Prove that x²+y²+z² isn't prime

[anemone]

Let $x,\,y,\,z$ be nonzero integers, $x\ne z$ such that $\dfrac{x}{z}=\dfrac{x^2+y^2}{y^2+z^2}$.

Prove that $x^2+y^2+z^2$ cannot be a prime.

 

[mente oscura]

Let $x,\,y,\,z$ be nonzero integers, $x\ne z$ such that $\dfrac{x}{z}=\dfrac{x^2+y^2}{y^2+z^2}$.

Prove that $x^2+y^2+z^2$ cannot be a prime.

Hello.

Spoiler

$$\dfrac{x}{z}-1=\dfrac{x^2+y^2}{y^2+z^2}-1$$

$$\dfrac{x-z}{z}=\dfrac{x^2-z^2}{y^2+z^2}$$

$$\dfrac{1}{z}=\dfrac{x+z}{y^2+z^2}$$

$$y^2+z^2=xz+z^2 \rightarrow{}y^2=xz$$1º) $$Let \ d \in{\mathbb{Z}}/ \ d|x \ and \ d|z \rightarrow{ } d|(x^2+y^2+z^2)$$2º) $$For \ x,z \ coprime \rightarrow{}y^2=a^2b^2/ a,b \in{\mathbb{Z}}/ \ x=a^2 \ and \ z=b^2$$

$$x^2+y^2+z^2=a^4+a^2b^2+b^4=(a^2+ab+b^2)(a^2-ab+b^2)$$

Regards.

 

[anemone]

Good job, mente oscura and thanks for participating!:)