solve-inside-Z-3-x-2-y-2-z-2-2xyz-

Question Number 73047 by mathmax by abdo last updated on 05/Nov/19

s o l v e i n s i d e Z^{3} x^{2} + y^{2} + z^{2} = 2 x y z

Answered by mind is power last updated on 07/Nov/19

2 ∣ 2 xyz \Rightarrow 2 ∣ x^{2} + y^{2} + z^{2}

\Rightarrow x, y, z can not bee all odd

lets asum x = 2 \times 1 \Leftrightarrow

{4 a}^{2} + y^{2} + z^{2} = 4 ayz \Rightarrow y^{2} + z^{2} = 4 (x_{1} yz - x^{2} 1)

\Rightarrow 4 ∣ y^{2} + z^{2} \Rightarrow y = {2 y}_{1} & z = {2 z}_{1}

equation becom

4 (x_{1}^{2} + y_{1}^{2} + z_{1}^{2}) = {8 x}_{1} y_{1} z_{1}

\Leftrightarrow x_{1}^{2} + y_{1}^{2} + z_{1}^{2} = {2 x}_{1} y_{1} z_{1}

\Rightarrow 2 ∣ x_{1}, y_{1}, z_{1}

so \forall n \in IN 2^{n} ∣ x, y, z \Rightarrow x = y = z = 0