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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48
votes
3answers
2k views

Any odd number is of form $a+b$ where $a^2+b^2$ is prime

This conjecture is tested for all odd natural numbers less than $10^8$: If $n>1$ is an odd natural number, then there are natural numbers $a,b$ such that $n=a+b$ and $a^2+b^2\in\mathbb P$. $\...
27
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3answers
1k views

Numbers that are the sum of the squares of their prime factors

A number which is equal to the sum of the squares of its prime factors with multiplicity: $16=2^2+2^2+2^2+2^2$ $27=3^2+3^2+3^2$ Are these the only two such numbers to exist? There has to be an easy ...
26
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9answers
16k 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 ...
26
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3answers
3k 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 ...
23
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1answer
732 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 ...
20
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2answers
228 views

Sequences where $\sum\limits_{n=k}^{\infty}{a_n}=\sum\limits_{n=k}^{\infty}{a_n^2}$

I was recently looking at the series $\sum_{n=1}^{\infty}{\sin{n}\over{n}}$, for which the value quite cleanly comes out to be ${1\over2}(\pi-1)$, which is a rather cool closed form. I then wondered ...
19
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3answers
17k 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 ...
19
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1answer
612 views

All elements in $\mathbb{Z}/n\mathbb{Z}$ are representable as sum of a square and a cube?

Is it true that all elements in $\mathbb{Z}/n\mathbb{Z}$ are representable as the sum of a square and a cube? Example: ($n=7$) $0 \equiv 0^2+0^3 \left( \text{mod } 7 \right)$ $1 \equiv 1^2+0^3 \...
18
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5answers
3k views

Integral solutions of $x^2+y^2+1=z^2$

I am interested in integral solutions of $$x^2+y^2+1=z^2.$$ Is there a complete theory comparable to the one for $x^2+y^2=z^2?$
18
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2answers
567 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 ...
16
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1answer
338 views

Integer solutions to $\prod\limits_{i=1}^{n}x_i=\sum\limits_{i=1}^{n}x_i^2$

Given integers $x_1,\dots,x_n>1$. Let's assume WLOG that ${x_1}\leq\ldots\leq{x_n}$. I want to prove that the only integer solutions to any equation of this type are: $x_{1,2,3 }=3\implies\prod\...
15
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4answers
2k views

Value of $(\alpha^2+1)(\beta^2+1)(\gamma^2+1)(\delta^2+1)$ if $z^4-2z^3+z^2+z-7=0$ for $z=\alpha$, $\beta$, $\gamma$, $\delta$

Let $\alpha$, $\beta$, $\gamma$, $\delta$ be the roots of $$z^4-2z^3+z^2+z-7=0$$ then find value of $$(\alpha^2+1)(\beta^2+1)(\gamma^2+1)(\delta^2+1)$$ Are Vieta's formulas appropriate?
15
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1answer
241 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 ...
14
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5answers
13k views

Can an integer of the form $4n+3$ written as a sum of two squares?

Let $u$ be an integer of the form $4n+3$, where $n$ is a positive integer. Can we find integers $a$ and $b$ such that $u = a^2 + b^2$? If not, how to establish this for a fact?
14
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3answers
3k views

Prove that if $p$ is an odd prime that divides a number of the form $n^4 + 1$ then $p \equiv 1 \pmod{8}$

Problem Prove that if $p$ is an odd prime that divides a number of the form $n^4 + 1$ then $p \equiv 1 \pmod{8}$ My attempt was, Since $p$ divides $n^4 + 1 \implies n^4 + 1 \equiv 0 \pmod{p} \...
13
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4answers
4k views

If a prime can be expressed as sum of square of two integers, then prove that the representation is unique.

If a prime can be expressed as sum of two squares, then prove that the representation is unique. My attempt: If $a^2+b^2=p$, then it is obvious that $a,b$ of different parity. Now, I assume the ...
12
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3answers
1k views

Can we prove a stronger claim?

Here it is asked whether for every $k>1$, there is a prime $p$ and $w_1,w_2,\cdots ,w_k>1$, such that $p$ divides $w_1^2+w_2^2+\cdots +w_k^2$, but none of the summands. Can we prove there is ...
11
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2answers
646 views

(Non?)-uniqueness of sums of squares

(I've had almost no exposure to number theory, so please keep answers as elementary as possible.) Write $\mathbb{N} = \{0,1,2,3,\ldots\}$ for the natural numbers. Then every element of $\mathbb{N}$ ...
11
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1answer
237 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 ...
11
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1answer
156 views

Sums of squares (Proof) [duplicate]

Prove that $n^2 + (n + 1)^2 = m^3$ does not have solutions in the positive integers. I guess that the proof is by contradiction, but if I suppose it, I can't find the contradiction. Thanks for your ...
10
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9answers
2k views

Prove that if $3\mid a^2+b^2$ then $3\mid a$ and $3\mid b$.

For elements $a$ and $b$ in the ring $\Bbb{Z}$ prove that if $3\mid a^2+b^2$ then $3\mid a$ and $3\mid b$. I tried proving it but I just don't manage to. Maybe I am missing some basic claims in the ...
10
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3answers
1k views

Show that $x^2+y^2+z^2=999$ has no integer solutions

The question is asking us to prove that $x^2+y^2+z^2=999$ has no integer solutions. Attempt at a solution: So I've noticed that since 999 is odd, either one of the variables or all three of the ...
10
votes
4answers
692 views

Find all $x,y,z$ such that $x^2 + y^2 + z^2 = 3^{10}$

By Legendre's 3-squares theorem, a number $n = x^2 + y^2 + z^2$ can be written as the sum of three squares if $n \neq 4^a(8b+7)$. In my case, I am choosing $$n = 3^{10} \equiv (3^2)^5 \equiv 1 \mod 8$...
10
votes
2answers
185 views

$ 7\mid x \text{ and } 7\mid y \Longleftrightarrow 7\mid x^2+y^2 $

Show that $$ 7\mid x \text{ and } 7\mid y \Longleftrightarrow 7\mid x^2+y^2 $$ Indeed, First let's show $7\mid x \text{ and } 7\mid y \Longrightarrow 7\mid x^2+y^2 $ we've $7\mid x \...
10
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1answer
300 views

Coeff. of $x^{97}$ in $f(x) = (x-1)\cdot (x-2)\cdot (x-3)\cdot (x-4)\cdot …(x-100)$

If $f(x) = (x-1)\cdot (x-2)\cdot (x-3)\cdot (x-4)\cdot ........(x-100)\;,$ Then Coefficient of $x^{99}$ and Coefficient of $x^{98}$ and Coefficient of $x^{97}$ in $f(x).$ $\bf{My\; try::}$ We can ...
10
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2answers
618 views

Finding all integers such that $a^2+4b^2 , 4a^2+b^2$ are both perfect squares

Are there any integers $a,b$, such that: $$a^2+4b^2 , 4a^2+b^2$$ are both perfect squares?
9
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2answers
538 views

How do I prove that powers of 5 are the sum of two squares using mathematical induction [duplicate]

Using mathematical induction I need to prove $$5^n=a_n^2 +b_n^2$$ What I've tried P(1): $$5^1 = 1^2 +2^2$$ Which is true P(2): $$5^2 = 3^2 +4^2$$ Which is also true Now for the induction ...
9
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1answer
3k views

Rabin and Shallit Algorithm

I want to implement Rabin and Shallit algorithm for representing a positive integer as a sum of three squares. Can anyone give me a rough sketch of the method? I searched through the internet but ...
9
votes
2answers
1k views

Pythagorean Quadruples:

Consider the set of integers $x_1, x_2, x_3, x_4$ Such that: $$x_1^2 + x_2^2 + x_3^2 = x_4^2$$ How does one compute all the solutions to this system? I have the following method in place for ...
9
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1answer
1k 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+...
9
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1answer
484 views

Weighted sum of squares, in a finite field

Let $\mathbb{F}$ be a finite field. Let $a_1,\dots,a_n \in \mathbb{F}$ be given. I want to know whether there exists $x_1,\dots,x_n \in \mathbb{F}$ such that $$a_1 x_1^2 + a_2 x_2^2 + \dots + a_n ...
9
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1answer
282 views

An Explicit Solution to $a^2+b^2=p$

$\def \op {\operatorname*} \def \C#1#2 {\begin{pmatrix} #1\\#2 \end{pmatrix}}$ I got the following theory from the internet and I seek a proof: Let $p=4k+1 (k \in \Bbb{Z}^+)$ be a given prime. Assume ...
9
votes
0answers
287 views

Looking for all sequences such that $a_i^2+a_j^2=a_k^2+a_l^2$ whenever $i^2+j^2=k^2+l^2$

I'm working in a difficult functional equation, and I have reduced the problem to the following question ($\mathbb{N}$ denotes the set of non negative integers $0,1,2,3,4\cdots$) Question: Can we ...
8
votes
14answers
18k 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 ...
8
votes
3answers
326 views

$ \exists a, b \in \mathbb{Z} $ such that $ a^2 + b^2 = 5^k $

I saw this problem recently and found an elegant solution to it, and was curious to see if anybody would think of something else. Nice solutions to nice problems are fun to see! Problem: Prove that, ...
8
votes
3answers
2k views

Every number is the sum of three squares with signs [duplicate]

The question. Can every $n\in \mathbb N$ can be written: $$n=a^2\pm b^2\pm c^2$$ where $\pm$ are signs of your choice? We know with Lagrange's four-square theorem that every integer can be written as ...
8
votes
7answers
12k views

Find all solutions: $x^2 + 2y^2 = z^2$

I'm use to finding the solutions of linear Diophantine equations, but what are you suppose to do when you have quadratic terms? For example consider the following problem: Find all solutions in ...
8
votes
2answers
539 views

Conjecture: All but 21 non-square integers are the sum of a square and a prime

Update on 6/19/2020. This discussion led to deeper and deeper results on the topic. The last findings are described in my new post (including my two answers), here. I came up with the following ...
8
votes
4answers
328 views

Can every natural number be written as a sum of signed odd squares?

Let $c_n \in \{ -1,1\}$. Here, it is stated that every natural number may be written as $$\sum c_kk^2$$ Where $k$ runs from $1$ to some finite number. I am wondering whether every natural number $n$ ...
8
votes
1answer
464 views

Numbers represented as two different sums of squares

This is an interesting question I came across, and it does not look that easy: $365$ can be written as a sum of $2$ consecutive squares and also $3$ consecutive squares: $$ \large 365 = 14^2 + 13^2 = ...
8
votes
3answers
527 views

My formula for sum of consecutive squares series?

I stumbled upon a specific series, who's Sum of squares of consecutive integers equals the sum of squares of the continuation of that consecutive integers. For exmaple, this first number in the ...
8
votes
1answer
798 views

Algorithm to find solution to $ax^2 + by^2 = 1$ in a finite field

Let $\mathbb{F}$ be a finite field, and let $a,b \in \mathbb{F}$ be given, subject to $a\ne 0, b \ne 0$. Consider the equation $$ax^2 + by^2 = 1.$$ It is guaranteed that there exists a solution to ...
8
votes
1answer
81 views

Is it possible to put all of the $\frac{1}{n}$ side squares into the unit square? [duplicate]

It's well known that $\displaystyle\underset{n\geq1}\sum\frac{1}{n^2}=\frac{\pi^2}{6},$ which is less than $2$, hence $\displaystyle\underset{n\geq2}\sum\frac{1}{n^2}<1$. However, this inequality ...
8
votes
2answers
232 views

Most even numbers is a sum $a+b+c+d$ where $a^2+b^2+c^2=d^2$

I have no idea how hard this conjecture is to prove: Any even number $n\ge 36$ can be written as $n=a+b+c+d$ where $a^2+b^2+c^2=d^2$ and $a,b,c,d\in\mathbb Z^+$. Small exceptions are $n=2, 4, 6, 10, ...
7
votes
3answers
1k views

Let $p$ be a prime so $p\equiv3\pmod4$. If $p|a^2+b^2$, then $p|a,b$

Let $p$ be a prime so $p\equiv3\pmod4$. If $p| a^2+b^2$, then $p| a,b$ How do I prove this small theorem? I know that it's quite useful. Are there other small theorems like this one? I am mostly ...
7
votes
1answer
50 views

If -1 is a sum of squares, then any element is a sum of squares

We have a field $F$ of characteristic different from $2$ in which there exist $a_1, \dots, a_n$ such that $$ a_1^2 + \dots + a_n^2 = -1. $$ Prove that for any $c \in F$, there exist $b_1, \dots, b_k$ ...
7
votes
2answers
233 views

Are there identities which show that every odd square is the sum of three squares?

I am looking for algebraic identities of the form $$ (2n+1)^2 = f(n)^2 + g(n)^2 + h(n)^2, $$ where the functions are polynomials in $n$. EDIT: Evidently $(6k)^2 = 36k^2$ is trivially the sum of ...
7
votes
3answers
2k views

Find the smallest positive integer which can be written as the sum of the squares of two positive integers in two different ways

Find the smallest positive integer which can be written as the sum of the squares of two positive integers in two different ways. I took extremely long to solve this I got $50= 7^2 + 1^2 $ $50= ...
7
votes
2answers
106 views

$\{a+b|a,b\in\mathbb N^+\wedge ma^2+nb^2\in\mathbb P\}=\{k>1|\gcd(k,m+n)=1\}$

Conjecture: $\{a+b|a,b\in\mathbb N^+\wedge ma^2+nb^2\in\mathbb P^{>2}\}=\{k>2|\gcd(k,m+n)=1\}$ if $m,n\in \mathbb N^+$ and $\gcd(m,n)=1$. This is a generalization of Any odd number is of form $...
7
votes
2answers
398 views

Solution to Lagrange's Four Square Theorem with Fewest Terms

Lagrange's four square theorem states that any natural number $n$ can be written as the sum of the square of 4 other integers. For most values of $n$, there are multiple square combinations that work. ...

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