- #1

evinda

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We suppose that the integers $x,y,z$ satisfy $x^2+2y^2=z^2$ and $(x,y)=1$ . I want to show that $(x,z)=(y,z)=1$, and that $x$ is odd and $y$ even.

I have tried the following:

Let $(x,z)=d>1$. Then there exists a prime number $p$ such that $p \mid d$.

Since $d \mid x$ and $d \mid z$, we get that $p \mid x$ and $p \mid z$. So $p \mid x^2$, $p \mid z^2$.

Thus $p \mid z^2-x^2=2y^2$. But then how can we deduce that $p \mid y^2$, so that we could get a contradiction? (Thinking)