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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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 ...
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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?
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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 ...
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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?
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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 ...
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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 ...
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46 views

How often can a number be written as a linear combination of the squares of its prime divisors?

Peter asked here "Can a number be equal to the sum of the squares of its prime divisors?" and, it seems clear that if $$n=p_1^{a_1}\cdots p_k^{a_k},$$ and $$f(n):=p_1^2+\cdots+p_k^2$$ that then $n=f(n)...
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The sum of an infinite series containing a finite series in each denominator [duplicate]

Evaluate $$\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{6^2}+\frac{1}{10^2}+\cdots+\frac{1}{\left[\frac{k(k+1)}{2}\right]^2}+\cdots$$ to $\infty$, where $k$ is the $k$th term of the series. Using ...
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65 views

How to work out sum of Square Number from given numbers?

Which of the following cannot be written as the sum of two distinct square numbers? A.106 B. 109 C. 112 D. 117 What would be the correct answer here and can someone explain in detail please.
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28 views

Pattern in Squared Numbers and their Digit Sum

So this has been boggling my mind for some time now. On spring break I was really bored and started messing around with numbers when I noticed something. I was squaring each number (1-9) when I ...
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54 views

Prove that if $m^2+n^2=0$ then $m=0$ and $n=0$

Prove that if $m^2+n^2=0$ then $m=0$ and $n=0$. Given $m^2+n^2=0$ then $m^2= -n^2$. Because $m$ and $n$ are real numbers, then $m^2 \geq 0$, $n^2 \geq 0$. Therefore, $m=0$ and $n=0$. Is that ...
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50 views

If $ a, b, c$ are real numbers such that $a^2 + b^2 + c^2 = 1$, then show $ab+bc+ca> \frac{-1}{2}$

If $ a, b, c$ are real numbers such that $a^2 + b^2 + c^2 = 1$, then show that $ab+bc+ca\ge \frac{-1}{2}$ If figured out that if I put $(a+b+c)^2 = 0$ then I will get the above answer, but $(a+b+c)^...
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57 views

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 ...
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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: $(n^2+m^2)' \equiv 0 \mod 4$ ?
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33 views

Proof that every sum of exponents can be represented as a polynomial. I am missing an inital idea.

$$ s_n(p)=\sum_{k=1}^n k^p $$ Show: For every $q \geq 1$ exist rational numbers $ a_{k,q} , 1 \leq k \leq q-1 $, such that $$ s_n(q)= \frac 1 {q+1} n^{q+1}+ \frac 1 2 n^q + \sum_{k=1}^{q-1} a_{k,q}...
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1answer
55 views

Quadratic Congruence modulo square-free integer

If $m$ is a square-free integer, show that $x^{2} + y^{2} \equiv k\pmod{m}$ has a solution $\forall k\in\mathbb{N}$. This means that we need to prove existence of such $m$ for all $k\in\mathbb{N}$. ...
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53 views

Sum of two squares theorem using complex numbers

Show that if $M$ can be written as the sum of squares of two integers, so can $2M, 5M, 8M, 10M, 13M$ and so on.. So I have figured out this question for the most part, if $M=a^2+b^2$ then I can use ...
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103 views

Can $5^n+1$ be sum of two squares?

I want to determine whether or not $5^n+1$ , $n\in\mathbb{N}$ can be written as sum of two squares. Obviously, the real problem is when $n$ is odd. I am aware of the known results about numbers ...
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35 views

Gauss' Proof of Lagrange Four-Square Theorem?

I read recently that Gauss provided a proof of Lagrange's Four-Square Theorem using his ideas about equivalence classes of quadratic forms (i.e. linear substitution of variables) somehow applied to $w^...
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Constructing a sum of squares in $\mathbb R[x]$ with given complex valuation

Fix a polynomial $g(x)\in\mathbb R[x]$ and a complex number $u\in\mathbb C\setminus\mathbb R$. My main question is How can we construct a polynomial $s(x)\in\mathbb R[x]$ such that $s(x)$ is a sum ...
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683 views

How much of an infinite board can a N-mover reach?

This question is inspired by the question on codegolf.SE: N-movers: How much of the infinite board can I reach? A N-mover is a knight-like piece that can move to any square that has a Euclidean ...
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29 views

ternary quadratic form as a sum two squares of linear forms

Let $\phi(x,y)=ax^2+2bxy+cy^2$ be a binary quadratic form with integer coefficients. Pall proved in 1951 that $\phi(x,y)$ can be written as the sum of two squares of two linear forms if and only if $...
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21 views

What is a closed form partial sum formula for the q-digamma function?

It is known that the partial sum of the digamma function can be expressed in closed form, but what of the q-digamma function? $$\sum_{x=1}^n\psi_q{\small(x)}\ =\ ?$$ $$q\in\mathbb N$$
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185 views

Show that if $a,b,c\in \mathbb{N}$ and ${a^2+b^2+c^2}\over{abc+1}$ is an integer it is the sum of two nonzero squares

I was reading about the fascinating problem in the IMO ($1988 $ #$6$) that asks: Let $a$ and $b$ be positive integers such that $(ab+1) | (a^2+b^2)$. Show that ${a^2+b^2}\over{ab+1}$ is a perfect ...
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107 views

Let a, b, c, d be any four distinct integers such that $a>b>c>d>1$. Show that if $ad=bc$, then $a^2+b^2+c^2+d^2$ is composite. [closed]

Let $a, b, c, d$ be any four distinct integers such that $a>b>c>d>1$. Show that if $ad=bc$, then $a^2+b^2+c^2+d^2$ is composite.
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136 views

Can factoring with the sum of 4 squares be made more efficient?

We have seen that it was possible to use the sum of two squares to factor numbers (see Can the sum of two squares be used to factor large numbers? ) The main drawback is the fact that the method ...
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51 views

Adding sequence of square roots [closed]

How to add sequence of square roots from square root 2 till square root 99 and how to add the sequence of their reciprocal here is the original problem
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1answer
105 views

Hypergeometric function 3F2 with unit argument

Recently I obtained the following expression $${}_3F_2(-n,a - b ,1-b-n; b + 1, 1-a-n; 1), $$ with $b>a>0$ and $n\in\mathbb{N}$. My question is: If someone knows a closed form solution to the ...
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Understanding some proofs-without-words for sums of consecutive numbers, consecutive squares, consecutive odd numbers, and consecutive cubes

I understand how to derive the formulas for sum of squares, consecutive squares, consecutive cubes, and sum of consecutive odd numbers but I don't understand the visual proofs for them. For the ...
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93 views

Numbers that are the sum of 2 distinct nonzero squares in exactly 1 way

Is there a formula that says if a number is the sum of 2 distinct nonzero squares ? This is the sequence I'm trying to emulate: 5, 10, 13, 17, etc.. http://oeis.org/A004431 (Invalid it contains 85, ...
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How to calculate $\sum_{i=0}^N\lfloor ai\rfloor^2$ for $0<a<1$ and $N\approx10^{16}$?

I have an irrational number $0<a<1$ for which I need to calculate the sum $$\sum_{i=0}^N\lfloor ai\rfloor^2\mbox{ mod }m.$$ Here $m$ is an $8$-digit number and $N$ is a $15$-digit number. The ...
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135 views

Primality test using the sum of 2 squares representation of a prime $p^2= a^2+b^2$

We have seen that the sum of two squares (2 sq rep) of an integer $N^2$ can be used to factor $N$. Can the sum of two squares be used to factor large numbers? Here we use the same method to show ...
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28 views

Find the next least number N that is N+1 = X^2 and (N/2)+1 = y^2?

To go with this following task to create a proper equation ? Many numbers, especially from smaller ones, can rightly claim that they are of particular interest to scientists. In the kingdom of ...
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295 views

Can the sum of two squares be used to factor large numbers?

The method described below is different from Euler sum of two squares factorization method. The method below does not require 2 sum of two squares representations to factor a number. This post was ...
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225 views

Can the sum of two squares be used to determine if a number is square free?

Mathword (http://mathworld.wolfram.com/Squarefree.html) stated that "There is no known polynomial time algorithm for recognizing squarefree integers or for computing the squarefree part of an integer."...
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If $xy$ divides $x^2 + y^2$ show that $x=\pm y$ [duplicate]

Problem statement : Let $x,y$ be integers, show that if $xy$ divides $x^2 + y^2$ then $x=\pm y.$ What I have tried: I can reduce this to the case where $\gcd(x,y)=1$, since if $x$ and $y$ have a ...
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$\exists\infty$ many pairs of consecutive squares s.t. their sum is also a square

First of all, the term "pairs" is two of them, I assume (question's formulation is rather difficult to understand for me). So I guess this is the statement: $$\exists\infty\text{ many pairs of ...
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29 views

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

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. ...
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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 ...
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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, ...
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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 ...
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115 views

Odd of the form $a^2+b^2+c^2+ab+ac+bc$.

The computation below shows that (for $a,b,c \in \mathbb{N}$) the form $$a^2+b^2+c^2+ab+ac+bc$$ covers every odd integer less than $10^5$ except those in $$I= \{ 5, 15, 23, 29, 41, 53, 59, 65, ...
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68 views

Sum of four Squares relation

Cheers, As is well known due to Lagrange the number of ways to write $n$ as a sum of four squares is given by $$ r_4(n)=8\sum_{d|n} d $$ if $n$ is odd. Now define $$ \tilde{r}_4(n)=\#\left\{1 + \...
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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 $\sum_{i=1}^{t}\sqrt{k_i}$.
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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 ...
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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 $$A^\dagger=(A^TA)^{-1}A^T$$ ...
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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 $a, b , c , d$...
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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 ...
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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 ...