Elementary cases of Mihailescu theorem

Question

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)

Answer 1

Of course, you can just consider the case when $p$ and $q$ are primes. A good reference for your question is, I think, Schoof's monograph on Catalan's equation. The case $q = 2$ is solved in Ch. II, and it involves some arithmetic in the ring of Gaussian integers (the argument goes back to V.A. Lebesgue). The case $p = 2$ is discussed in Chs III, for $q \ge 5$ (basic algebra in the ring $\mathbb Z[\sqrt{y}]$), and IV, for $q = 3$ (some arithemtic in the ring $\mathbb Z[\sqrt[3]{2}]$): This is just a little bit more laborious than the case $q = 2$, but still rather "elementary". In the appendix, you can also find Euler's original proof for $(p,q) = (2,3)$, which is essentially based on the same descent argument, though in a disguised form, used to prove that the group of rational points on the elliptic curve $x^2-y^3=1$ is finite and has order $6$. Some others of your cases follow at once from Cassel's theorem (Ch. VI), whose proof in the book relies on Runge's method (Ch. V).