I am trying to solve and understand Catalan's conjecture from this source.
As it is known, the conjecture states that the only solution in terms of consecutive numbers for$$x^p-y^q=1 \tag*{(1)}$$ is $(x,y,p,q)=(3,2,2,3)$.
Now, to solve $(1)$ in the general sense, the source (Page no. 4) does the following, $$x^p-1=y^q$$ $$(x-1)\frac{x^p-1}{x-1}=y^q \tag*{(2)}$$
then the source goes on to consider $x^p$ as $((x-1)+1)^p$. Later they conclude that the GCD of the terms on the left hand side of $(2)$ is either $p$ or $1$.
How do they find the GCD here? And how can there be two GCDs (i.e., $p$ or $1$)?
Any kind of help will be appreciated.
Cheers!
$$ \frac{x^p - 1}{x - 1} = \sum_{k = 0}^{p - 1} x^k \equiv \sum_{k = 0}^{p - 1} 1^k = p \pmod{x - 1}, $$ so$$ \left( \frac{x^p - 1}{x - 1}, x - 1 \right) = (p, x - 1) = 1 \text{ or }p. $$