Questions tagged [sums-of-squares]

For questions concerning various representation of integers as sums of squares, which are studied in number theory.

Filter by
Sorted by
Tagged with
3
votes
2answers
127 views

Find two arithmetic progressions of three square numbers

I want to know if it is possible to find two arithmetic progressions of three square numbers, with the same common difference: \begin{align} \ & a^2 +r = b^2 \\ & b^2 +r = c^2 \\ & a^...
2
votes
3answers
152 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 ...
0
votes
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?
2
votes
3answers
163 views

Finding Pythagorean triplet given the hypotenuse

I have a number $c$ which is an integer and can be even or odd. It is the hypotenuse of a right angled triangle. How can I find integers $a,b$ such that $$ a^2 + b^2 = c^2 $$ What would be the ...
2
votes
3answers
177 views

In ℕ⁺, can the sum of three squares equal the sum of two squares?

Are there any examples where: $a² + b² + c² = p² + q²\qquad {a, b, c, p, q ∈ ℕ⁺}\tag{1}$ If not, can $(1)$ be disproven?
0
votes
1answer
1k views

How to compute SSR with just residuals and Xi?

How do we calculate SSR? I know SSE is the square of residuals all added together, but SSR is a subtraction between prediction for each observation and the population mean. Not sure how calculate SSR. ...
4
votes
4answers
455 views

Pythagorean triples

So I am given that $65 = 1^2 + 8^2 = 7^2 + 4^2$ , how can I use this observation to find two Pythagorean triangles with hypotenuse of 65. I know that I need to find integers $a$ and $b$ such that $a^...
1
vote
1answer
37 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 ...
0
votes
0answers
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)...
17
votes
8answers
9k views

Diophantine equation $a^2+b^2=c^2+d^2$

I was reasonably certain I've seen this before, but I was wondering how to solve the Diophantine equation $$a^2+b^2=c^2+d^2$$ I tried a web search and found nothing on this one. I'm trying to avoid ...
11
votes
1answer
186 views

Standalone proof of a conditional part of Lagrange’s Four-Square Theorem?

Lagrange’s Four-Square Theorem — a special case of the Fermat-Cauchy Polygonal Number Theorem (FCPNT) — states that every natural number can be written as the sum of the squares of at most four ...
2
votes
3answers
93 views

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 ...
0
votes
4answers
1k views

How can I write $1105$ as the sum of two squares other than $33^2 + 4^2$? [duplicate]

How can I write $1105$ as the sum of 2 squares other than $1105 = 33^2 + 4^2$? Could someone explain to me a procedure for doing this? I know that it has at least 2 other representations as a sum of ...
0
votes
1answer
24 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: $(n^2+m^2)' \equiv 0 \mod 4$ ?
0
votes
3answers
46 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.
1
vote
1answer
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 ...
1
vote
1answer
51 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 ...
3
votes
11answers
9k views

Natural number which can be expressed as sum of two perfect squares in two different ways?

Ramanujan's number is $1729$ which is the least natural number which can be expressed as the sum of two perfect cubes in two different ways. But can we find a number which can be expressed as the sum ...
0
votes
1answer
49 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)^...
1
vote
2answers
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 ...
16
votes
1answer
231 views

A conjecture about an unlimited path

Conjecture: For all primes of the form $a^2+b^2$, there are natural numbers $s,t,u,v$ such that $\quad s^2+t^2,u^2+v^2$ are primes $\quad a+b=s+t$ $\quad u+v=s+t+2$ $\quad |s-u|+|t-v|=2$ This ...
18
votes
2answers
501 views

Exploiting a Diophantine approximation of $\pi^4$ into giving a series of rationals for $\pi^4$

A note about this question: The original question asked seems likely impossible so I am really asking if we can exploit the technique below into giving us a 'nice' form for $\pi^4$. By nice form I ...
1
vote
1answer
32 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}...
8
votes
1answer
954 views

The sum of the first $n$ squares is a square: a system of two Pell-type-equations

This question comes from trying to see why 24 is the only non-trivial value of $n$ for which $$1^2+2^2+3^2+\cdots+n^2$$ is a perfect square. To this end, let $m,n \in \mathbb N$ be such that $1^2+2^2+...
1
vote
1answer
52 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 $k\in\mathbb{N}$. ...
0
votes
1answer
763 views

Put quadratic form into sum of squares

Is there a method or process that doesn't require a matrix to put quadratic forms into a sum of squares ? Two examples that I find extremely challenging. i) $q(x, y, z) = (x − y) ^2 + (y − z) ^2 − ...
1
vote
2answers
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 ...
1
vote
3answers
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 ...
0
votes
0answers
33 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^...
2
votes
1answer
213 views

Decomposition into three squares [closed]

Doing a coding assignment. And it's basically having a user enter $n$. Then I need to provide (If it exists) $$n = x^2 + y^2 + z^2.$$ Not really sure how to approach this. Any ideas?
16
votes
3answers
14k views

Number of integer solutions of $x^2 + y^2 = k$

I'm looking for some help disproving an answer provided on a StackOverflow question I posted about computing the number of double square combinations for a given integer. The original question is ...
1
vote
1answer
35 views

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 ...
-2
votes
2answers
331 views

Sum of two squares equal to $2018^{2019}+2018$ [closed]

$$x^2+y^2 = 2018^{2019}+2018$$ is expressed as sum of two perfect squares. Any pair of perfect squares can satisfy?
4
votes
2answers
63 views

Does $\exists n : 6 + \sum_{i=2}^n p_i = x^6+y^6$ for $p_i$ the $i^{\text{th}}$ prime and $x,y\in\mathbb{Z}$?

I noticed something about the prime numbers: Pick the number $2$. Then, add the first odd prime, namely $3$. The result is $5 = 1^2+2^2$. Notice that the exponents are also the number we picked. ...
23
votes
1answer
678 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 ...
0
votes
0answers
27 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 $...
0
votes
0answers
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? $$\sum_{x=1}^n\psi_q{\small(x)}\ =\ ?$$ $$q\in\mathbb N$$
1
vote
2answers
180 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 ...
-1
votes
3answers
106 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.
0
votes
0answers
131 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 ...
-1
votes
2answers
41 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
1
vote
1answer
97 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 ...
24
votes
3answers
2k views

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 ...
1
vote
0answers
92 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, ...
0
votes
0answers
15 views

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 ...
2
votes
0answers
291 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 ...
0
votes
0answers
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 ...
5
votes
1answer
333 views

How to prove that all primes of the form $4k+1$ can be represented by the sum of two squares in only one way regardless of the order?

I am reading a book about Number Theory as a new learner. The book has proved that all primes of the form $4k+1$ can be represented by the sum of two squares. This question is given as exercise and ...
2
votes
0answers
222 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."...
0
votes
0answers
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 ...