Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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)

share|improve this question
add comment

1 Answer

up vote 2 down vote accepted

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).

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.