From: utzoo!decvax!harpo!zeppo!whuxlb!thf
Newsgroups: net.general
Title: Number theory problem
Article-I.D.: whuxlb.348
Posted: Sat Jul 17 19:37:12 1982
Received: Sun Jul 18 04:49:16 1982

Does anyone know whether the following famous problem due to 
Collatz has been solved? For a positive integer n let f(n)
= n/2 if n is even and 3n+1 if n is odd. Show that for each
n there exist a k such that the k-th iterate of f applied to
n is 1.
example: f(5) = 16, f(f(5)) = 8, f(f(f(5))) = 4, f(f(f(f(5)))) = 2, f(f(f(f(f(5))))) = 1, so k = 5 in this case.
The book Fundamentals of Data Structures by Horowitz and 
Sahni has this as problem 29 in chapter 1  (with a hint to use induction) 
which would
lead me to believe that the problem has been solved.
another recent source says that for n<2^50 it is known to be true.