Questions tagged [sums-of-squares]

Question 1

I was trying to find bounds for the infinite sum $S_N=\sum_{x^2+y^2\leq N}\frac{1}{\sqrt{x^2+y^2}}$, where $(x,y)\in\mathbb Z^2$. This could be interpreted as the sum of the reciprocal of the ...

Question 2

I am having trouble locating any information about the following series (probably because I don’t know what to search for). I am looking for an expression that counts the number of unique ways the ...

Question 3

Problem. Let $a,\,b,\,c,\,d$ be real numbers. Denote $$u = \frac{a+b+c+d}{4}, \quad v = \frac{ab+ac+ad+bc+bd+cd}{6}, \quad w = \frac{abc+bcd+cda+dab}{4}.$$ Prove that $$3uv-2u^{3} - 2\left(u^{2}-v\...

Question 4

From Representing as sum of squares of polynomials I know that not every nonnegative polynomial is a sum of squares of polynomials. A classic counterexample is the Motzkin polynomial $$ M(x,y) = x^4 y^...

Question 5

I'm looking for some ideas to solve the following inequality. Problem. For any non-negative real numbers $a,b,c$ with $ab+bc+ca+abc=4,$ prove that$$\color{blue}{\sqrt{\frac{a}{bc+8}}+\sqrt{\frac{b}{...

Question 6

I was self studying a book on proofs and in the chapter on induction, the following was an exercise problem: Prove that, for every $n \in \mathbb N$, there are $n$ distinct natural numbers $a_1, a_2,…,...

Question 7

Let $a,\,b,\,c$ be non-negative real numbers. Prove that $$36(2+abc-ab-bc-ca) \geqslant [7(a^2+b^2+c^2)+ab+bc+ca-12](4-abc-a^2-b^2-c^2).$$ Note. For this inequality, if $a^2+b^2+c^2+abc=4,$ then we ...

Question 8

Number theory problem from a collection of 1220 composed by Amir Hossein. Prove that there exist infinitely many positive integers which can't be represented as sum of less than $10$ odd positive ...

Question 9

This is a problem from the 13th Yau Contest in 2022. Let $p$ be a prime number and $F$ a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers. (a) Suppose $p \neq 2$. Prove that every ...

Question 10

I have recently seen two formulae using Gauss sums which give the Solution to the equation $a^2+b^2=p$: $a=(X(p)-p)/2$ where $X(p)$ is the no. of solutions to the equation $y^3+16=x^2 \bmod p$. A ...

Question 11

Question: Why does this pattern of prime numbers appear in the magic sum of pseudo-magic squares of squares, and does this represent an infinite "family" of solutions? Consider a magic ...

Question 12

A well-known formula by Jacobi says that the number of ways to express a given number as a sum of two squares is $$ r_2(n) = 4 \sum_{2 \nmid d | n} (-1)^{(d-1) / 2} $$ which also gives a short proof ...

Question 13

How proof that every positive prime $p$ can be represent as a sum of squares of three integers in following way: $$2p=a^2+b^2+(2c)^2$$ This seems to be true for all prime numbers... (but I'm not 100% ...

Question 14

I. Quadruples In this post, we saw how infinitely many Pythagorean triples such as, $$3^2+4^2-5^2=0\quad$$ can lead to 4th power equalities, $$2^4+2^4+4^4+3^4+4^4=5^4\quad$$ It turns out that ...

Question 15

I have problem with proving this inequality. It is from an old book but without answers and I struggled with this question for a long time. I tried using induction, QM-AM-GM-HM inequalities but ...

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Questions tagged [sums-of-squares]

Growth of $\sum_{x^2+y^2\leq N}\frac{1}{\sqrt{x^2+y^2}}$

How many ways can $k$ natural numbers squared sum to $N$?

An upper and lower bound of $abc+bcd+cda+dab$

Are there semidefinite trivariate polynomials that cannot be written as sums of squares even if cubic roots are allowed?

How can I prove this inequality

Do there exist $n$ distinct natural numbers such that the sum of their squares is also a perfect square?

Prove that $36(2+abc-ab-bc-ca) \geqslant [7(a^2+b^2+c^2)+ab+bc+ca-12](4-abc-a^2-b^2-c^2)$

Prove that there exist infinitely many positive integers which can't be represented as sum of less than $10$ odd positive integers' perfect squares.

Sum of squares over $p$-adic field [closed]

Is there a deep relation of elliptic curves with quadratic forms?

Prime Number Pattern in Magic Sum of Pseudo / Semi Magic Squares of Squares

Sum of squares in $\mathbb{F}_p[T]$

$2p=a^2+b^2+(2c)^2$ [closed]

Finding sixth powers like $65^6 + 52^6 + 15^6 = 36^6 + 67^6 + 37^6$ using Pythagorean quadruples $65^2 = 52^2 + 15^2 + 36^2$?

Prove that $1+ 1/2 + 1/3 + 1/4 + \cdots + 1/n$ is less then $\sqrt n$ for $n\geq7$ [duplicate]

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