On 9/13/06, Marc LeBrun <mlb@well.com> wrote:
Heh! Actually, I've found a marvelous cycle of length 442783582734897230450239994058234089230450023451234951 but the margin is too small to contain the start value!
More seriously, it's always seemed to me that Collatz might be a member of some parametric family--including others like 3n-1, 3n+5, 17n+7, 105n+69...say. Then a "real" answer would be a theory that predicts the cycle structure given the parameters--as opposed to just this one instance. Or, perhaps at least a negative result such as that it's incomputably chaotic...
Conway proved it's undecidable. Any FRACTRAN program can be converted to a Collatz-like form, thus the general problem of predicting cycles is as hard as the Halting problem. Of course, that doesn't say anything about the original Collatz problem, and I'd be very surprised if it's universal for computation (only one known non-halting program!)
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