Some idea about why Wiles's work doesn't solve Beal conjecture: a soft question

Question

I know that Wiles provided a proof of Fermat's Last Theorem, as you can see for example from Wikipedia. But I know it (the fact that Wiles solved the problem) from an informative point of view. On the other hand I know a related/similar problem to Fermat's Last Theorem, that is the so-called Beal conjecture, this Wikipedia.

Then the situation (from this my point of view, from the ignorance) is that a very elaborated and complicated theory (if I'm right about modular forms and elliptic curves) provide us a proof of Fermat's Last Theorem but it doesn't provide us a proof of the Beal conjecture.

Question. Can you provide me an explanation (the detail*) about why the theory developed in Wiles's work doesn't solve Beal conjecture? Many thanks.

*I'm asking it as a soft question, and of course I understand that Fermat's Last Theorem and Beal conjecture are different problems, thus I am asking to get some idea/feedback (the mentioned detail) about why Wiles's work doesn't solve Beal conjecture, if such explanation is feasible in easy words.

Answer 1

The way Wiles did it was by proving the Taniyama-Shimura Conjecture for semistable elliptic curves. This meant that all elliptic curves (that mattered for this number-theoretic question) were secretly modular (or, more technically, had a modular parametrisation through a Galois representation).

Why was this important? Because of the Frey Curve - this essentially was an elliptic curve that, assuming there was a counter-example to FLT, would exist and would seemingly be not modular. This was proven as Ribet's Theorem which stated further that if there were no non-modular elliptic curves of a specific kind (the semi-stable variety), then there would be no counterexamples to FLT (i.e. FLT was true).

So, while I don't know enough about the subject to know "why" creating an analagous curve to Frey's that had more general counterexamples to things referenced in Beal's Conjecture didn't happen/was impossible, I know the reason why Wiles' Proof was not related was because he simply was "stopping Frey's Curve from existing", which spoke only on FLT.

Answer 2

From a little reading up on it, I would say there are two key reasons:

1. Fermat's Last Theorem is a special case of Beal's conjecture. Beal's asks about $a^x+b^y=c^z$, while FLT only concerns $a^x+b^x=c^x$
2. (and the more likely reason why Wiles' work doesnt cover Beal). FLT states there are $no$ solutions to $a^x+b^x=c^x$, however, Beal's regards that there $are$ solutions to $a^x+b^y=c^z$, and wants to know what qualities those solutions have. This disparity is most likely why Wiles' work on Fermat's couldn't bridge the gap.