The 3x+1 Conjecture is simple to state and apparently intractably hard to solve. It shares these properties with other iteration problems, for example that of aliquot sequences (see Guy [36], Problem B6) and with celebrated Diophantine equations such as Fermat's last theorem. Paul Erdös commented concerning the intractability of the 3x+1 problem: ``Mathematics is not yet ready for such problems.'' Despite this doleful pronouncement, study of the 3x+1 problem has not been without reward. It has interesting connections with the Diophantine approximation of and the distribution of the sequence , with questions of ergodic theory on the 2-adic integers , and with computability theory --- a generalization of the 3x+1 problem has been shown to be a computationally unsolvable problem. In this paper I describe the history of the problem and survey all the literature I am aware of about this problem and its generalizations.

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