# 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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### 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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### 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 ...
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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 ...
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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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### 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 ...
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### 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?
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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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### (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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### 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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### 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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### 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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### 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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### $\exists a, b \in \mathbb{Z}$ 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### Can the sum of the first n squares of primes be a perfect square?

So far, all I have found is for $n=1$, for which the square is 4. (Rather trivial) To give some context, I recently watched the Numberphile video Squared Squares. Basically, a squared square is a ...
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