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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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The number of ways of writing $k$ as a sum of the squares of "not so big" two elements

This question arises from the attempt to compute the Euler characteristic of a space using a Morse function. We fix a positive integer $n$. For each integer $k$ which satisfies the condition $$1\leq k ...
Yasuhiko Kamiyama's user avatar
1 vote
1 answer
82 views

Expressing $x^2-2y^2$ as sum of 2 squares: is the equation $169^2 - 2 \cdot 80^2 = 119^2 + 40^2$ a coincidence?

Is there a general method for expressing $x^2 - 2y^2$ as a sum of 2 squares when we know for some reason that it must be possible? I was solving a problem for which once you get to the end, you're ...
Display name's user avatar
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how to calculate $\sum\limits_{n=0}^{\infty}\sum\limits_{m=-n}^{n}j_n(r)^2Y_n^m(\theta,\psi)^2$

I am wondering how to calculate $\sum\limits_{n=0}^{\infty}\sum\limits_{m=-n}^{n}j_n(r)^2Y_n^m(\theta,\psi)^2$, where $j_n(r)$ is the Spherical Bessel function, and $Y_n^m(\theta,\psi)$ is the ...
madao's user avatar
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calculate$\sum_{n=-\infty}^{\infty}J_n(\omega)^2e^{jn\psi}$

I am wondering how to calculate the following expression: $$\sum_{n=-\infty}^{\infty}J_n(\omega)^2e^{jn\psi}$$ I have tried to use the Jacobi-Anger Expansion, also the equation below: $$\sum_{n=-\...
madao's user avatar
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Finding the sum of a series using the binomial theorem [duplicate]

If $(1+x+x^2)^n = a_{0} + a_{1}x+a_{2}x^2 + ... + a_{2n}x^{2n}$, prove that $a_{0}^2-a_{1}^2+a_{2}^2+...+(-1)^{n-1}a_{n-1}^2= \frac{1}{2}a_{n}(1-(-1)^na_{n})$. I was able to find the value of $a_{0}^2-...
CallousCalculus's user avatar
1 vote
2 answers
74 views

How do I get this $Q(x,y)$ into a sum of squares without matrices

The bivariate quadratic polynomial $Q(x,y)$ is: $$Q(x,y)=x^2+y^2+xy-a(2x+y)$$ to get it into a sum of squares, is there a method without any rotation of matrices involved? I can kind of can get it to ...
Ivy's user avatar
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1 vote
1 answer
137 views

Sum of three squares equalling a different sum of three squares

Assume $x_1, x_2, x_3, y_1, y_2, y_3 \in \mathbb{N}_{> 0}$. I am trying to figure out if it is possible to find all solutions where $$x_1^2 + x_2^2 + x_3^2 = y_1^2 + y_2^2 + y_3^2.$$ I know the ...
jmath's user avatar
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4 votes
1 answer
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How to write $\displaystyle \sum\limits_{\text{cyc}}(a-b)(a-c)(a-d)(a-e)$ as the sum of squares?

How to write $\displaystyle \sum\limits_{\text{cyc}}(a-b)(a-c)(a-d)(a-e)$ as a sum of squares? This problem comes from 1971 IMO problem 1,which is stated as follows. Prove that the following ...
grj040803's user avatar
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How can I express two positive integers as the real and imaginary part of an exponential sum

In the following problem, suppose you have two positive integers $A$, $B$, $A$ odd, $B$ even and let $A^2+B^2=p$ a prime. Let $g$ be a primitive root modulo $n$. For any $b$ in $\mathbb{Z}^{\times}_n=\...
3809525720's user avatar
6 votes
4 answers
1k views

Relationship between the squares of first n natural numbers and first n natural odd numbers.

Here's a question from high school mathematics. If $ 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + \dots + 100^2 = x $, then ($1^2 + 3^2 + 5^2 + \dots + 99^2$) is equal to ? Options were: (a) $\frac{x}{2}-2525$ (b) ...
Ishant's user avatar
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3 votes
1 answer
275 views

What numbers can be written uniquely as a sum of two squares?

What numbers can be written uniquely as a sum of two squares? I was looking at sequence A125022, which shows the numbers that can be uniquely written as a sum of two squares. Here are a few things ...
huh's user avatar
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1 answer
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Integers that are sums of two squares [duplicate]

It is easy to decide if a given integer $n$ is the sum of two squares, and in fact there is a simple formula (based on the prime factorization) to compute the number of ways that $n$ can be written as ...
Math101's user avatar
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Consequence following from the hypothesis $ : 7$ divides $a^2 + b^2$ with $ a, b \in \mathbb Z$ [duplicate]

Not HM, simply self teaching. Source : Alain Troesch (Louis-le-Grand High School, Paris) , Exercices $2022-2023$, Polycopié des exercices , page $99$, Ex. $21.10$ http://alain.troesch.free.fr/ ...
Vince Vickler's user avatar
1 vote
1 answer
67 views

Sequence of squares which can't be written as the sum of a smaller non-zero square and twice a triangular number

Are there infinitely many squares which cannot be written as the sum of a smaller non-zero square and twice a triangular number? In other words, is the list given at https://oeis.org/A230312 infinite? ...
Ok-Virus2237's user avatar
3 votes
2 answers
355 views

Writing 2024 as the sum of 3 and 4 squares

I'm currently taking a course in number theory and we've just seen that any number can be written as the sum of the 4 squares, and that numbers can be written as the sum of 3 if they aren't of a ...
Skark123's user avatar
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2 votes
1 answer
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$L^1$ and $L^4$ norm relation to fixed $L^2$ norm

Hello math stack exchange! Suppose I have some set of nonnegative values $x_1, x_2, \cdots, x_n$ and I have that the sum of their squares is a constant so $x_1^2+x_2^2+\cdots+x_n^2=\sum_{i=1}^n x_i^2 =...
Dhruv Bhandarkar Pai's user avatar
3 votes
1 answer
111 views

If the sum of four squares is a square, what is the equivalent sum of three squares?

Let’s say I have \begin{align} \tag{1} x_1^2 = y_1^2+y_2^2+y_3^2+y_4^2, \end{align} where all the numbers are integers, and $x_1$ is odd. Evidently, $x_1^2$ is not of the form $4^k(8m+7)$, and so [by ...
Kieren MacMillan's user avatar
16 votes
3 answers
729 views

Odd square as sum of $9$ distinct odd squares

I'm interested in representing odd squares as sum of $9$ distinct odd squares. So, let $n\in\mathbb{N}$ be odd and $x_1,\,x_2,\,...,\,x_9\in\mathbb{N}$ be odd and pairwise distinct. The question is, ...
summingsummer's user avatar
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1 answer
96 views

Derivation of $SSE =S_{yy} - S_{xy}^2/S_{xx}$

I have seen this result and I am trying to figure out how to derive it from $SSE = \sum(Y_i - \hat{Y})^2$. I know that $r = \frac{s_{xy}}{\sqrt{s_{xx}s_{yy}}} $ and I have seen online to use and ...
Jackanap3s's user avatar
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0 answers
27 views

Finding $\sin(a+b)^{2k} = \sum a_i \sin^{2 n_i}(b_i a) \cos^{2 m_i}(c_i a) \sin^{2 l_i}(d_i b) \cos^{2 j_i}(e_i b)$

I am looking for trig identities of the form $\sin(a+b)^{2k} = \sum a_i \sin^{2 n_i}(b_i a) \cos^{2 m_i}(c_i a) \sin^{2 l_i}(d_i b) \cos^{2 j_i}(e_i b)$ where $k,n_i,m_i,l_i,j_i$ are integers $> -...
mick's user avatar
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2 votes
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How do I calculate the prediction interval for a data set in python?

I have a data set taken from real measurements that I have modeled with a simple univariate linear regression in python using scipy.stats.linregress, so the model ...
Boone's user avatar
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2 answers
181 views

Show that the sum is between (1,2) [closed]

Show that the sum $$1+\frac{1}{2^2}+\frac{1}{3^2}+\dots+ \frac{1}{2022^2}$$ is between $(1,2)$, I mean is bigger than $1$ and less than $2$. What I tried: honestly I never knew or understood how to ...
Usee0927's user avatar
2 votes
1 answer
63 views

Sum of squares $\pmod p$

For a prime $p>3$ the sum of the squares $\pmod p$ is $0$. Indeed, we have $$\sum_{k=1}^{p-1} k^2=\dfrac{p(p-1)(2p-1)}6\equiv 0\pmod p$$ But for $p\equiv 1\pmod 4$ we have that $-1$ is a square $\...
ajotatxe's user avatar
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1 vote
1 answer
140 views

Factorisation of the sum of $n$ squares using hypercomplex numbers

After finding out that you could factor $a^2 + b^2$ as $(a+bi)(a-bi)$ using complex numbers I wondered if there were any useful factoring tricks using the quaternions or octonions and after some ...
hefe's user avatar
  • 35
11 votes
0 answers
150 views

Sum of three rational squares is the sum of their reciprocals

Question: What are the solutions to $$a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}$$ where $a,b,c\in \mathbb{Q}$ are rational numbers. There is the easy solution $a=b=c=\pm 1$. And in fact, ...
Chris Wolird's user avatar
3 votes
2 answers
129 views

Asymptotic growth of sum-of-two-squares function

Let $r(n)$ denote the sum-of-two-squares function for $n \in \mathbb N$, that is, the number of pairs $(i,j) \in \mathbb Z^2$ such that $i^2+j^2=n$. The paper Trigonometric polynomials and lattice ...
Luis Mendo's user avatar
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3 votes
1 answer
158 views

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(\...
tomos's user avatar
  • 1,662
0 votes
1 answer
76 views

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) $$...
happyle's user avatar
  • 173
0 votes
0 answers
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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^...
vvg's user avatar
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1 vote
2 answers
66 views

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 $P=\sqrt{2}.$ ...
Dragon boy's user avatar
0 votes
1 answer
49 views

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 ...
Dragon boy's user avatar
1 vote
1 answer
74 views

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 ...
Dragon boy's user avatar
0 votes
0 answers
47 views

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 $b=s+t+u+v$. ...
vvg's user avatar
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0 votes
0 answers
50 views

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 ...
Sgg8's user avatar
  • 1,488
10 votes
10 answers
4k views

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 ...
Mark Thomas's user avatar
4 votes
1 answer
173 views

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 ...
uvdose's user avatar
  • 107
1 vote
0 answers
45 views

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 $...
DesmondMiles's user avatar
  • 2,803
1 vote
0 answers
71 views

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 ...
vvg's user avatar
  • 3,341
3 votes
1 answer
325 views

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(...
Nicco's user avatar
  • 2,813
1 vote
0 answers
95 views

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 ...
ElementX's user avatar
  • 972
1 vote
1 answer
106 views

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 ...
Kieren MacMillan's user avatar
4 votes
1 answer
118 views

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 ...
Little Jonny's user avatar
1 vote
0 answers
55 views

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 ...
Kieren MacMillan's user avatar
0 votes
2 answers
62 views

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 ...
Squirrel-Power's user avatar
2 votes
5 answers
163 views

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 ...
Hector's user avatar
  • 219
6 votes
1 answer
252 views

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 ...
Peter's user avatar
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3 votes
1 answer
95 views

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 ...
Jean Du Plessis's user avatar
2 votes
1 answer
121 views

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....
youthdoo's user avatar
  • 1,475
1 vote
1 answer
77 views

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}} (\...
Saeed's user avatar
  • 175
3 votes
0 answers
68 views

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 ...
jvc's user avatar
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