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Collatz conjecture
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Does the Collatz sequence eventually reach 1 for all positive integer initial values?
(more unsolved problems in mathematics)
The Collatz conjecture is a
conjecture
in
mathematics
that concerns
sequences
defined as follows: start with any
positive integer
n. Then each term is obtained from the previous term as follows: if the previous term is
even
, the next term is one half of the previous term. If the previous term is odd, the next term is 3 times the previous term plus 1. The conjecture is that no matter what value of n, the sequence will always reach 1.
The conjecture is named after
Lothar Collatz
, who introduced the idea in 1937, two years after receiving his doctorate.^{}
[1]
It is also known as the 3n + 1 problem, the 3n + 1 conjecture, the Ulam conjecture (after
Stanisław Ulam
), Kakutani’s problem (after
Shizuo Kakutani
), the Thwaites conjecture (after Sir
Bryan Thwaites
), Hasse’s algorithm (after
Helmut Hasse
), or the Syracuse problem.^{}
[2]
^{}
[4]
The sequence of numbers involved is sometimes referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like
hailstones
in a cloud),^{}
[5]
^{}
[6]
or as wondrous numbers.^{}
[7]
Paul Erdős
said about the Collatz conjecture: “Mathematics may not be ready for such problems.”^{}
[8]
He also offered US$500 for its solution.^{}
[9]
Jeffrey Lagarias
stated in 2010 that the Collatz conjecture “is an extraordinarily difficult problem, completely out of reach of present day mathematics.”^{}
[10]
Statement of the problem[
edit
]
Consider the following operation on an arbitrary
positive integer
:
 If the number is even, divide it by two.
 If the number is odd, triple it and add one.
In
modular arithmetic
notation, define the
function
f as follows:
 f(n)={n2if n≡0(mod2)3n+1if n≡1(mod2).{displaystyle f(n)={begin{cases}{frac {n}{2}}&{text{if }}nequiv 0{pmod {2}}\[4px]3n+1&{text{if }}nequiv 1{pmod {2}}.end{cases}}}
Now form a sequence by performing this operation repeatedly, beginning with any positive integer, and taking the result at each step as the input at the next.
In notation:
 ai={nfor i=0f(ai−1)for i>0{displaystyle a_{i}={begin{cases}n&{text{for }}i=0\f(a_{i1})&{text{for }}i>0end{cases}}}
(that is: a_{i} is the value of f applied to n recursively i times; a_{i} = f^{i}(n)).
The Collatz conjecture is: This process will eventually reach the number 1, regardless of which positive integer is chosen initially.
If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence that does not contain 1. Such a sequence would either enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found.
The smallest i such that a_{i} < a_{0} is called the stopping time of n. Similarly, the smallest k such that a_{k} = 1 is called the total stopping time of n.^{}
[3]
If one of the indexes i or k doesn’t exist, we say that the stopping time or the total stopping time, respectively, is infinite.
The Collatz conjecture asserts that the total stopping time of every n is finite. It is also equivalent to saying that every n ≥ 2 has a finite stopping time.
Since 3n + 1 is even whenever n is odd, one may instead use the “shortcut” form of the Collatz function
This definition yields smaller values for the stopping time and total stopping time without changing the overall dynamics of the process.
Empirical data[
edit
]
For instance, starting with n = 12, one gets the sequence 12, 6, 3, 10, 5, 16, 8, 4, 2, 1.
The number n = 19 takes longer to reach 1: 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.
The sequence for n = 27, listed and graphed below, takes 111 steps (41 steps through odd numbers, in bold), climbing as high as 9232 before descending to 1.
 27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1 (sequence
A008884
in the
OEIS
)
Numbers with a total stopping time longer than that of any smaller starting value form a sequence beginning with:
 1, 2, 3, 6, 7, 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, 327, 649, 703, 871, 1161, 2223, 2463, 2919, 3711, 6171, … (sequence
A006877
in the
OEIS
).
The starting values whose maximum trajectory point is greater than that of any smaller starting value are as follows:
 1, 2, 3, 7, 15, 27, 255, 447, 639, 703, 1819, 4255, 4591, 9663, 20895, 26623, 31911, 60975, 77671, 113383, 138367, 159487, 270271, 665215, 704511, … (sequence
A006884
in the
OEIS
)
Number of steps for n to reach 1 are
 0, 1, 7, 2, 5, 8, 16, 3, 19, 6, 14, 9, 9, 17, 17, 4, 12, 20, 20, 7, 7, 15, 15, 10, 23, 10, 111, 18, 18, 18, 106, 5, 26, 13, 13, 21, 21, 21, 34, 8, 109, 8, 29, 16, 16, 16, 104, 11, 24, 24, … (sequence
A006577
in the
OEIS
)
The starting value having the largest total stopping time while being
 less than 10 is 9, which has 19 steps,
 less than 100 is 97, which has 118 steps,
 less than 1000 is 871, which has 178 steps,
 less than 10^{4} is 6171, which has 261 steps,
 less than 10^{5} is 77031, which has 350 steps,
 less than 10^{6} is 837799, which has 524 steps,
 less than 10^{7} is 8400511, which has 685 steps,
 less than 10^{8} is 63728127, which has 949 steps,
 less than 10^{9} is 670617279, which has 986 steps,
 less than 10^{10} is 9780657630, which has 1132 steps,^{ [11] }
 less than 10^{11} is 75128138247, which has 1228 steps,
 less than 10^{12} is 989345275647, which has 1348 steps,
 less than 10^{13} is 7887663552367, which has 1563 steps,
 less than 10^{14} is 80867137596217, which has 1662 steps,
 less than 10^{15} is 942488749153153, which has 1862 steps,
 less than 10^{16} is 7579309213675935, which has 1958 steps,
 less than 10^{17} is 93571393692802302, which has 2091 steps and
 less than 10^{18} is 931386509544713451, which has 2283 steps.^{ [12] }
These numbers are the lowest ones with the indicated step count, but not necessarily the only ones below the given limit. As an example, 9780657631 has 1132 steps, as does 9780657630.
The starting values having the smallest total stopping time with respect to their number of digits (in base 2) are the
powers of two
since 2^{n} is halved n times to reach 1, and is never increased.
Visualizations[
edit
]

Directed graph showing the orbits of the first 1000 numbers.

The x axis represents starting number, the y axis represents the highest number reached during the chain to 1. This plot shows a restricted y axis: some x values produce intermediates as high as 2.7×10^{7} (for x = 9663)

The tree of all the numbers having fewer than 20 steps (click to enlarge).
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Supporting arguments[
edit
]
Although the conjecture has not been proven, most mathematicians who have looked into the problem think the conjecture is true because experimental evidence and heuristic arguments support it.
Experimental evidence[
edit
]
As of 2020^{}
[update]
, the conjecture has been checked by computer for all starting values up to 2^{68} ≈ 2.95×10^{20}. All initial values tested so far eventually end in the repeating cycle (4;2;1) of period 3.^{}
[13]
This computer evidence is not sufficient to prove that the conjecture is true for all starting values. As in the case of some disproved conjectures, like the
Pólya conjecture
,
counterexamples
might be found when considering very large numbers.
However, such verifications may have other implications. For example, one can derive additional constraints on the period and structural form of a nontrivial
cycle
.^{}
[14]
^{}
[15]
^{}
[16]
A probabilistic heuristic[
edit
]
If one considers only the odd numbers in the sequence generated by the Collatz process, then each odd number is on average 3/4 of the previous one.^{}
[17]
(More precisely, the geometric mean of the ratios of outcomes is 3/4.) This yields a heuristic argument that every Hailstone sequence should decrease in the long run, although this is not evidence against other cycles, only against divergence. The argument is not a proof because it assumes that Hailstone sequences are assembled from uncorrelated probabilistic events. (It does rigorously establish that the
2adic
extension of the Collatz process has two division steps for every multiplication step for almost all 2adic starting values.)
Stopping times[
edit
]
As proved by Terras, almost every positive integer n has a finite stopping time.^{}
[18]
In other words, almost every Collatz sequence reaches a point that is strictly below its initial value. The proof is based on the distribution of
parity vectors
and uses the
central limit theorem
.
In 2019,
Terence Tao
considerably improved this result by showing, using logarithmic
density
, that almost all Collatz orbits are descending below any given function of the starting point, provided that this function diverges to infinity, no matter how slowly. Responding to this work,
Quanta Magazine
wrote that Tao “came away with one of the most significant results on the Collatz conjecture in decades.”^{}
[19]
^{}
[20]
Lower bounds[
edit
]
In a
computeraided proof
, Krasikov and Lagarias showed that the number of integers in the interval [1,x] that eventually reach 1 is at least equal to x^{0.84} for all sufficiently large x.^{}
[21]
Cycles[
edit
]
In this part, consider the shortcut form of the Collatz function
 f(n)={n2if n≡0(mod2),3n+12if n≡1(mod2).{displaystyle f(n)={begin{cases}{frac {n}{2}}&{text{if }}nequiv 0{pmod {2}},\[4px]{frac {3n+1}{2}}&{text{if }}nequiv 1{pmod {2}}.end{cases}}}
A
cycle
is a sequence (a_{0}, a_{1}, …, a_{q}) of distinct positive integers where f(a_{0}) = a_{1}, f(a_{1}) = a_{2}, …, and f(a_{q}) = a_{0}.
The only known cycle is (1,2) of period 2, called the trivial cycle.
Cycle length[
edit
]
The length of a nontrivial cycle is known to be at least 17087915.^{}
[15]
In fact, Eliahou (1993) proved that the period p of any nontrivial cycle is of the form
 p=301994a+17087915b+85137581c{displaystyle p=301994a+17087915b+85137581c}
where a, b and c are nonnegative integers, b ≥ 1 and ac = 0. This result is based on the
continued fraction
expansion of ln 3/ln 2.
A similar reasoning that accounts for the recent verification of the conjecture up to 2^{68} leads to the improved lower bound 114208327604 (or 186265759595 without the “shortcut”). This lower bound is consistent with the above result, since 114208327604 = 17087915 × 361 + 85137581 × 1269.
kcycles[
edit
]
A kcycle is a cycle that can be partitioned into 2k contiguous subsequences: k increasing sequences of odd numbers alternating with k decreasing sequences of even numbers.^{}
[16]
For instance, if the cycle consists of a single increasing sequence of odd numbers followed by a decreasing sequence of even numbers, it is called a 1cycle.
Steiner (1977) proved that there is no 1cycle other than the trivial (1;2).^{}
[22]
Simons (2004) used Steiner’s method to prove that there is no 2cycle.^{}
[23]
Simons & de Weger (2005) extended this proof up to 68cycles: there is no kcycle up to k = 68.^{}
[16]
For each k beyond 68, this method gives an upper bound for the smallest term of a kcycle: for example, if there is a 77cycle, then at least one element of the cycle is less than 38137×2^{50}.^{}
[16]
Along with the verification of the conjecture up to 2^{68}, this implies the nonexistence of a nontrivial kcycle up to k = 77.^{}
[13]
As exhaustive computer searches continue, larger k values may be ruled out. To state the argument more intuitively: we need not look for cycles that have at most 77 circuits, where each circuit consists of consecutive ups followed by consecutive downs.
Other formulations of the conjecture[
edit
]
In reverse[
edit
]
There is another approach to prove the conjecture, which considers the bottomup method of growing the socalled Collatz graph. The Collatz graph is a
graph
defined by the inverse
relation
 R(n)={{2n}if n≡0,1,2,3,5{2n,n−13}if n≡4(mod6).{displaystyle R(n)={begin{cases}{2n}&{text{if }}nequiv 0,1,2,3,5\[4px]left{2n,{frac {n1}{3}}right}&{text{if }}nequiv 4end{cases}}{pmod {6}}.}
So, instead of proving that all positive integers eventually lead to 1, we can try to prove that 1 leads backwards to all positive integers. For any integer n, n ≡ 1 (mod 2)
if and only if
3n + 1 ≡ 4 (mod 6). Equivalently, n − 1/3 ≡ 1 (mod 2) if and only if n ≡ 4 (mod 6). Conjecturally, this inverse relation forms a
tree
except for the 1–2–4 loop (the inverse of the 4–2–1 loop of the unaltered function f defined in the
Statement of the problem
section of this article).
When the relation 3n + 1 of the function f is replaced by the common substitute “shortcut” relation 3n + 1/2, the Collatz graph is defined by the inverse relation,
 R(n)={{2n}if n≡0,1{2n,2n−13}if n≡2(mod3).{displaystyle R(n)={begin{cases}{2n}&{text{if }}nequiv 0,1\[4px]left{2n,{frac {2n1}{3}}right}&{text{if }}nequiv 2end{cases}}{pmod {3}}.}
For any integer n, n ≡ 1 (mod 2) if and only if 3n + 1/2 ≡ 2 (mod 3). Equivalently, 2n − 1/3 ≡ 1 (mod 2) if and only if n ≡ 2 (mod 3). Conjecturally, this inverse relation forms a tree except for a 1–2 loop (the inverse of the 1–2 loop of the function f(n) revised as indicated above).
Alternatively, replace the 3n + 1 with n′/H(n′) where n′ = 3n + 1 and H(n′) is the highest
power of 2
that divides n′ (with no
remainder
). The resulting function f maps from
odd numbers
to odd numbers. Now suppose that for some odd number n, applying this operation k times yields the number 1 (that is, f^{k}(n) = 1). Then in
binary
, the number n can be written as the concatenation of
strings
w_{k}w_{k−1} … w_{1} where each w_{h} is a finite and contiguous extract from the representation of 1/3^{h}.^{}
[24]
The representation of n therefore holds the
repetends
of 1/3^{h}, where each repetend is optionally rotated and then replicated up to a finite number of bits. It is only in binary that this occurs.^{}
[25]
Conjecturally, every binary string s that ends with a ‘1’ can be reached by a representation of this form (where we may add or delete leading ‘0’s to s).
As an abstract machine that computes in base two[
edit
]
Repeated applications of the Collatz function can be represented as an
abstract machine
that handles
strings
of
bits
. The machine will perform the following three steps on any odd number until only one “1” remains:
 Append 1 to the (right) end of the number in binary (giving 2n + 1);
 Add this to the original number by binary addition (giving 2n + 1 + n = 3n + 1);
 Remove all trailing “0”s (i.e. repeatedly divide by two until the result is odd).
Example[
edit
]
The starting number 7 is written in base two as 111. The resulting Collatz sequence is:
111 1111 1011010111 100010100011 11010011011 1010001011 10000
As a parity sequence[
edit
]
For this section, consider the Collatz function in the slightly modified form
 f(n)={n2if n≡03n+12if n≡1(mod2).{displaystyle f(n)={begin{cases}{frac {n}{2}}&{text{if }}nequiv 0\[4px]{frac {3n+1}{2}}&{text{if }}nequiv 1end{cases}}{pmod {2}}.}
This can be done because when n is odd, 3n + 1 is always even.
If P(…) is the parity of a number, that is P(2n) = 0 and P(2n + 1) = 1, then we can define the Collatz parity sequence (or parity vector) for a number n as p_{i} = P(a_{i}), where a_{0} = n, and a_{i+1} = f(a_{i}).
Which operation is performed, 3n + 1/2 or n/2, depends on the parity. The parity sequence is the same as the sequence of operations.
Using this form for f(n), it can be shown that the parity sequences for two numbers m and n will agree in the first k terms if and only if m and n are equivalent modulo 2^{k}. This implies that every number is uniquely identified by its parity sequence, and moreover that if there are multiple Hailstone cycles, then their corresponding parity cycles must be different.^{}
[3]
^{}
[18]
Applying the f function k times to the number n = 2^{k}a + b will give the result 3^{c}a + d, where d is the result of applying the f function k times to b, and c is how many increases were encountered during that sequence (e.g. for 2^{5}a + 1 there are 3 increases as 1 iterates to 2, 1, 2, 1, and finally to 2 so the result is 3^{3}a + 2; for 2^{2}a + 1 there is only 1 increase as 1 rises to 2 and falls to 1 so the result is 3a + 1). When b is 2^{k} − 1 then there will be k rises and the result will be 2 × 3^{k}a − 1. The factor of 3 multiplying a is independent of the value of a; it depends only on the behavior of b. This allows one to predict that certain forms of numbers will always lead to a smaller number after a certain number of iterations, e.g. 4a + 1 becomes 3a + 1 after two applications of f and 16a + 3 becomes 9a + 2 after 4 applications of f. Whether those smaller numbers continue to 1, however, depends on the value of a.
As a tag system[
edit
]
For the Collatz function in the form
f(n)={n2if n≡03n+12if n≡1.(mod2){displaystyle f(n)={begin{cases}{frac {n}{2}}&{text{if }}nequiv 0\[4px]{frac {3n+1}{2}}&{text{if }}nequiv 1.end{cases}}{pmod {2}}}
Hailstone sequences can be computed by the extremely simple
2tag system
with production rules
 a → bc, b → a, c → aaa.
In this system, the positive integer n is represented by a string of n copies of a, and iteration of the tag operation halts on any word of length less than 2. (Adapted from De Mol.)
The Collatz conjecture equivalently states that this tag system, with an arbitrary finite string of a as the initial word, eventually halts (see
Tag system#Example: Computation of Collatz sequences
for a worked example).
Extensions to larger domains[
edit
]
Iterating on all integers[
edit
]
An extension to the Collatz conjecture is to include all integers, not just positive integers. Leaving aside the cycle 0 → 0 which cannot be entered from outside, there are a total of 4 known cycles, which all nonzero integers seem to eventually fall into under iteration of f. These cycles are listed here, starting with the wellknown cycle for positive n:
Odd values are listed in large bold. Each cycle is listed with its member of least absolute value (which is always odd) first.
Cycle  Oddvalue cycle length  Full cycle length 

1 → 4 → 2 → 1 …  1  3 
−1 → −2 → −1 …  1  2 
−5 → −14 → −7 → −20 → −10 → −5 …  2  5 
−17 → −50 → −25 → −74 → −37 → −110 → −55 → −164 → −82 → −41 → −122 → −61 → −182 → −91 → −272 → −136 → −68 → −34 → −17 …  7  18 
The generalized Collatz conjecture is the assertion that every integer, under iteration by f, eventually falls into one of the four cycles above or the cycle 0 → 0. The 0 → 0 cycle is often regarded as “trivial” by the argument, as it is only included for the sake of completeness.
Iterating on rationals with odd denominators[
edit
]
The Collatz map can be extended to (positive or negative) rational numbers which have odd denominators when written in lowest terms. The number is taken to be ‘odd’ or ‘even’ according to whether its numerator is odd or even. Then the formula for the map is exactly the same as when the domain is the integers: an ‘even’ such rational is divided by 2; an ‘odd’ such rational is multiplied by 3 and then 1 is added. A closely related fact is that the Collatz map extends to the ring of
2adic integers
, which contains the ring of rationals with odd denominators as a subring.
When using the “shortcut” definition of the Collatz map, it is known that any periodic
parity sequence
is generated by exactly one rational.^{}
[26]
Conversely, it is conjectured that every rational with an odd denominator has an eventually cyclic parity sequence (Periodicity Conjecture^{}
[3]
).
If a parity cycle has length n and includes odd numbers exactly m times at indices k_{0} < ⋯ < k_{m−1}, then the unique rational which generates immediately and periodically this parity cycle is

3m−12k0+⋯+302km−12n−3m.{displaystyle {frac {3^{m1}2^{k_{0}}+cdots +3^{0}2^{k_{m1}}}{2^{n}3^{m}}}.}
(1)
For example, the parity cycle (1 0 1 1 0 0 1) has length 7 and four odd terms at indices 0, 2, 3, and 6. It is repeatedly generated by the fraction
 3320+3222+3123+302627−34=15147{displaystyle {frac {3^{3}2^{0}+3^{2}2^{2}+3^{1}2^{3}+3^{0}2^{6}}{2^{7}3^{4}}}={frac {151}{47}}}
as the latter leads to the rational cycle
 15147→25047→12547→21147→34047→17047→8547→15147.{displaystyle {frac {151}{47}}rightarrow {frac {250}{47}}rightarrow {frac {125}{47}}rightarrow {frac {211}{47}}rightarrow {frac {340}{47}}rightarrow {frac {170}{47}}rightarrow {frac {85}{47}}rightarrow {frac {151}{47}}.}
Any cyclic permutation of (1 0 1 1 0 0 1) is associated to one of the above fractions. For instance, the cycle (0 1 1 0 0 1 1) is produced by the fraction
 3321+3222+3125+302627−34=25047.{displaystyle {frac {3^{3}2^{1}+3^{2}2^{2}+3^{1}2^{5}+3^{0}2^{6}}{2^{7}3^{4}}}={frac {250}{47}}.}
For a onetoone correspondence, a parity cycle should be irreducible, i.e., not partitionable into identical subcycles. As an illustration of this, the parity cycle (1 1 0 0 1 1 0 0) and its subcycle (1 1 0 0) are associated to the same fraction 5/7 when reduced to lowest terms.
In this context, assuming the validity of the Collatz conjecture implies that (1 0) and (0 1) are the only parity cycles generated by positive whole numbers (1 and 2, respectively).
If the odd denominator d of a rational is not a multiple of 3, then all the iterates have the same denominator and the sequence of numerators can be obtained by applying the “3n + d” generalization^{}
[27]
of the Collatz function
 Td(x)={x2if x≡0(mod2),3x+d2if x≡1(mod2).{displaystyle T_{d}(x)={begin{cases}{frac {x}{2}}&{text{if }}xequiv 0{pmod {2}},\[4px]{frac {3x+d}{2}}&{text{if }}xequiv 1{pmod {2}}.end{cases}}}
2adic extension[
edit
]
The function
 T(x)={x2if x≡0(mod2)3x+12if x≡1(mod2){displaystyle T(x)={begin{cases}{frac {x}{2}}&{text{if }}xequiv 0{pmod {2}}\[4px]{frac {3x+1}{2}}&{text{if }}xequiv 1{pmod {2}}end{cases}}}
is welldefined on the ring ℤ_{2} of
2adic integers
, where it is continuous and
measurepreserving
with respect to the 2adic measure. Moreover, its dynamics is known to be
ergodic
.^{}
[3]
Define the parity vector function Q acting on ℤ_{2} as
 Q(x)=∑k=0∞(Tk(x)mod2)2k{displaystyle Q(x)=sum _{k=0}^{infty }left(T^{k}(x)mod 2right)2^{k}}.
The function Q is a 2adic
isometry
.^{}
[28]
Consequently, every infinite parity sequence occurs for exactly one 2adic integer, so that almost all trajectories are acyclic in Z2{displaystyle mathbb {Z} _{2}}.
An equivalent formulation of the Collatz conjecture is that
 Q(Z+)⊂13Z.{displaystyle Qleft(mathbb {Z} ^{+}right)subset {tfrac {1}{3}}mathbb {Z} .}
Iterating on real or complex numbers[
edit
]
The Collatz map (with shortcut) can be viewed as the restriction to the integers of the
smooth
map
 f(z)=12zcos2(π2z)+3z+12sin2(π2z).{displaystyle f(z)={frac {1}{2}}zcos ^{2}left({frac {pi }{2}}zright)+{frac {3z+1}{2}}sin ^{2}left({frac {pi }{2}}zright).}
The
iterations
of this map on the
real line
lead to a
dynamical system
, further investigated by Chamberland.^{}
[29]
He showed that the conjecture does not hold for positive real numbers since there are infinitely many fixed points, as well as orbits escaping monotonically to infinity. The function f{displaystyle f} has two attracting cycles of period 2, (1;2){displaystyle (1;2)} and (1.1925…;2.1386…){displaystyle (1.1925…;2.1386…)}. Moreover, the set of unbounded orbits is conjectured to be of
measure
0.
Letherman, Schleicher, and Wood extended the study to the
complex plane
, where most of the points have orbits that diverge to infinity (colored region on the illustration).^{}
[30]
The boundary between the colored region and the black
components
, namely the
Julia set
of f{displaystyle f}, is a
fractal
pattern, sometimes called the “Collatz fractal”.
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Optimizations[
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]
Time–space tradeoff[
edit
]
The section
As a parity sequence
above gives a way to speed up simulation of the sequence. To jump ahead k steps on each iteration (using the f function from that section), break up the current number into two parts, b (the k least significant bits, interpreted as an integer), and a (the rest of the bits as an integer). The result of jumping ahead k steps can be found as:
 f^{k}(2^{k}a + b) = 3^{c(b)}a + d(b).
The c (or better 3^{c}) and d arrays are precalculated for all possible kbit numbers b, where d(b) is the result of applying the f function k times to b, and c(b) is the number of odd numbers encountered on the way.^{}
[31]
For example, if k = 5, one can jump ahead 5 steps on each iteration by separating out the 5 least significant bits of a number and using:
 c(0…31) = {0,3,2,2,2,2,2,4,1,4,1,3,2,2,3,4,1,2,3,3,1,1,3,3,2,3,2,4,3,3,4,5}
 d(0…31) = {0,2,1,1,2,2,2,20,1,26,1,10,4,4,13,40,2,5,17,17,2,2,20,20,8,22,8,71,26,26,80,242}.
This requires 2^{k}
precomputation
and storage to speed up the resulting calculation by a factor of k, a
space–time tradeoff
.
Modular restrictions[
edit
]
For the special purpose of searching for a counterexample to the Collatz conjecture, this precomputation leads to an even more important acceleration, used by Tomás Oliveira e Silva in his computational confirmations of the Collatz conjecture up to large values of n. If, for some given b and k, the inequality
 f^{k}(2^{k}a + b) = 3^{c(b)}a + d(b) < 2^{k}a + b
holds for all a, then the first counterexample, if it exists, cannot be b modulo 2^{k}.^{}
[14]
For instance, the first counterexample must be odd because f(2n) = n, smaller than 2n; and it must be 3 mod 4 because f^{2}(4n + 1) = 3n + 1, smaller than 4n + 1. For each starting value a which is not a counterexample to the Collatz conjecture, there is a k for which such an inequality holds, so checking the Collatz conjecture for one starting value is as good as checking an entire congruence class. As k increases, the search only needs to check those residues b that are not eliminated by lower values of k. Only an exponentially small fraction of the residues survive.^{}
[32]
For example, the only surviving residues mod 32 are 7, 15, 27, and 31.
Syracuse function[
edit
]
If k is an odd integer, then 3k + 1 is even, so 3k + 1 = 2^{a}k′ with k′ odd and a ≥ 1. The Syracuse function is the function f from the set I of odd integers into itself, for which f(k) = k′ (sequence
A075677
in the
OEIS
).
Some properties of the Syracuse function are:
 For all k ∈ I, f(4k + 1) = f(k). (Because 3(4k + 1) + 1 = 12k + 4 = 4(3k + 1).)
 In more generality: For all p ≥ 1 and odd h, f^{p − 1}(2^{p}h − 1) = 2 × 3^{p − 1}h − 1. (Here f^{p − 1} is
function iteration notation
.)
 For all odd h, f(2h − 1) ≤ 3h − 1/2
The Collatz conjecture is equivalent to the statement that, for all k in I, there exists an integer n ≥ 1 such that f^{n}(k) = 1.
Undecidable generalizations[
edit
]
In 1972,
John Horton Conway
proved that a natural generalization of the Collatz problem is algorithmically
undecidable
.^{}
[33]
Specifically, he considered functions of the form
 g(n)=ain+bi,wheren≡i(modP){displaystyle {g(n)=a_{i}n+b_{i}};;,{text{where}};{nequiv i{pmod {P}}}}
and a_{0}, b_{0},…,a_{P − 1}, b_{P − 1} are rational numbers which are so chosen that g(n) is always an integer.
The standard Collatz function is given by P = 2, a_{0} = 1/2, b_{0} = 0, a_{1} = 3, b_{1} = 1. Conway proved that the problem:
 Given g and n, does the sequence of iterates g^{k}(n) reach 1?
is undecidable, by representing the
halting problem
in this way.
Closer to the Collatz problem is the following universally quantified problem:
 Given g does the sequence of iterates g^{k}(n) reach 1, for all n > 0?
Modifying the condition in this way can make a problem either harder or easier to solve (intuitively, it is harder to justify a positive answer but might be easier to justify a negative one). Kurtz and Simon^{}
[34]
proved that the above problem is, in fact, undecidable and even higher in the
arithmetical hierarchy
, specifically Π^{0}
_{2}complete. This hardness result holds even if one restricts the class of functions g by fixing the modulus P to 6480.^{}
[35]
In popular culture[
edit
]
In the movie
Incendies
, a graduate student in pure mathematics explains the Collatz conjecture to a group of undergraduates. She puts her studies on hold for a time to address some unresolved questions about her family’s past. Late in the movie, the Collatz conjecture turns out to have foreshadowed a disturbing and difficult discovery that she makes about her family.^{}
[36]
^{}
[37]
Xem thêm: Chuyển đổi qua lại giữa Although / though despite / in spite of
See also[
edit
]

3x + 1 semigroup

Arithmetic dynamics

Modular arithmetic

Residueclasswise affine group
Further reading[
edit
]
 The Ultimate Challenge: the 3x + 1 problem:
 This volume,^{ [10] edited by Jeffrey Lagarias and published by the American Mathematical Society , is a compendium of information on the Collatz conjecture, methods of approaching it and generalizations. It includes two survey papers by the editor and five by other authors, concerning the history of the problem, generalizations, statistical approaches and results from the theory of computation . It also includes reprints of early papers on the subject (including an entry by Lothar Collatz).}
References[
edit
]

^
O’Connor, J.J.; Robertson, E.F. (2006).
“Lothar Collatz”
. St Andrews University School of Mathematics and Statistics, Scotland.

^
Maddux, Cleborne D.; Johnson, D. Lamont (1997). Logo: A Retrospective. New York: Haworth Press. p. 160.
ISBN
0789003740
.
The problem is also known by several other names, including: Ulam’s conjecture, the Hailstone problem, the Syracuse problem, Kakutani’s problem, Hasse’s algorithm, and the Collatz problem.
 ^
^{a}
^{b}
^{c}
^{d}
^{e}
^{f}
^{g}
Lagarias, Jeffrey C. (1985). “The 3x + 1 problem and its generalizations”.
The American Mathematical Monthly
. 92 (1): 3–23.
doi
:
10.1080/00029890.1985.11971528
.
JSTOR
2322189
.

^
According to Lagarias (1985),^{}
[3]
p. 4, the name “Syracuse problem” was proposed by Hasse in the 1950s, during a visit to
Syracuse University
.

^
Pickover, Clifford A. (2001).
Wonders of Numbers
. Oxford: Oxford University Press. pp.
116
–118.
ISBN
0195133420
.

^
“Hailstone Number”
. MathWorld. Wolfram Research.

^
Hofstadter, Douglas R.
(1979).
Gödel, Escher, Bach
. New York: Basic Books. pp.
400–2
.
ISBN
0465026850
.

^
Guy, Richard K.
(2004).
““E17: Permutation Sequences““
. Unsolved problems in number theory (3rd ed.).
SpringerVerlag
. pp. 336–7.
ISBN
0387208607
.
Zbl
1058.11001
.

^
Guy, R. K. (1983). “Don’t try to solve these problems”. Amer. Math. Monthly. 90 (1): 35–41.
doi
:
10.2307/2975688
.
JSTOR
2975688
. By this Erdos means that there aren’t powerful tools for manipulating such objects.
 ^
^{a}
^{b}
Lagarias, Jeffrey C.
, ed. (2010). The Ultimate Challenge: the 3x + 1 problem.
American Mathematical Society
.
ISBN
9780821849408
.
Zbl
1253.11003
.

^
Leavens, Gary T.; Vermeulen, Mike (December 1992). “3x + 1 Search Programs”. Computers & Mathematics with Applications. 24 (11): 79–99.
doi
:
10.1016/08981221(92)90034F
.

^
Roosendaal, Eric.
“3x+1 Delay Records”
. Retrieved 14 March 2020. (Note: “Delay records” are total stopping time records.)
 ^
^{a}
^{b}
Barina, David (2020). “Convergence verification of the Collatz problem”. The Journal of Supercomputing. 77 (3): 2681–2688.
doi
:
10.1007/s1122702003368x
.
S2CID
220294340
.
 ^
^{a}
^{b}
Garner, Lynn E. (1981). “On the Collatz 3n + 1 Algorithm”. Proceedings of the American Mathematical Society. 82 (1): 19–22.
doi
:
10.2307/2044308
.
JSTOR
2044308
.
 ^
^{a}
^{b}
Eliahou, Shalom (1993). “The 3x + 1 problem: new lower bounds on nontrivial cycle lengths”. Discrete Mathematics. 118 (1): 45–56.
doi
:
10.1016/0012365X(93)90052U
.
 ^
^{a}
^{b}
^{c}
^{d}
Simons, J.; de Weger, B. (2005).
“Theoretical and computational bounds for mcycles of the 3n + 1 problem”
(PDF). Acta Arithmetica. 117 (1): 51–70.
Bibcode
:
2005AcAri.117…51S
.
doi
:
10.4064/aa11713
.

^
Lagarias (1985),^{}
[3]
section “
A heuristic argument”
.
 ^
^{a}
^{b}
Terras, Riho (1976).
“A stopping time problem on the positive integers”
(PDF). Acta Arithmetica. 30 (3): 241–252.
doi
:
10.4064/aa303241252
.
MR
0568274
.

^
Tao, Terence (10 September 2019).
“Almost all Collatz orbits attain almost bounded values”
. What’s new. Retrieved 11 September 2019.

^
Hartnett, Kevin.
“Mathematician Proves Huge Result on ‘Dangerous’ Problem”
. Quanta Magazine. Retrieved 26 December 2019.

^
Krasikov, Ilia;
Lagarias, Jeffrey C.
(2003).
“Bounds for the 3x + 1 problem using difference inequalities”
. Acta Arithmetica. 109 (3): 237–258.
arXiv
:
math/0205002
.
Bibcode
:
2003AcAri.109..237K
.
doi
:
10.4064/aa10934
.
MR
1980260
.
S2CID
18467460
.

^
Steiner, R. P. (1977). “A theorem on the syracuse problem”. Proceedings of the 7th Manitoba Conference on Numerical Mathematics. pp. 553–9.
MR
0535032
.

^
Simons, John L. (2005).
“On the nonexistence of 2cycles for the 3x + 1 problem”
. Math. Comp. 74: 1565–72.
Bibcode
:
2005MaCom..74.1565S
.
doi
:
10.1090/s0025571804017284
.
MR
2137019
.

^
Colussi, Livio (9 September 2011).
“The convergence classes of Collatz function”
. Theoretical Computer Science. 412 (39): 5409–5419.
doi
:
10.1016/j.tcs.2011.05.056
.

^
Hew, Patrick Chisan (7 March 2016).
“Working in binary protects the repetends of 1/3^{h}: Comment on Colussi’s ‘The convergence classes of Collatz function‘“
. Theoretical Computer Science. 618: 135–141.
doi
:
10.1016/j.tcs.2015.12.033
.

^
Lagarias, Jeffrey (1990).
“The set of rational cycles for the 3x+1 problem”
. Acta Arithmetica. 56 (1): 33–53.
doi
:
10.4064/aa5613353
.
ISSN
00651036
.

^
Belaga, Edward G.; Mignotte, Maurice (1998).
“Embedding the 3x+1 Conjecture in a 3x+d Context”
. Experimental Mathematics. 7 (2): 145–151.
doi
:
10.1080/10586458.1998.10504364
.

^
Lagarias, Jeffrey C.; Bernstein, Daniel J. (1996). “The 3x + 1 Conjugacy Map”. Canadian Journal of Mathematics. 48 (6): 1154–1169.
doi
:
10.4153/CJM1996060x
.
ISSN
0008414X
.

^
Chamberland, Marc (1996). “A continuous extension of the 3x + 1 problem to the real line”. Dynam. Contin. Discrete Impuls Systems. 2 (4): 495–509.

^
Letherman, Simon; Schleicher, Dierk; Wood, Reg (1999). “The (3n + 1)Problem and Holomorphic Dynamics”. Experimental Mathematics. 8 (3): 241–252.
doi
:
10.1080/10586458.1999.10504402
.

^
Scollo, Giuseppe (2007).
“Looking for Class Records in the 3x + 1 Problem by means of the COMETA Grid Infrastructure”
(PDF). Grid Open Days at the University of Palermo.

^
Lagarias (1985),^{}
[3]
Theorem D.

^
Conway, John H. (1972). “Unpredictable Iterations”. Proc. 1972 Number Theory Conf., Univ. Colorado, Boulder. pp. 49–52.

^
Kurtz, Stuart A.; Simon, Janos (2007).
“The Undecidability of the Generalized Collatz Problem”
. In Cai, J.Y.; Cooper, S. B.; Zhu, H. (eds.). Proceedings of the 4th International Conference on Theory and Applications of Models of Computation, TAMC 2007, held in Shanghai, China in May 2007. pp. 542–553.
doi
:
10.1007/9783540725046_49
.
ISBN
9783540725039
. As
PDF

^
BenAmram, Amir M. (2015). “Mortality of iterated piecewise affine functions over the integers: Decidability and complexity”. Computability. 1 (1): 19–56.
doi
:
10.3233/COM150032
.

^
Emmer, Michele (2012). Imagine Math: Between Culture and Mathematics.
Springer Publishing
. pp. 260–264.
ISBN
9788847024267
.

^
Mazmanian, Adam (19 May 2011).
“MOVIE REVIEW: ‘Incendies‘“
.
The Washington Times
. Retrieved 7 December 2019.
External links[
edit
]

Keith Matthews’ 3x + 1 page: Review of progress, plus various programs
.

Distributed computing
(
BOINC
)
project
that verifies the Collatz conjecture for larger values.
 An ongoing
distributed computing
project
by Eric Roosendaal verifies the Collatz conjecture for larger and larger values.
 Another ongoing
distributed computing
project
by Tomás Oliveira e Silva continues to verify the Collatz conjecture (with fewer statistics than Eric Roosendaal’s page but with further progress made).

Weisstein, Eric W.
“Collatz Problem”
.
MathWorld
.

Collatz Problem
at
PlanetMath
..
 Nochella, Jesse.
“Collatz Paths”
.
Wolfram Demonstrations Project
.

Mathematicians Are So Close to Cracking This 82YearOld Riddle
Categories
:

Conjectures

Arithmetic dynamics

Integer sequences
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