Jeff Lagarias
AT&T Bell Laboratories
Murray Hill, New Jersey
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 and with celebrated Diophantine equations such as Fermat's last theorem. Paul Erdos 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 the binary logarithm of 3 and the distribution mod 1 of the sequence {(3/2)^k : k = 1, 2, ...}, 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 3x+1 problem and survey all the literature I am aware of about this problem and its generalizations.