# 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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1answer
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### Let a and b be natural numbers such that $2a - b, a - 2b$ and $a + b$ are all distinct squares. What is the smallest possible value of $b$?

Let a and b be natural numbers such that $2a - b, a - 2b$ and $a + b$ are all distinct squares. What is the smallest possible value of $b$? Let, $2a-b=k^2, a-2b=p^2, a+b=q^2$. $k^2=p^2+q^2$ after ...
1answer
71 views

### Primes of the form $a^2+b^2+c^2$, $0<a<b<c$

Are there results showing which prime numbers that can be expressed as the sum of three different integers greater than zero?
1answer
92 views

### Any positive integer can be written as sum / difference of consecutive squares

How should one go about proving that $x \in \mathbb{N}$ can be written (with the right combination of signs) as $\pm 1^2 \pm 2^2 \pm \ldots \pm n^2$ for any $x : x, n \in \mathbb N$? I have tried for ...
3answers
59 views

### Prove that $x^2+2y^2+3z^2=10a^2$ has no integer solutions aside from all of them being 0

I got this equation while I was trying to solve a certain math Olympiad problem. I tried modulus and whatnot, but I haven't got anywhere. Is there a way to prove this?
3answers
164 views

### Solutions to a system of three equations with Pythagorean triples

Is there any solution to this system of equations where $x,y,z,s,w,t\in\mathbb{Z}$, none are $0$. \begin{align*} x^2+y^2=z^2\\\ s^2+z^2=w^2\\\ x^2+t^2=w^2 \end{align*} EDIT: Thank you zwim for the ...
1answer
40 views

### Sum of harmonic numbers $H_{n+k}$

I'm trying to take that sum: $$\sum_{k=1}^n H_{n+k}$$ So I transformed this sum to such: $\sum_{i=1}^n iH_{2n+1-i}$, unfortunately i can't make this sum out :( Hope You can help me, Thanks for ...
0answers
46 views

2answers
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### Finding an equivalence classes for the relation $(x_1,y_1)\sim(x_2,y_2)\ \iff\ x_1^2+y_1^2=x_2^2+y_2^2$.

Let R be the relation defined on the set of integer pairs by $(x_1,y_1)R(x_2,y_2)$ when $x_{1}^{2}$ + $y_{1}^{2}$ = $x_{2}^{2}$ + $y_{2}^{2}$. Prove that R is an equivalence relation and determine ...
1answer
25 views

### Arithmetic Derivative on sum of two perfect squares

Let $n,m \in \mathbb N$ and $n$ even, $m$ odd. If we take there squares and add them $n^2+m^2$, are there examples when we take the arithmetic derivative of the sum: $(n^2+m^2)' \equiv 0 \mod 4$ ?
1answer
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0answers
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### Does bounded version of Lagrange's 4-square theorem follow from this result?

Let $S_4(n)$ be the number of ways of representing $n$ as a sum of 4 integer squares and suppose I can give you a proof of the result that $\sum_{i=1}^nS_4(i) \sim \frac{\pi^2}{2}n^2$. Is this result ...
1answer
49 views

### $\lim_{n\to\infty}\frac{\sum_{i=1}^ni^{4\lambda}}{\Big(\sum_{i=1}^ni^{2\lambda}\Big)^2}=0$

Is it true that $\forall \lambda>0$ $$\lim_{n\to\infty}\frac{\sum_{i=1}^ni^{4\lambda}}{\Big(\sum_{i=1}^ni^{2\lambda}\Big)^2}=0$$ I cannot find a way to prove it, nor can I find a counterexample. ...
4answers
48 views

### Diophantine equation: solving $a^2+4n=b^2$

I found myself working with diophantine equations but I have no experience at all with them. Given an integer $n$, can I find two integers, $a$ and $b$, such that $$a^2+4n=b^2$$ How would you guys ...
1answer
41 views

### Inequality regarding sum of squared probabilities

I'm working on a problem set for a course on Machine Learning and one the problems asks me to prove a given inequality. As an aid for that, the problem gives me the hint to use the following result, ...
2answers
47 views

### How to express a number as a sum of $k$ squares?

My question is the following: Show that for each integer $k \ge 5$, there is an integer $N(k)$ such that every integer $n \ge N(k)$ can be written as a sum of $k$ nonzero squares. What is the process ...
0answers
115 views

1answer
62 views

### Upper bound on Sum of square roots

Let $k_1,k_2\ldots k_t$ be integers and $\sum_{i=1}^{t}{k_i}=k$ where $k$ is fixed. What is the maximum value of $\sum_{i=1}^{t}\sqrt{k_i}$.
1answer
34 views

### Quasiconvexity of a specific quartic bivariate polynomial family

I have a quartic polynomial of shape $$(a\cdot x+b+c\cdot xy-d\cdot y+y^2\big)^2$$ where $a,b,c,d\in\mathbb Z_{>0}$ are coefficients. The polynomial is not convex. However is it quasiconvex for ...
2answers
376 views

### How do the normal equations *always* have a solution?

My professor says that "the normal equations always have a solution", even when $A$ is not full rank. HOwever, this does not make sense to me. The normal equations are $$A^\dagger=(A^TA)^{-1}A^T$$ ...
2answers
186 views

### Product of two numbers which are expressable as the sum of two perfect squares

Problem: "Let m and n be integers such that each can be expressed as the sum of two perfect squares. Show that mn has this property as well" Proof: Let $m =a^2+b^2$ , $n=c^2+d^2$ ,where $a, b , c , d$...
0answers
30 views

### What is the error propagation for $r = \sqrt{a^2+b^2}$ if the error in $\delta a$ and $\delta b$ are known?

Suppose I have some value $r$ which is defined as $$r = \sqrt{a^2+b^2}$$ where $a$ and $b$ have errors of $\delta a$ and $\delta b$, respectively. What is $\delta r$? Using "standard" error ...
0answers
39 views

### How to find how similar two vectors are, while giving weight to their lengths?

I asked a question about this yesterday and got a really good response! Apparently I should use the euclidian (sum of squares) distance between the two vectors. This works well, but I'm having a bit ...