## Collatz Problem

A problem posed by L. Collatz in 1937, also called the 3x+1 Mapping, Hasse's Algorithm, Kakutani's Problem, Syracuse Algorithm, Syracuse Problem, Thwaites Conjecture, and Ulam's Problem (Lagarias 1985). Thwaites (1996) has offered a 1000 reward for resolving the Conjecture. Let be an Integer. Then the Collatz problem asks if iterating

 (1)

always returns to 1 for Positive . This question has been tested and found to be true for all numbers (Leavens and Vermeulen 1992), and more recently, (Vardi 1991, p. 129). The members of the Sequence produced by the Collatz are sometimes known as Hailstone Numbers. Because of the difficulty in solving this problem, Erdös commented that mathematics is not yet ready for such problems'' (Lagarias 1985). If Negative numbers are included, there are four known cycles (excluding the trivial 0 cycle): (4, 2, 1), (, ), (, , ), and (, , , , , , , , , , ). The number of tripling steps needed to reach 1 for , 2, ... are 0, 0, 2, 0, 1, 2, 5, 0, 6, ... (Sloane's A006667).

The Collatz problem was modified by Terras (1976, 1979), who asked if iterating

 (2)

always returns to 1 for initial integer value . If Negative numbers are included, there are 4 known cycles: (1, 2), (), (, , ), and (, , , , , , , , , , ). It is a special case of the generalized Collatz problem'' with , , , , and . Terras (1976, 1979) also proved that the set of Integers has stopping time has a limiting asymptotic density , such that if is the number of such that and , then the limit
 (3)

exists. Furthermore, as , so almost all Integers have a finite stopping time. Finally, for all ,
 (4)

where
 (5) (6) (7)

(Lagarias 1985).

Conway proved that the original Collatz problem has no nontrivial cycles of length . Lagarias (1985) showed that there are no nontrivial cycles with length . Conway (1972) also proved that Collatz-type problems can be formally Undecidable.

A generalization of the Collatz Problem lets be a Positive Integer and , ..., be Nonzero Integers. Also let satisfy

 (8)

Then
 (9)

for defines a generalized Collatz mapping. An equivalent form is
 (10)

for where , ..., are Integers and is the Floor Function. The problem is connected with Ergodic Theory and Markov Chains (Matthews 1995). Matthews (1995) obtained the following table for the mapping
 (11)

where .

 # Cycles Max. Cycle Length 0 5 27 1 10 34 2 13 118 3 17 118 4 19 118 5 21 165 6 23 433

Matthews and Watts (1984) proposed the following conjectures.

1. If , then all trajectories for eventually cycle.

2. If , then almost all trajectories for are divergent, except for an exceptional set of Integers satisfying

3. The number of cycles is finite.

4. If the trajectory for is not eventually cyclic, then the iterates are uniformly distribution mod for each , with
 (12)
for .

Matthews believes that the map
 (13)

will either reach 0 (mod 3) or will enter one of the cycles or , and offers a $100 (Australian?) prize for a proof. See also Hailstone Number References Applegate, D. and Lagarias, J. C. Density Bounds for the Problem 1. Tree-Search Method.'' Math. Comput. 64, 411-426, 1995. Applegate, D. and Lagarias, J. C. Density Bounds for the Problem 2. Krasikov Inequalities.'' Math. Comput. 64, 427-438, 1995. Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972. Burckel, S. Functional Equations Associated with Congruential Functions.'' Theor. Comp. Sci. 123, 397-406, 1994. Conway, J. H. Unpredictable Iterations.'' Proc. 1972 Number Th. Conf., University of Colorado, Boulder, Colorado, pp. 49-52, 1972. Crandall, R. On the ' Problem.'' Math. Comput. 32, 1281-1292, 1978. Everett, C. Iteration of the Number Theoretic Function , .'' Adv. Math. 25, 42-45, 1977. Guy, R. K. Collatz's Sequence.'' §E16 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 215-218, 1994. Lagarias, J. C. The Problem and Its Generalizations.'' Amer. Math. Monthly 92, 3-23, 1985. http://www.cecm.sfu.ca/organics/papers/lagarias/. Leavens, G. T. and Vermeulen, M.  Search Programs.'' Comput. Math. Appl. 24, 79-99, 1992. Matthews, K. R. The Generalized Mapping.'' http://www.maths.uq.oz.au/~krm/survey.ps. Rev. Mar. 30, 1999. Matthews, K. R. A Generalized Conjecture.'' [$100 Reward for a Proof.] ftp://www.maths.uq.edu.au/pub/krm/gnubc/challenge.

Matthews, K. R. and Watts, A. M. A Generalization of Hasses's Generalization of the Syracuse Algorithm.'' Acta Arith. 43, 167-175, 1984.

Sloane, N. J. A. Sequence A006667/M0019 in An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Terras, R. A Stopping Time Problem on the Positive Integers.'' Acta Arith. 30, 241-252, 1976.

Terras, R. On the Existence of a Density.'' Acta Arith. 35, 101-102, 1979.

Thwaites, B. Two Conjectures, or How to win 1100.'' Math.Gaz. 80, 35-36, 1996.

Vardi, I. `The Problem.'' Ch. 7 in Computational Recreations in Mathematica. Redwood City, CA: Addison-Wesley, pp. 129-137, 1991.