MHB Show that the diophantine equation has only these solutions

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The diophantine equation \(x^2+y^2=2\) has only four integer solutions: \((1,1), (1,-1), (-1,1), (-1,-1)\). The reasoning involves recognizing that both \(x^2\) and \(y^2\) must be non-negative and less than or equal to 2, leading to possible integer values of \(\pm 1\) and 0. However, only the combinations involving \(\pm 1\) satisfy the equation. The suggestion to use modulo reasoning is unnecessary, as the integer solutions can be derived straightforwardly. The incorrect solutions \((\pm 2, 0)\) do not satisfy the equation, confirming the validity of the identified solutions.
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Hello! (Wave)

I want to show that the diophantine equation $x^2+y^2=2$ has only these solutions:
$$(x,y)=(1,1),(1,-1), (-1,1), (-1,-1)$$

That's what I have tried:

$$x^2+y^2=2 \Rightarrow x^2=2-y^2$$

$$x^2>0 \Rightarrow 2-y^2>0 \Rightarrow y^2<2 \Rightarrow -\sqrt{2}<y< \sqrt{2} \Rightarrow y=\pm 1,0$$

  • $y=0: x= \pm \sqrt{2} \notin \mathbb{Z}$
  • $y=\pm 1: x= \pm 1$

So, the solutions are $(\pm 1, \pm 1), (\pm 2,0)$.

Is it right? Could I also find the solutions, using modulo?

Also, how can I show that these solutions are the only ones? (Thinking)
 
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Are you sure you are asked to find integer solutions to $x^2 + y^2 = 2$? Is there any chance *rational* solutions were asked?

In the case of integers, it's trivially easy and you're overcomplicating it. $x^2 + y^2 = 2$ and $x^2, y^2$ are both positive (squares can't take negative values). Hence $x^2 \leq 2$ and $y^2 \leq 2$. The only possible integer values of $x$ and $y$ satisfying these are $(x, y) = (\pm 1, \pm 1), (\pm 1, 0), (0, \pm 1), (0, 0)$

And from all of them, only the first four $(\pm 1, \pm 1)$, i.e., $(1, 1), (-1, 1), (1, -1), (-1, -1)$ satisfies the equation. No modulo argument is needed.

EDIT : Oh I observed that you enumerated the wrong solutions $(x, y) = (\pm 2, 0)$. There doesn't work as $x^2 + y^2 = 4 + 0 \neq 2$.
 
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