# 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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### Squaring complex number equation with absolute values

I don't understand how you go from the first to the second line in this problem : $$|(a-k)+i(7-2a)|=|(a-2)+i(9-2a)|$$ $$(a-k)^2+(7-2a)^2=(a-2)^2+(9-2a)^2.$$ Firstly, squaring i should make it -1 I ...
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### show that there are arbitrarily long sequences of consecutive positive integers which are not sums of two squares

Let $p_1, p_2, . . . , p_k$ be different prime numbers. By the Chinese Remainder theorem, show that for each $k ∈ N$ there exists an integer n such that $p_1 |n+1; p_2 |n+2; ...; p_k |n+k$ but $p^2_k$ ...
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### Sum of squares and primes

Let $S$ be the set of the integers that can be represented as the sum of two squares. $S={0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, 32, ...}$. I have made this hypothesis: If an integer ...
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### Will iterating the sum of the squares of the digits of any number ultimately give $1$?

I was doing a program where I had to figure out whether the sum of square of digits of a number will ultimately give $1$. For eg: $$68 \;\to\; 6^2+8^2=36+64=100 \;\to\; 1^2+0^2+0^2=1$$ So I wanted to ...
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### sum of square of three squares function $r_3(n)$ II

From sum of square of cube function $r_3(n)$, I believed that, as $r_k(n)$ denotes the number of ways that $n$ is the sum of $k$ squares, $$\sum_{n\le X} r_3(n)^2\asymp X^{3/2}\log^{C}X,...(*)$$ but ...
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### Find the signature of a bilinear form given by a matrix

I'm trying to complete the bilinear form given by the matrix $$M=\left(\begin{array}{ccc}1 & -1 & 2 \\ -1 & 3 & 1 \\ 2 & 1 & 1\end{array}\right)$$ into squares to find the ...
1 vote
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### convergence of sum of squares over squared sum

If we have $\frac{\sum_{i=1}^{n} a^2_i}{(\sum_{i=1}^{n} a_i)^2}$ where $a_i$ is a positive sequence where each element is finite with probability 1, under which set of conditions does the division ...
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### SOSTOOLS in MATLAB: findbound gives wrong output

Here is the MATLAB code that uses SOSTOOLS toolbox: ...
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### Finding a tighter bound given sum and sum of squares

Let $x_1, ..., x_{25}>0$ be such that $\sum_{i=1}^{25}{x_i} = 4350$ and $\sum_{i=1}^{25}{x_i^2} = 757770.25$. From the first equality alone, we know that at least one of the $x_i$'s must be less ...
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### Finding $(a,b)\in\mathbb{N}^2$ such that $\dfrac{a^2+b^2+1}{a+b} \in \mathbb{N}$.

A pair $(a,b)\in\mathbb{N}^2$ is called good if $a < b$ and $$\frac{a^2+b^2+1}{a+b}\in\mathbb{N}.$$ I think I've shown that there are infinitely many good pairs. However, the family of good pairs ...
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### Prove that $\frac{a-c}{k}=\frac{d+b}{n}$, where $N=a^2+b^2=c^2+d^2$ and $k=\gcd(a-c,d-b)$, $n=\gcd(a+c, d+b)$

Let $N$ be odd and $N = a^2 + b^2 = c^2 + d^2$, where $a, b, c, d \in \mathbb{N}$ and WLOG let $a, c$ be odd, $b, d$ be even, $a > c$, and $b < d$. Prove that $\frac{a-c}{k}=\frac{d+b}{n}$. I ...
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### How do I efficiently calculate the total sum with respect to a changing v in a square root?

How do I efficiently calculate the total sum with respect to a changing $\space v \space\space$in a square root $\sqrt{\dfrac {\hbar{((v)/(6.25E34))}}{4G}}r\space\large{?}$ The sum is based on the ...
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### Bound on ratio of sums

I would am interested in upper (or lower) bounds of the following ratio: $$\frac{\sum_{j=1}^k a_j^2}{\sum_{j=1}^k a_j b_j}$$ where $a_j$ are non-negative integers, and $b_j$ are non-negative reals. As ...
1 vote
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### Difference between increasing integer values of $C=x^2+y^2+z^2$

Given are non-negative integer variables $x$, $y$ and $z$. I am trying to deduce the absolute difference between a certain value of $C=x^2+y^2+z^2$ and the very next smallest increase in $C$ possible. ...
1 vote
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### Is it the case that for every $n$, $m$, there is an integer expressible as a sum of $m$ distinct positive integer squares in $n$ distinct ways?

I am not a mathematician, but I learn from various sources that we know that, for every n, there is an integer that can be expressed as a sum of two positive integer squares in n distinct ways, and ...
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### How to show this inequality $a^{14}-a^{13}+a^8-a^5+a^2-a+1>0$

Given the real $a$. Prove that $$a^{14}-a^{13}+a^8-a^5+a^2-a+1>0$$ I tried to factor it as $$(a-1)\Bigl(a^{13}+a^5(a^2+a+1)+a\Bigr)+1$$ I think it should be written as a sum of squares. Any idea ...
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### Is there a prime $a^2+b^2$ for all $a \in \mathbb N$?

I am wondering whether it is a known result whether for every natural number $a \geq 1$, there is at least one natural number $b$ (of any size) such that $a^2+b^2$ is prime. This seems empirically ...
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### For integers $x<y<z$, why are these cases impossible for Mengoli's Six-Square Problem?

I got inspired by this question "Four squares such that the difference of any two is a square?" and rewrote zwim's program that is provided by his answer to the question "Solutions to a ...
1 vote
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1 vote
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### Intuition to faulhabers sum of k-th power of n first integrals

In another post I made, an answer pointed me toward Faulhabers formula for sum of k-th powers of the n first integers. The answer I got of how to reach a formula for $P(n)=\sum_{k=0}^{n}k^2$ looks ...
1 vote
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### Variance of the Square

Suppose $X_1, \cdots, X_n$ are a sample of independent variables taken from a normally distributed population with mean $\mu$ and variance $\sigma^2$. I would like to determine the variance of the ...
1 vote
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### Is there another formula which generates Pythagoras' triples such that the largest $2$ of the triple differ by $3$?

I was thinking about Pythagoras' triples recently, and I wondered if I could find a formula that generated Pythagorean triples such that the largest of $2$ numbers of the triple differ by $1$, and I ...
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### Sums of Squares algorithm

I want to express a large number as the sum of two squares, given that it is possible and given its prime factors. Let's say the number is $273097$. It's prime factors are $11^2, 37$ and $61$. Here ...
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### Proving a Sum Identity for Variance Estimator

I'm trying to prove the following identity: for a set of numbers: $$x_1, ... ,x_n$$ $$\frac{1}{n(n-1)} \sum_{i=1}^n (x_i- \bar x)^2 = \bar x^2 - \frac{1}{n(n-1)} \sum_{i \neq j} x_ix_j$$ but I cant ...
If $\sum_{i = 1}^n a_i = x$ for $a_i \geq 0$, then is it possible to find upper bound to $\sum_{i=1}^n a_i^2$? I know that the lower bound can be easily determined using Cauchy- Schwarz Inequality. ...
I have the equation $(4n+3)^2-48m^2=1$ that I changed into $X^2-48m^2=1$ so that it would be a Pell's equation, that has infinitely many solutions, that I found being \$\left\{\begin{align} x_k = \...