What would one write in a paper describing a counterexample to a published proof?

Question

Inspired by What steps should I take when contacting another researcher after finding possible errors in their work? where the OP describes something similar.

Let's say I've found a counterexample to Fermat's Last Theorem. The theorem is easily understood, and any counterexample can be quickly verified. It's also been proven as true by Andrew Wiles in 1994, so a counterexample would imply an error in Andrew Wiles' proof.

What should the paper I write giving the counterexample look like? Do I need to give a historical background of the theorem? Do I need to find the error in Andrew Wiles' proof?

I'm looking for an example where the counterexample is easily understood with high school mathematics (as in the linked question), and where the conjecture/theorem has a published proof. The closest paper I'm aware of seems to have been abnormally short, and Euler's conjecture did not have a 'proof'. I vaguely remember reading about a proof that the cube root of some number is irrational that was proven wrong by a counterexample, I believe in one of Martin Gardner's columns for Scientific American, but this seems to imply that I remembered wrong. Other counterexample papers I found from a Google search all seem to be quite complex, and it takes real effort to prove that the counterexample is a counterexample. If there is no such example, a "best practice" description would also be helpful.

Footnote: I don't actually have a counterexample to Fermat's Last Theorem.

Answer 1

A famous counterexample in knot theory is the observation by Ken Perko that two knots long thought to be distinct are in fact equivalent; On the classification of knots, Proc. Amer. Math. Soc. 45 (1974), 262—266. The paper is mostly about classification of the ten-crossing knots according to their linking numbers, but along the way Perko notes:

This single result is certainly elementary, in that anyone could look at the sequence of pictures - possibly aided by a piece of string - and confirm that the manoueuvers work. Although it is not explicitly stated in the paper, the observation also disproves a "theorem" of C. N. Little in Non-alternating +/- knots, Trans. Roy. Soc. Edinburgh 39 (1900), who claimed (p774):

Theorem: The total twist of a reduced knot is constant for all forms in which the knot can be projected.

or in modern language, that the writhe is an invariant.

An answer on Math Overflow by Daniel Moskovich gives more detail.

Perko does not explore exactly why Little's proof went wrong, perhaps because it was from an early era of knot theory where much of the approach had already been superseded, or regarded as insufficiently rigorous in general.

An FLT counterexample would be at the other end of the spectrum since Wiles's proof has been closely scrutinized, and connects with many other mathematical results that are well-studied: so it would be more surprising to find out that it was wrong. In such a case it would be very interesting for mathematicians to understand exactly why the original proof went astray. But I expect that because of the theorem's prominence, a counterexample would still be publishable without such an account.

But in general, I'd expect a counterexample paper to say more than "here is the counterexample". For example, if it was found by computer search then there may be interesting things to say about how the search was done. There could also be results like "any counterexample must have such-and-such form", or "the original theorem is still true for quasiregular bloboids, but fails for the general case". Even if the eventual counterexample is easy to check by elementary means, these additional bits of context can account for some of the complexity you've noticed. (Similarly, the main results of Perko's paper involve words like "autohomeomorphism" even though the counterexample is in terms of "look at this picture".) This is fundamentally because a paper showing a counterexample is still a mathematical paper to be judged by the same standards as any other, regarding noteworthiness, motivation, rigour, etc. These factors also relate to where a paper might appear.