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So immediately I just want to say that all these integers n must themselves be squares. Then, I know the following from class:

  1. primes that are the sums of two squares are 1 (mod 4)
  2. n, an integer, is the sum of two squares if it's prime factorization has primes congruent congruent to 3 (mod 4) occurring only with even powers
  3. products of sums of two squares are also the sums of two squares
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    $\begingroup$ You want $n=x^2+y^2$ and $n=z^2$, right? So we have all Pythagorean triples. $\endgroup$ – Dietrich Burde Nov 16 '19 at 20:05
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    $\begingroup$ wrong @Dieterich we want $$n=x^2+y^2\land (x=0\lor y=0)$$ $\endgroup$ – user645636 Nov 16 '19 at 22:28
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Let $n=x^{2}+y^{2}$ be such an integer. if $x \neq 0$ and $y \neq 0$ we have a contradiction. So we may assume that $y = 0$.

Therefore $n = x^{2}$.$ \ $ Hence $n$ is a square.

Assume $n>1$, since $n$ can't be written as as the sum of two nonzero squares, all of its odd prime factors must be $3$ mod $4$.

If not, assume $p | n$ and $p = 1 $ mod $4$

Then we may write:$ \ \ \ p=a^{2}+b^{2} \Rightarrow p^2 = (a^{2}-b^{2})^{2}+(2ab)^{2}.$

$n = \frac{x^{2}}{p^{2}} p^{2} = [\frac{x}{p}(a^{2}-b^{2})]^{2}+[\frac{x}{p}(2ab)]^{2}$ which is the sum of 2 non-zero squares.

Therefore $n$ is either $0, 1$ or a number of the form: $\ \ \ \ 2^{2k}\prod_{ì=1}^{s}p_{i}^{2k_{i}},$

Where each $p_{i}$ is $3$ mod $4$.

On the other hand, if $n$ can be written in such a way, and $ \ n = x^{2}+y^{2}$,

$\Rightarrow n = (x + yi)(x-yi)$, By unique factorization of $\mathbb{Z}[i]$, there are 2 cases to consider:

Either one of the $p_{i}$ would factor non trivially in $\mathbb{Z}[i]$, but it doesn't as $p_{i}$ is $3$ mod $4$.

The only other possibility is that $\ \ 2^{2k}=a^{2}+b^{2}, \ a \neq 0, \ b \neq 0$

but you can check that this is not possible by taking congruences mod $4$.

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