# 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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### 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 $x^2+y^2=z^2?$
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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 ...
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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 ...
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### 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 ...
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### (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 $\mathbb{N}$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $\exists a, b \in \mathbb{Z}$ 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $n$ ...
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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 $50= 7^2 + 1^2$ $50= ... 2answers 106 views ###$\{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$. This is a generalization of Any odd number is of form$...
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. ...