Can't understand part of proof of Catalan conjecture

Question

So I'm trying to understand this proof or at least part of it and I'm stumped at a point that seems pretty basic.

In the ring $Z[i]$ we have $x^p = y^2 + 1 = (y - i)(y + i)$. It is known that $Z[i]$ is a unique factorisation domain. Now there is no prime $\pi \in Z[i]$ such that $\pi|y + i$ and $\pi|y - i$. For suppose there is such a prime., then it has to divide $y + i - (y - i) = 2i$, so $\pi$ divides 2, since $i$ is a unit. It follows that 2 divides $x$, which we have proven to be impossible.

that last step is what confuses me ("it follows that 2 divides x"). I can see how that fact would follow if 2 divided $\pi$, as then, since $\pi$ divides, say $y - i$, it divides $x^p$ and hence x itself. But how the contradiction follows from the fact that $\pi$ divides 2 eludes me. I can also see how that would follow if $\pi$ dividing 2 meant that $\pi$ equals 2. And that would be the case in the normal ring of integers but since these are Gaussian integers, I'm not sure that's the case...

Answer 1

Since $\pi \mid y+i$ then $N(\pi) \mid N(y+i) = y^2+1 = x^p$

But $\pi \mid 2$ implies $\pi \sim 1+i$, so $N(\pi) = 2$

Answer 2

Norms show you that the only possible primes dividing $2$ are $\pm (1 \pm i)$, and since $x$ is real, if one of these primes divides $x$, so too must its conjugate. That's because complex conjugation is a ring automorphism. If both $1+i$ and $1-i$ divide $x$, then so too must their product, so $2=(1+i)(1-i) = (-1+i)(-1-i) \vert x$.