Marc writes: << 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...

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In fact, Jeff Lagarias has painstakingly annotated every paper on the subject he could get his hands on (and at least listed the others!) in two bibliographies, one up through 2000, and another 2001-2006, which are easily googlable with the string lagarias collatz and freely downloadable in several formats. A number of suggestions related to Marc's have been given some attention. (Note: replacing 3n+1 by 3n-1 is equivalent to interchanging the roles of positive and negative integers.) --Dan