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.