Hi all; I just ended to write a file which collects some cases of Mihailescu theorem that are solvable directly with elementary tools, and that can be useful to a student following math contests; in particular, given the equation in integer $x^p-y^q=1$, the following cases are studied: $2\mid p$, $2\mid q$ (both are historically known), $y\mid x-1$ (that is a kind of generalization of class of problems, like $y$ prime), $x\mid q$ and $\text{gcd}(y,p)=1$ s.t. $y\le 2^p$ (last two ones are original, as far as I know).

My questions are:

Are last two cases really not known, or there exists some kind of generalization solvable with elementary tools? Can it be suitable of publication somewhere? In case, I tought about Mathematical Reflections, or Mathematical Magazine, but I don't know other ones

(In case, I can upload the actual version of the file; please, edit the tags below, if there are better ones)