The 3x+1 Problem


"The 3x+1 problem, also known as the Collatz problem, the Syracuse problem, Kakutani's problem, Hasse's algorithm, and Ulam's problem, concerns the behavior of the iterates of the function which takes odd integers n to 3n+1 and even integers n to n/2. The 3x+1 Conjecture asserts that, starting from any positive integer n, repeated iteration of this function eventually produces the value 1.

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."

From the introduction of an article by Jeff Lagarias: "The 3x+1 problem and its generalizations", American Mathematical Monthly, Volume 92, 1985, 3 - 23.

The book The Ultimate Challenge: The 3x+1 Problem appeared in 2010.

A conference on the 3x+1 problem (with proceedings).

A Continuous Extension of the 3x+1 Problem to the Real Line , Dynamics of Continuous, Discrete and Impulsive Systems, Volume 2, (1996), pp. 495-509.

My 2003 update on the 3x+1 problem first appeared as "Una actualizacio del problema 3x+1" (Catalan, translated by Toni Guillamon i Grabolosa), Butlleti de la Societat Catalana de Matematiques, v.22, 2003, 1-27.

Eric Roosendaal's page with LOTS of 3x+1 material, including links to on-line calculators.