Can you spot the mistake here?

Question

assume that : $$x^{3} + y^{3} = z^{3}$$ so : $$z^{3} - y^{3} = x^{3}$$ $$z^{3} ≡ y^{3}\bmod x$$ $$z ≡ y\bmod x $$ $$z-y = x $$ and now let's plug this result to the original equation: $$(z-y)^{3} + y^{3} = z^{3}$$ $$z^{3}-3z^{2}y+3zy^{2}-y^{3}+y^{3}=z^3$$ by simplfying, we get this: $$3zy^{2} = 3z^{2}y$$ dividing by $3zy$: $$y=z$$ so this equation has solutions if and only if $y = z$, and now let's plug our results to the original equation: $$x^{3}+y^{3} = y^{3}$$ $$x^{3} = 0$$ $$x = 0$$ Therefore :There is no solution when: $$x,y,z\in\mathbb{N}$$ and by mathematical induction we can prove it (instead of the power 3) for all $ n\in\mathbb{N}$

Answer 1

$z \equiv y \pmod x$ does not imply $z-y=x$. We could have $z=y$ or $z-y=2x$ or many other possibilities.