Mathematics Stack Exchange
Mathematics Stack Exchange

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:

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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 with…
Lehs
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Can a number be equal to the sum of the squares of its prime divisors?

If $$n=p_1^{a_1}\cdots p_k^{a_k},$$ then define $$f(n):=p_1^2+\cdots+p_k^2$$ So, $f(n)$ is the sum of the squares of the prime divisors of $n$. For which natural numbers $n\ge 2$ do we have $f(n)=n$ ? It is clear that $f(n)=n$ is true for the…
Peter
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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 second and third images, I am completely lost. For…
user8358234
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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
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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
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Four squares such that the difference of any two is a square?

I. This post asks to find $4$ integers $a,b,c,d$ such that the difference between any two is a square. As mentioned by my answer, it is equivalent to finding $3$ squares such that the difference of any two is also a square. With the positive answer…
Tito Piezas III
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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…
infmagic2047
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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…
volcanrb
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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
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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(…
Ryan
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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
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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…
Mason
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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
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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
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Odd square as sum of $9$ distinct odd squares

I'm interested in representing odd squares as sum of $9$ distinct odd squares. So, let $n\in\mathbb{N}$ be odd and $x_1,\,x_2,\,...,\,x_9\in\mathbb{N}$ be odd and pairwise distinct. The question is, for which $n$ exists a solution for…
summingsummer
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