- "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 oddintegers n to 3n+1 and even integers n to n/2.The 3x+1 Conjecture asserts that, starting from any positiveinteger n, repeated iteration of this function eventually produces thevalue 1.
The 3x+1 Conjecture is simple to state and apparentlyintractably hard to solve.It shares these properties with other iteration problems, for examplethat of aliquot sequences and with celebrated Diophantine equations such as Fermat's lasttheorem.Paul Erdos commented concerning the intractability of the3x+1 problem: "Mathematics is not yet ready for such problems."Despite this doleful pronouncement, study of the 3x+1 problem hasnot been without reward.It has interesting connections with the Diophantine approximation ofthe binary logarithm of 3 and the distribution mod 1 of the sequence {(3/2)^k : k = 1, 2, ...}, with questions of ergodictheory on the 2-adic integers, and with computability theory- a generalization of the 3x+1 problem has been shown to be acomputationally unsolvable problem."
The video The Simplest Impossible Problem is accessible to non-mathematicians.
The bookThe 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.
ONLINE Calculator! Enter your choice of starting numbers!
