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Was this what led to proving Fermat’s last theorem? Have those intersections and connections been explored and used, yet? It seems to me that the proof required a machine to complete it (or am I incorrect). Machine proofs were only fairly recently accepted as proving anything. In any event, It was a huge step forward.
Yes, fermat is a special case of the Langlands program.
No, wiles' proof did not require computers other than what is fairly common for ultra complicated theoretical proofs: for verification purposes.
A famous proof that was achieved with computers is the map colouring one.
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As far as I know, the computer proofs run an enormous amount of iterations of the problem, perhaps regressively, until they reach a solution or contradiction. There was contention over whether the computer proofs were mathematically sound or not. I don’t have the expertise to weigh in on that controversy. I think it was settled that the computer proofs are acceptable to mathematicians.
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