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
$$1\leq k ...
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Expressing $x^2-2y^2$ as sum of 2 squares: is the equation $169^2 - 2 \cdot 80^2 = 119^2 + 40^2$ a coincidence?
Is there a general method for expressing $x^2 - 2y^2$ as a sum of 2 squares when we know for some reason that it must be possible?
I was solving a problem for which once you get to the end, you're ...
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how to calculate $\sum\limits_{n=0}^{\infty}\sum\limits_{m=-n}^{n}j_n(r)^2Y_n^m(\theta,\psi)^2$
I am wondering how to calculate $\sum\limits_{n=0}^{\infty}\sum\limits_{m=-n}^{n}j_n(r)^2Y_n^m(\theta,\psi)^2$,
where $j_n(r)$ is the Spherical Bessel function, and $Y_n^m(\theta,\psi)$ is the ...
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calculate$\sum_{n=-\infty}^{\infty}J_n(\omega)^2e^{jn\psi}$
I am wondering how to calculate the following expression:
$$\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=-\...
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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-...
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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 ...
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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 ...
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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 ...
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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=\...
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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) ...
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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 ...
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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 ...
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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/
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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?
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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 ...
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