Korva is a scroll of the internet’s best ideas.
Lightning bolts, mountains, and hustle
Lightning bolts
A lightning bolt connects expertise in one field to expertise in another. The Erdős problem numbered 1196, about primitive sets, had that character: a Markov chain approach, working probabilistically from the bottom up, combined with the von Mangoldt function. A mathematician in the know hears that and can run with it, so these proofs are very human-parsable.
Mountains
A mountain is a new body of theory. Fermat's Last Theorem needed elliptic curves and modular forms, two mountains of ideas that both had to be built before the right question connecting them could be asked. The abc conjecture is the closest example of a mathematician in Japan producing a whole new way of thinking, called inter-universal geometry. It took mathematicians a long time to parse, and it probably is not a correct solution.
Raw hustle
Raw hustle is a thousand-page proof with no new theories, which no one gets much out of. That one calls for a digestion process that squeezes it into the succinct, compressed version of its ideas that humans can understand.
How did Fermat's Last Theorem finally fall?
In 1986, Ken Ribet showed that a proof of the Taniyama-Shimura conjecture about elliptic curves and modular forms would imply Fermat's Last Theorem. Andrew Wiles spent about seven years proving enough of that conjecture, and announced a proof in 1993. A gap was found and repaired with Richard Taylor, and the final proof appeared in 1995.
Why does the abc conjecture matter so much?
The abc conjecture relates the prime factors of two numbers that add to a third. Many results in number theory follow from it, including a version of Fermat's Last Theorem for all large exponents. It also implies Faltings's theorem on rational points on curves in a new way. Joseph Oesterle and David Masser formulated it in the mid-1980s.
What is the longest proof ever written?
The classification of finite simple groups is usually named as the longest. Its proof is spread over hundreds of journal papers by more than a hundred authors, totaling tens of thousands of pages. Work on it ran from the 1950s into the early 2000s, with additional revision efforts continuing afterward.
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