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collatz-conjecture
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A child can run it and nobody can prove it. Terence Tao showed in 2019 that almost all orbits reach almost bounded values, the strongest result so far. Conway showed that natural generalizations are undecidable, which hints that the specific numbers 3 and 1 matter. A counterexample would be a number a computer can check, which makes this one of the few famous problems where a finite search could still end it.
Posed 1937, Lothar Collatz.
What would settle it
Lean statement in formal-conjectures