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And when it can't, attempt to classify which subset of the integers can be reached. E.g. Legendre's three-square theorem.
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math.stackexchange.com/questions/700235/is-there-an-easy-proof-for-the-classification-of-6-transitive-finite-groups says there aren't any non-boring ones.
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bbchallenge.org/story#what-is-known-about-bb lists some (all?) cool examples,
- BB(15): Erdős' conjecture on powers of 2, which has some relation to Collatz conjecture
- BB(27): Goldbach's conjecture
- BB(744): Riemann hypothesis
- BB(748): independent from the Zermelo-Fraenkel axioms
- BB(7910): independent from the ZFC
wiki.bbchallenge.org/wiki/Cryptids contains a larger list. In June 2024 it was discovered that BB(6) is hard.
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