Many months ago, I have found this paper: https://hal.archives-ouvertes.fr/file/index/docid/678031/filename/ethcatalan2.pdf which is supposed to give an elementary proof of Catalan's conjecture, a.k.a. Mihailescu theorem. Back then I wasn't really able to follow the paper (many transitions involve ad hoc defined numbers and their definitions are all over the paper) but did not question its correctness.
Today I have found it again, and when looking at it I started to think there is something fishy going on. Having heard a little bit about background theory of original proof I have really hard time believing that such simple tools can prove this. Paper is still hard to follow, so I haven't identified any error, but what supports my belief is that most of the proof doesn't even use the fact that the numbers involved are integers, so same method can give some false results about real powers.
Is this paper a valid elementary proof of Mihailescu theorem? If not, where is the flaw?
This worries me even more because this paper is one of the top hits on Google when looking for this theorem's proof…
Thanks in advance.
Best Answer
I had a little go at reading it. The paper is a mess with like $20$ help-variables together with the original $4$ defined randomly throughout making it almost impossible to follow. Finding the mistake(s) is not easy, but I think I have found a serious flaw.
The main part of the argument (his first proof) starts at page 4 (before it is just the special case $c,c' = 1$). Without using anywhere the fact that the variables have to be integers and by just defining new variables and making simple manipulations the author arrives at the result (mid-page 6) that if $Y^p = X^q + 1$ then $X^p = 4$. The only assumption is that $c = \frac{X^p - 1}{Y^{p/2}}\not= 1$ and $c' =\frac{7-X^p}{Y^{p/2}}\not= 1$. By the method used this should also hold if $X,Y,p,q$ are real numbers which is obviously wrong.
I should note that there is a possibillity that the author have been using hidden assumptions / deductions (i.e. using divisibillity properties without saying so). If this is the case then it becomes impossible to follow so at best we can say that the proof is flawed. Another issue is that the author uses cases without naming them so we have arguments like “But .. thus .. and .. hence .. thus .. or .. also .. and .. hence .. or .. and .. or .. and .. and .. hence”. What is in each case / subcase is hard to read from this. To reach the conclusion above I have tried to read the cases and sub cases as they are most naturally interpreted.