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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Sums of squares using the circle method
Let's say we want to count solutions to $$N=n_1^2+...+n_4^2$$ using the circle method, so we write the number of solutions as an integral $$\int _0^1S(\alpha )^4e(-\alpha N)d\alpha \hspace {15mm}S(\...
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Sum of square of parts, and sum of binomials over integer partition
Let $n$ be positive integer. Consider its integer partitions denoting as $(m_1,\cdots,m_k)$, where $m_1+\cdots+m_k=n$ and the order does not matter.
I am interested in the following two quantity
(1) $$...
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On divisors of $N$ appearing in the set of curvatures of an Apollonian gasket
Let $A$ be an Apollonian gasket generated by the quadruple $(k_1, k_2, k_3, k_4)$. The curvatures of the circles that form the gasket satisfy Descartes Theorem:
$$
(k_1+k_2+k_3+k_4)^2 = 2(k_1^2 + k_2^...
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What is min $T=\sqrt{\frac{1-bc}{2b^2+5bc+2c^2}}+\sqrt{\frac{1-ac}{2a^2+5ac+2c^2}}+\sqrt{\frac{1-ba}{2b^2+5ba+2a^2}}$
Find the minimal value $$P=\sqrt{\frac{1-bc}{2b^2+5bc+2c^2}}+\sqrt{\frac{1-ac}{2a^2+5ac+2c^2}}+\sqrt{\frac{1-ba}{2b^2+5ba+2a^2}}$$
when $a,b,c\ge 0: ab+bc+ca=1.$
I set $a=b=1;c=0$ I got $P=\sqrt{2}.$
...
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How to find minimum $T= a+b+(a-b)(c-b)$ for $ab+bc+ca=3.$
Let $a \ge b\ge c\ge 0$ such that $ab+bc+ca=3.$ Find minimum $$T= a+b+(a-b)(c-b).$$
For $a=b=c=1,$ T get minimal value is equal to $2.$ Hence, I tried to prove $a+b+(a-b)(c-b)\ge 2.$
My ugly proof ...
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How to prove $\frac{\sqrt{ab+bc+ca}}{2}\ge\frac{a^3(b+c)+b^3(c+a)+c^3(a+b)}{(a^2+b^2+c^2)(a+b+c)-2abc}.$?
If $a,b,c\ge 0: a+b+c,>0$ then prove
$$\frac{\sqrt{ab+bc+ca}}{2}\ge\frac{a^3(b+c)+b^3(c+a)+c^3(a+b)}{(a^2+b^2+c^2)(a+b+c)-2abc}.$$
I'm looking for a simple proof which student could full it in ...
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On Cauchy's lemma and Lagrange four squares representation
Cauchy’s Lemma. Let $a$ and $b$ be odd positive integers satisfying $b^2\lt4a$ and $3a \lt b^2+2b+4$. Then, there exist non-negative integers $s,t,u,v$ such that $a=s^2+t^2+u^2+v^2$ and $b=s+t+u+v$.
...
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Sums of squares of relatively prime natural numbers
Let $x, y, z, t$ be natural pairwise relatively prime numbers such that
$$xy+yz+zt = xt$$
Prove that the sum of squares of two of them is exactly twise the sum of squares of the other two numbers.
I ...
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Algebraically why must a single square root be done on all terms rather than individually?
Let's assume we know that $x+9=10$.
I understand this is illegal:
$$\sqrt[]{x} + \sqrt[]{9} = \sqrt[]{10}.$$
And this is correct:
$$\sqrt[]{x + 9} = \sqrt[]{10}.$$
Is there an intuitive way to ...
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Sequences $(a_n)$ such that $i^2+j^2=k^2+\ell^2\Longrightarrow a_i+a_j=a_k+a_{\ell}$
In trying to solve the problem posed in this old message, I asked myself the following question :
Which sequences $(a_n)_{n\in\mathbb{N}}$ (with $\forall n\in\mathbb{N}\,,\,a_n\in\mathbb{Z}$) satisfy ...
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Which are all polynomials in $\mathbb{F}_p[x]$ which are a sum of two squares?
Let $p$ be a prime. Determine all polynomials $f(x) \in \mathbb{F}_p[x]$ for which there exist polynomials $A(x), B(x) \in \mathbb{F}_p[x]$ such that $f(x) = A(x)^2 + B(x)^2$.
If we were working with $...
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The maximum weight in a weighted sum of Fibonacci squares representation of a positive integer
Theorem. Every positive integer can be represented as a weighted sum of Fibonacci squares.
Proof. Start with a postive integer $N$. Obtain its unique sum of non-consecutive Fibonacci numbers ...
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Numbers with a unique partition as a sum of two squares
The well known Ramanujan tau function $\tau(n)$ is defined as the nth Fourier coefficient of the modular discriminant
$\displaystyle \Delta(q)=q\prod_{m=1}^\infty (1-q^m)^{24} = \sum_{n=1}^\infty \tau(...
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Is a non-negative quadratic polynomial always a sum of squares?
I am studying positive semi-definite matrices. A natural problem arises when we study the properties of the corresponding quadratic form of such matrices: Can we always represent such quadratic forms ...
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Are there ways to generate alternative representations of a number as the sum of three squares?
Assume we have a positive integer $n$ which is the sum of the squares of three integers — for example,
$$
n = 451 = 15^2 + 15^2 + 1^2.
$$
As it turns out, this particular $n$ has are several other ...
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If $m$ is an odd number whose prime factors are all $\equiv 1\pmod{4}$, then $m=a^2+b^2$, where $\gcd(a,b)=1$.
This is exercise 24.5.(a) from the book A Friendly Introduction To Number Theory:
If $m$ is odd and if every prime dividing $m$ is congruent to 1 modulo 4, prove that $m$ can be written as a sum of ...
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Do eight-square representations have patterns like four-square representations?
It is well-known that every positive integer can be written as the sum of the squares of four integers. More nuanced patterns are also known — for example, every odd positive integer has a ...
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Number of ways to represent an integer $n$ in a finite field $\mathbb F_{p^m}$ as a sum of two squares [closed]
We know that in any finite field $\mathbb F$, any element $n \in \mathbb F$ can be written as a sum of two (integer) squares. I am wondering if there is an explicit formula of the number of ways to ...
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Formula for finding sum of three squares that equal a fourth square
I am investigating the sum of three squares that equal a fourth square value.
I am only interested in positive integers greater than zero.
Specifically i am collecting formulae that identify values of ...
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Question about the divisibility of a sum
In this post , the function $$f(n):=\sum_{j=1}^n j!^2$$ is mentioned. $f(n)$ seems to be squarefree for every positive integer $n$.
Do we have $n+1\mid f(n)$ for some positive integer $n$ ? The ...
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When can a sum over the sum of squares function be replaced by an integral?
Say I have some function $f(n,x)$. For many "nice" functions, I find (numerically) that
\begin{equation}
\sum_{n=1}^\infty r_2(n) f(n,x)\approx\pi\int_0^\infty dn f(n,x).
\end{equation}
Here ...
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For how many numbers $1\le k\le n$ does there exist $a$, $b\in\Bbb Z$, so that $k=a^2+b^2$?
For how many numbers $1\le k\le n$ does there exist $a$, $b\in\Bbb Z$, so that $k=a^2+b^2$? Give an approximation, focus more on the lower bound.
Let the number be $f(n)$. First, I know the following....
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An Algebraic Understanding of Degrees of Freedom
I am looking at Simple Linear Regression where our model has two parameters, i.e.
$$ E[Y|x_{i}] = b_{0} + b_{1}x_{i} $$
I am having trouble understanding where the idea that $$ \frac{SSR}{\sigma^{2}}\...
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Simplifying square of partial sums to sum of squares and products
Assume the square of sum of $S$ of $v_1, v_2, \dots, v_N$ means the square of the sum of all possible combinations of $S$ values among $v_1, v_2, \dots, v_N$, i.e.,
$$
\sum_{i=1}^{{N \choose S}} (\...
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Asymptotic of $\sum_{0<|n = (n_1,n_2) |\leq \mu} \big(|n|^{-2\beta} - \mu^{-2\beta}\big)^{\frac12}$
I am interested in the asymptotic behavior of
$$S_{\beta}(\mu) := \sum_{0<|n|\leq \mu} \big(|n|^{-2\beta} - \mu^{-2\beta}\big)^{\frac12} \sim ~ ?~,~ \mu \to \infty $$ where the sum runs over the $n ...
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Is there a hidden relation between the sum of squares and the factorial?
Is there a hidden relation between the sum of squares and the factorial?
Two cases will be considered: the sum of odd squares and that of the even squares. We will not consider the classical sum of ...
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Write $7 \cdot 10^{100} + 7$ as a sum of four squares
How do you write $7 \cdot 10^{100} +7$ as a sum of four squares?
I know that you can write it as a sum of four squares by the Lagrange's Four Squares Theorem, but I don't know how to write such a big ...
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How many ways are there to write $173$ as a sum of squares?
Q. How many ways are there to write $173$ as a sum of squares of natural numbers?
A friend of mine recently asked me this question, and I haven't been able to figure it out fully. I know that $$2^2+3^...
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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 ...
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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 ...
user1135823
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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 ...
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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 ...
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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 ...
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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 ...
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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 $c$ ...
user1135823
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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 ...
user1135823
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BMO2 1994 Q1 - Need Help
Find the first number such that the average of the sum of the squares from $1$ to $n$ (where $n > 1$) equals $k^2$.
Here is what I have done so far:
The sentence is equivalent to saying that $\frac{...
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Find $\mathop {\lim }\limits_{n \to \infty } \,n\left( {\frac{{{1^{2022}} + {2^{2022}} + ... + {n^{2022}}}}{{{n^{2023}}}} - \frac{1}{{2023}}} \right)$ [duplicate]
It's ok to find the limit with integral
$$\mathop {\lim }\limits_{n \to \infty } \frac{{{1^{2022}} + {2^{2022}} + ... + {n^{2022}}}}{{{n^{2023}}}} = \frac{1}{{2023}}$$
Put ${u_n} = \frac{{{1^{2022}} + ...
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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 ...
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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 ...
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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 ...
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Show divergence of infinite sum outside of $(-1,1)\backslash\{0\}$.
I have to show that this dreadfull thing is infinite $\forall a\in(-1,1)\backslash\{0\}$
$$f(a)=\sum_{n=0}^\infty \left(\frac{1-(\pm a/2)^{n+1}}{(2\mp a) a^{n+1}}\right)^2$$
I can show that if $0<...
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Expectation of a single squared term from sum of squared terms
Let $A \in \mathbb{R}^{N \times n}$ be a full row rank matrix and $b \in \mathbb{R}^{N}$ where all entries are drawn randomly, identically, and independently from $\mathcal{N}(0, 1)$. Let $f$ be the ...
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Solve for vector $r$ in an equation involving sum of the square elements of $r$ and the sum of the square elements of $Mr$ where $M$ is a known matrix
Hello stackexchange community.
I have a system of equations where the aim is finding two vectors $r$ and $c$ of lengths $t$ and $m$ respectively. There is a known matrix $M$ of shape $m\times t$ and ...
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Prove that there exist infinitely many values of $n$ such that $z_n, z_{n+1}, …, z_{n+2019}$ are all odd.
Let all numbers of form $x^2 +y^2$ where $x, y$ are coprime integers be arranged in a sequence $z_1 < z_2 < z_3 < . . ..$
(So the sequence begins $z_1 = 2 = 1^2 + 1^2 , z_2 = 5 = 1^2 + 2^2 , ...
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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 ...
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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 ...
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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 ...
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Way to obtain root of sum of squares of three timeseries
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 ...
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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 ...
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