with the squares the entire ...