Turing machine that halts if and only if Collatz conjecture is false
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Turing machine that halts if and only if Collatz conjecture is false by 
Ciro Santilli 34 Updated 2024-11-15 Created 1970-01-01
Ciro Santilli 34 Updated 2024-11-15 Created 1970-01-01mathoverflow.net/questions/309044/is-there-a-known-turing-machine-which-halts-if-and-only-if-the-collatz-conjectur suggests one does not exist. Amazing.
Intuitively we see that the situation is fundamentally different from the Turing machine that halts if and only if the Goldbach conjecture is false because for Collatz the counter example must go off into infinity, while in Goldbach conjecture we can finitely check any failures.
Amazing.