number theory - On elementary proofs of Fermat's Last Theorem

I came across this one of many claimed elementary proofs of FLT. It looked credible, and I felt surprised seeing that it wasn't drawing much attention from anyone. I investigated and I ended up finding this argument ruling out any chances for this kind of "proofs" to be correct. Now, my question shall be organized in three steps:

• I have a very basic understanding of the "trick". I get its underlying logic, but unluckily I have a ridiculous knowledge of rings and fields, and in particular I know almost nothing of $p$-adic numbers. Can you confirm that, reasoning in analogy with the familiar number sets, I can safely assume that having a solution in $Q_p$ would imply a counterpart in $Z_p$? Is there a way to make it comprehensible how a solution made of $p$-adic integers would look like?




• This is what I'm interested in most. Is it possible to understand, at least at an intuitive level, in lay-person's terms, what is the characteristic of the ring of familiar integers that makes it different from other factorial rings? In other words, what is (or are) the features typical only of our beloved usual numbers that make FLT hold for them? In further other words, what properties have been involved by the advanced mathematical tools used to prove FLT?




• Finally, if we can spot such a characteristic, and it necessarily requires the use of instruments that Fermat didn't have, isn't this definitive evidence that Fermat could not have a proof? Why is this still sometimes questioned? Did he have any chance to perform something that did not fall under the disproof of the "trick", something that wouldn't apply to other rings like that of $p$-adic integers?

Answer

I think the claim of that thread is blatantly overstated. For one thing, there are lots of properties that $\mathbb{Z}$ has, but $Z_p$ does not have. First and foremost, well ordering of the positive elements, which is heavily used as a key to solving many diophantine equations.

Now consider Fermat's elementary proof that
$$x^4 + y^4 = z^4$$
has no solutions for $(x,y,z) \in \mathbb{Z}$ with $xyz \ne 0$.

I'm not sure whether or not there is a solution in $p$-adic integers, for some prime $p$, but if such solutions exist, it's an example showing that the existence of qualifying $p$-adic solutions doesn't imply the existence of qualifying integer solutions.

As I indicated in the comments, I suspect that most flawed attempts, at least the ones where the solver knows some Number Theory and is not obviously crazy, would include steps for which there is no analogue in $Z_p$, so the $Z_p$ criterion would be useless for invalidating those attempts.

For a given proposed proof, a more common way of quickly demonstrating that there must be a mistake$\,-\,$without actually pinpointing the error, is to observe that the argument would still work for the equation $x^2 + y^2 = z^2$, or, alternatively, to apply the argument line by line for the equation $x^3+y^3=z^3$, and see if the proof at least works for that case.