A problem posed by L. Collatz in 1937, also called the 
mapping, 
problem, 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 one form of Collatz problem asks
if iterating
 |
(1)
|
always returns to 1 for positive 
. (If negative numbers are included,
there are four known cycles (excluding the trivial 0 cycle): (4, 2, 1), (
, 
), (
, 
, 
, 
, 
), and (
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
).)
The members of the sequence produced by the Collatz are sometimes known as hailstone numbers. 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. Kurtz and Simon (2007)
proved that a natural generalization of the Collatz problem is undecidable; unfortunately,
this proof cannot be applied to the original Collatz problem.
The Collatz algorithm has been tested and found to always reach 1 for all numbers 
(Oliveira e Silva 2008), improving the earlier results of 
(Vardi 1991, p. 129) and 
(Leavens and Vermeulen 1992). Because of the
difficulty in solving this problem, Erdős commented that "mathematics is
not yet ready for such problems" (Lagarias 1985).
The following table gives the sequences obtained for the first few starting values (OEIS A070165).
 |  ,  ,  , ... |
| 1 | 1 |
| 2 | 2, 1 |
| 3 | 3, 10, 5, 16, 8, 4, 2, 1 |
| 4 | 4, 2, 1 |
| 5 | 5, 16, 8, 4, 2, 1 |
| 6 | 6, 3, 10, 5, 16, 8, 4, 2, 1 |
The numbers of steps required for the algorithm to reach 1 for 
, 2, ... are 0, 1, 7, 2, 5, 8, 16, 3, 19, 6, 14, 9, 9,
17, 17, 4, 12, 20, 20, 7, ... (OEIS A006577;
illustrated above). Of these, the numbers of tripling steps are 0, 0, 2, 0, 1, 2,
5, 0, 6, ... (OEIS A006667), and the number
of halving steps are 0, 1, 5, 2, 4, 6, 11, 3, 13, ... (OEIS A006666).
The smallest starting values of 
that yields a Collatz sequence containing 
, 2, ... are 1, 2, 3, 3, 3, 6, 7, 3, 9, 3, 7, 12, 7, 9, 15,
3, 7, 18, 19, ... (OEIS A070167).
The Collatz problem can be implemented as an 8-register machine (Wolfram 2002, p. 100), quasi-cellular
automaton (Cloney et al. 1987, Bruschi 2005), or 6-color one-dimensional
quasi-cellular automaton with local rules but which wraps first and last digits around
(Zeleny). In general, the difficulty in constructing true local-rule cellular automata
arises from the necessity of a carry operation when multiplying by 3 which, in the
worst case, can extend the entire length of the base-
representation of digits (and thus require propagating information
at faster than the CA's speed of light).
The Collatz problem was modified by Terras (1976, 1979), who asked if iterating
 |
(2)
|
always returns to 1 for initial integer value 
(e.g., Lagarias 1985, Cloney et al. 1987). This is
simply the original statement above but combining the division by two into the addition
step if 
is odd, thus compressing the number of steps. The following table gives the sequences
for the first few starting values 
, 2, ... (OEIS A070168).
 |  ,  , ... |
| 1 | 1 |
| 2 | 2, 1 |
| 3 | 3, 5, 8, 4, 2, 1 |
| 4 | 4, 2, 1 |
| 5 | 5, 8, 4, 2, 1 |
| 6 | 6, 3, 5, 8, 4, 2, 1 |
| 7 | 7, 11, 17, 26, 13, 20, 10, 5, 8, 4, 2, 1 |
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
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).
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 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
 |
(12)
|
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
 |
(13)
|
for 
.
Matthews believes that the map
 |
(14)
|
will either reach 0 (mod 3) or will enter one of the cycles 
or 
, and offers a $100 (Australian?) prize for a proof.
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.Bruschi, M. "Two Cellular Automata for the 
Map." 
Problem: A Quasi Cellular Automaton." Complex Sys. 1,
349-360, 1987.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.De Mol, L. "Tag Systems and Collatz-Like Functions."
Theor. Comput. Sci. 390, 92-101, 2008.Everett, C. "Iteration
of the Number Theoretic Function 
, 
." Adv. Math. 25, 42-45, 1977.Guy,
R. K. "Collatz's Sequence." §E16 in 
Problem and Its Generalizations." Amer. Math. Monthly 92, 3-23,
1985.Leavens, G. T. and Vermeulen, M. "
Search Programs." Comput. Math. Appl. 24,
79-99, 1992.Margenstern, M. and Matiyasevich, Y. "A Binomial Representation
of the 
Problem." Acta Arith. 91, 367-378, 1999.Matthews,
K. R. "The Generalized 
Mapping." 
Conjecture." [$100 Reward for a Proof.] 
Problem: Computational Results."
Math. Comput. 68, 371-384, 1999.Oliveira e Silva, T. "Computational
Verification of the 
Conjecture." Sep. 19, 2008. 
Problem." Ch. 7 in 
Problem." Elem. Math. 55, 142-155, 2000.Wolfram,
S.