Questions tagged [sums-of-squares]

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

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

These topics include, for example:

654 questions
50
votes
3 answers

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 $a^2+b^2\in\mathbb P$. $\mathbb P$ is the set of prime numbers. I wish help…
27
votes
3 answers

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 second and third images, I am completely lost. For…
user8358234
  • 588
  • 4
  • 16
27
votes
9 answers

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 another library trip to a less than local library…
Mike
  • 12,679
  • 4
  • 21
  • 45
26
votes
3 answers

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 proof for this, but it seems to elude me. Thanks
barak manos
  • 42,243
  • 8
  • 51
  • 127
23
votes
1 answer

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 distance of $\sqrt{N}$ from its current square. That…
20
votes
5 answers

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 $x^2+y^2=z^2?$
TCL
  • 13,592
  • 5
  • 26
  • 73
20
votes
2 answers

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 what would happen to the value of the series if…
20
votes
3 answers

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 from the Facebook Hacker Cup Source: Facebook…
Jacob
  • 303
  • 1
  • 2
  • 7
19
votes
1 answer

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: ($n=7$) $0 \equiv 0^2+0^3 \left( \text{mod } 7 \right)$ $1 \equiv 1^2+0^3 \left( \text{mod } 7 \right)$ $2 \equiv 1^2+1^3 \left(…
18
votes
2 answers

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 mean an explicitly defined series of rationals. A…
16
votes
1 answer

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 …
barak manos
  • 42,243
  • 8
  • 51
  • 127
15
votes
4 answers

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?
Rohan Shinde
  • 9,392
  • 1
  • 13
  • 57
15
votes
1 answer

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 conjecture implies Any odd number is of form $a+b$…
Lehs
  • 13,268
  • 4
  • 23
  • 72
14
votes
4 answers

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 contraposition that the representation is not unique,…
Hawk
  • 6,150
  • 24
  • 64
14
votes
5 answers

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?
Saaqib Mahmood
  • 23,806
  • 12
  • 54
  • 172
1
2 3
43 44