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{{short description|Open problem on 3x+1 and x/2 functions}}
{{pp|small=yes}}
{{unsolved|mathematics|{{bulleted list|For even numbers, divide by 2;|For odd numbers, multiply by 3 and add 1.}}With enough repetition, do all positive integers converge to 1?}}
[[File:Collatz-graph-50-no27.svg|thumb|upright=0.6|[[Directed graph]] showing the [[Orbit (dynamics)|orbits]] of small numbers under the Collatz map, skipping even numbers. The Collatz conjecture states that all paths eventually lead to 1.]]
The '''Collatz conjecture'''{{efn|It is also known as the '''{{math|3''n'' + 1}} problem''' (or '''conjecture'''), the '''{{math|3''x'' + 1}} problem''' (or '''conjecture'''), the '''Ulam conjecture''' (after [[Stanisław Ulam]]), '''Kakutani's problem''' (after [[Shizuo Kakutani]]), the '''Thwaites conjecture''' (after [[Bryan Thwaites]]), '''Hasse's algorithm''' (after [[Helmut Hasse]]), or the '''Syracuse problem''' (after [[Syracuse University]]).<ref>{{cite book |last1=Maddux |first1=Cleborne D. |last2=Johnson |first2=D. Lamont |year=1997 |title=Logo: A Retrospective |publisher=Haworth Press |___location=New York |isbn=0-7890-0374-0 |page=160 |quote=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.}}</ref>{{refn|According to {{named ref|name=Lagarias (1985)}} p. 4, the name "Syracuse problem" was proposed by Hasse in the 1950s, during a visit to [[Syracuse University]].}}}} is one of the most famous [[List of unsolved problems in mathematics|unsolved problems in mathematics]]. The [[conjecture]] asks whether repeating two simple arithmetic operations will eventually transform every [[positive integer]] into 1. It concerns [[integer sequence|sequences of integers]] in which each term is obtained from the previous term as follows: if a term is [[Parity (mathematics)|even]], the next term is one half of it. If a term is odd, the next term is 3 times the previous term plus 1. The conjecture is that these sequences always reach 1, no matter which positive integer is chosen to start the sequence. The conjecture has been shown to hold for all positive integers up to {{val|2.36e21}}, but no general proof has been found.
It is named after the mathematician [[Lothar Collatz]], who introduced the idea in 1937, two years after receiving his doctorate.<ref>{{mactutor|title=Lothar Collatz|id=Collatz}}</ref> The sequence of numbers involved is sometimes referred to as the '''hailstone sequence''', '''hailstone numbers''' or '''hailstone numerals''' (because the values are usually subject to multiple descents and ascents like [[hailstones]] in a cloud),<ref>{{cite book |last=Pickover |first=Clifford A. |year=2001 |title=Wonders of Numbers |url=https://archive.org/details/wondersnumbersad00pick |url-access=limited |publisher=Oxford University Press |___location=Oxford |isbn=0-19-513342-0 |pages=[https://archive.org/details/wondersnumbersad00pick/page/n136 116]–118}}</ref> or as '''wondrous numbers'''.<ref>{{cite book |last=Hofstadter |first=Douglas R. |author-link=Douglas Hofstadter |year=1979 |title=Gödel, Escher, Bach |publisher=Basic Books |___location=New York |isbn=0-465-02685-0 |pages=[https://archive.org/details/godelescherbach00doug/page/400 400–2]|title-link=Gödel, Escher, Bach }}</ref>
[[Paul Erdős]] said about the Collatz conjecture: "Mathematics may not be ready for such problems."<ref name="Guy (2004)"/>
==Statement of the problem==
[[File:Collatz-stopping-time.svg|thumb|Numbers from 1 to 9999 and their corresponding total stopping time]]
[[File:CollatzStatistic100million.png|thumb|Histogram of total stopping times for the numbers 1 to 10<sup>8</sup>. Total stopping time is on the {{mvar|x}} axis, frequency on the {{mvar|y}} axis.]]
[[File:CollatzStatistic1billion.png|thumb|Histogram of total stopping times for the numbers 1 to 10<sup>9</sup>. Total stopping time is on the {{mvar|x}} axis, frequency on the {{mvar|y}} axis.]]
[[File:Collatz-10Million.png|thumb|Iteration time for inputs of 2 to 10<sup>7</sup>.]]
[[File:Collatz Gif.gif|alt=Total Stopping Time:
Consider the following operation on an arbitrary [[positive integer]]:
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In [[modular arithmetic]] notation, define the [[function (mathematics)|function]] {{mvar|f}} as follows:
<math display="block"> f(n) = \begin{cases} n/2 &\text{if } n \equiv 0 \pmod{2},\\
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:
<math display="block"> a_i = \begin{cases}n & \text{for } i = 0, \\ f(a_{i-1}) & \text{for } i > 0 \end{cases}</math>
(that is: {{math|''a<sub>i</sub>''}} is the value of {{mvar|f}} applied to {{mvar|n}} recursively {{mvar|i}} times; {{math|''a<sub>i</sub>'' {{=}} ''f''{{hsp}}{{isup|''i''}}(''n'')}}).
The Collatz conjecture is: ''This process will eventually reach the number 1, regardless of which positive integer is chosen initially. That is, for each'' <math>n</math>, there is some <math>i</math> with <math>a_i = 1</math>.
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 {{mvar|i}} such that {{math|''a<sub>i</sub>'' < ''a''<sub>0</sub> }} is called the '''stopping time''' of {{mvar|n}}. Similarly, the smallest {{mvar|k}} such that {{math|''a<sub>k</sub>'' {{=}} 1}} is called the '''total stopping time''' of {{mvar|n}}.<ref name="Lagarias (1985)"/> If one of the indexes {{mvar|i}} or {{mvar|k}} does not exist, we say that the stopping time or the total stopping time, respectively, is infinite.
Since {{math|3''n'' + 1}} is even whenever {{mvar|n}} is odd, one may instead use the "shortcut" form of the Collatz function:
<math display = "block"> f(n) = \begin{cases} \frac{n}{2} &\text{if } n \equiv 0 \pmod{2},\\ \frac{3n+1}{2} & \text{if } n\equiv 1 \pmod{2}. \end{cases}</math>
This definition yields smaller values for the stopping time and total stopping time without changing the overall dynamics of the process.
==
For instance, starting with {{math|''n'' {{=}} 12}}
The number {{math|''n'' {{=}} 19}}
<!-- NOTICE TO EDITORS... Please note that the number of steps is one less than the number of elements of the sequence! So the number of 111 steps is CORRECT for n=27. Thanks for paying attention to this factoid! -->
The sequence for {{math|''n'' {{=}} 27}}, listed and graphed below, takes 111 steps (41 steps through odd numbers, in
: {{CSG|27|bold= odd}}{{OEIS|id=A008884}}
[[File:Collatz5.svg|frameless|upright=2|center]]
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: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, ... {{OEIS|A006877}}.
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, ... {{OEIS|id=A006884}}
Line 58 ⟶ 63:
: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, ... {{OEIS|id=A006577}}
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,
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:less than 10<sup>6</sup> is {{val|837799}}, which has 524 steps,
:less than 10<sup>7</sup> is {{val|8400511}}, which has 685 steps,
:less than 10<sup>8</sup> is {{val|63728127}}, which has 949 steps,
:less than 10<sup>9</sup> is {{val|670617279}}, which has 986 steps,
:less than 10<sup>10</sup> is {{val|9780657630}}, which has 1132 steps,<ref>{{cite journal |last1=Leavens |first1=Gary T. |last2=Vermeulen |first2=Mike |date=December 1992 |title=3''x'' + 1
:less than 10<sup>11</sup> is {{val|75128138247}}, which has 1228 steps,
:less than 10<sup>12</sup> is {{val|989345275647}}, which has 1348 steps
These numbers are the lowest ones with the indicated step count, but not necessarily the only ones below the given limit. As an example, {{val|9780657631}} has 1132 steps, as does {{val|9780657630}}.
The starting values having the smallest total stopping time with respect to their number of digits (in base 2) are the [[Power of two|powers of two]],
==Visualizations==
<gallery
File:CollatzConjectureGraphMaxValues.jpg|The {{mvar|x}} axis represents starting number, the {{mvar|y}} axis represents the highest number reached during the chain to 1. This plot shows a restricted {{mvar|y}} axis: some {{mvar|x}} values produce intermediates as high as {{val|2.7e7}} (for {{math|''x'' {{=}} 9663}})
File:Collatz-max.png|The same plot as the previous one but on log scale, so all {{mvar|y}} values are shown. The first thick line towards the middle of the plot corresponds to the tip at 27, which reaches a maximum at 9232.
File:All Collatz sequences of a length inferior to 20.svg|The tree of all the numbers having fewer than 20 steps.
File:Collatz Conjecture 100M.jpg|alt=Collatz Conjecture 100M|The number of iterations it takes to get to one for the first 100 million numbers.
File:Collatz_conjecture_tree_visualization.png|Collatz conjecture paths for 5000 random starting points below 1 million.
</gallery>
==Supporting arguments==
Although the conjecture has not been proven, most mathematicians{{Citation needed|date=April 2025}} who have looked into the problem think the conjecture is true because experimental evidence and heuristic arguments support it.
===Experimental evidence===
This computer evidence is still not
However, such verifications may have other implications. Certain constraints on any non-trivial cycle, such as [[lower bound]]s on the length of the cycle, can be proven based on the value of the lowest term in the cycle. Therefore, computer searches to rule out cycles that have a small lowest term can strengthen these constraints.<ref name="Garner (1981)"/><ref name="Eliahou (1993)"/><ref name="Simons & de Weger (2005)"/>
===A probabilistic heuristic===
If one considers only the ''odd'' numbers in the sequence generated by the Collatz process, then each odd number is on average {{sfrac|3|4}} of the previous one.{{refn|{{named ref|name=Lagarias (1985)}} section "[http://www.cecm.sfu.ca/organics/papers/lagarias/paper/html/node3.html A heuristic argument"].}} (More precisely, the geometric mean of the ratios of outcomes is {{sfrac|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.
===Stopping times===
As proven by [[Riho Terras (mathematician)|Riho Terras]], almost every positive integer has a finite stopping time.{{efn|Here "almost every" means that the [[natural density]] of the set of integers with finite stopping times is 1.}}<ref name="Terras (1976)"/> 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 [[#As a parity sequence|parity vectors]] and uses the [[central limit theorem]].
In 2019, [[Terence Tao]]
===
In a [[computer-aided proof]], Krasikov and Lagarias showed that the number of integers in the interval {{math|[1,''x'']}} that eventually reach 1 is at least equal to {{math|''x''<sup>0.84</sup>}} for all sufficiently large {{mvar|x}}.<ref>{{Cite journal
| last1 = Krasikov | first1 = Ilia
| last2 = Lagarias | first2 = Jeffrey C. |
| year = 2003
| title = Bounds for the 3''x'' + 1 problem using difference inequalities
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| pages = 237–258| arxiv = math/0205002
| bibcode = 2003AcAri.109..237K
| s2cid = 18467460
}}</ref>
==Cycles==
In this part, consider the
A
The only known cycle is {{math|(1
===Cycle length===
where {{mvar|a}}, {{mvar|b}} and {{mvar|c}} are non-negative integers, {{math|''b'' ≥ 1}} and {{math|1=''ac''
==={{mvar|k}}-cycles===
A {{mvar|k}}-cycle is a cycle that can be partitioned into {{math|
Steiner (1977) proved that there is no 1-cycle other than the trivial {{math|(1; 2)}}.<ref name="Steiner (1977)"/> Simons (
==Other formulations of the conjecture==
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There is another approach to prove the conjecture, which considers the bottom-up
method of growing the so-called ''Collatz graph''. The ''Collatz graph'' is a [[Graph (discrete mathematics)|graph]] defined by the inverse [[relation (mathematics)|relation]]
<math display="block"> R(n) = \begin{cases} \{2n\} & \text{if } n\equiv 0,1,2,3,5 \\ \left\{2n,\frac{n-1}{3}\right\} & \text{if } n\equiv 4 \end{cases} \pmod 6. </math>
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 {{mvar|n}}, {{math|''n'' ≡ 1 (mod 2)}} [[if and only if]] {{math|3''n'' + 1 ≡ 4 (mod 6)}}. Equivalently, {{math|{{sfrac|''n'' − 1|3}} ≡ 1 (mod 2)}} if and only if {{math|''n'' ≡ 4 (mod 6)}}. Conjecturally, this inverse relation forms a [[tree (graph theory)|tree]] except for the 1–2–4 loop (the inverse of the 4–2–1 loop of the unaltered function {{mvar|f}} defined in the [[#Statement of the problem|Statement of the problem]] section of this article).
When the relation {{math|3''n'' + 1}} of the function {{mvar|f}} is replaced by the common substitute "shortcut" relation {{math|{{sfrac|3''n'' + 1|2}}}}, the Collatz graph is defined by the inverse relation,
<math display="block"> R(n) = \begin{cases} \{2n\} & \text{if } n\equiv 0,1 \\ \left\{2n,\frac{2n-1}{3}\right\} & \text{if } n\equiv 2 \end{cases} \pmod 3. </math>
For any integer {{mvar|n}}, {{math|''n'' ≡ 1 (mod 2)}} if and only if {{math|{{sfrac|3''n'' + 1|2}} ≡ 2 (mod 3)}}. Equivalently, {{math|{{sfrac|2''n'' − 1|3}} ≡ 1 (mod 2)}} if and only if {{math|''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 {{math|3''n'' + 1}}
===As an abstract machine that computes in base two===
Repeated applications of the Collatz function can be represented as an [[abstract machine]] that handles [[string (computer science)|strings]] of [[bit]]s.
# Append {{mono|1}} to the (right) end of the number in binary (giving {{math|2''n'' + 1}});
# Add this to the original number by binary addition (giving {{math|2''n'' + 1 + ''n'' {{=}} 3''n'' + 1}});
# Remove all trailing
====Example====
The starting number 7 is written in base two as {{mono|111}}.
<div style="font-family:
</div>
===As a parity sequence===
For this section, consider the
<math display="block"> f(n) = \begin{cases} \frac{n}{2} &\text{if } n \equiv 0 \\ \frac{3n + 1}{2} & \text{if } n \equiv 1 \end{cases} \pmod{2}.</math>
If {{math|P(...)}} is the parity of a number, that is {{math|P(2''n'') {{=}} 0}} and {{math|P(2''n'' + 1) {{=}} 1}}, then we can define the Collatz parity sequence (or parity vector) for a number {{mvar|n}} as {{math|''p<sub>i</sub>'' {{=}} P(''a<sub>i</sub>'')}}, where {{math|''a''<sub>0</sub> {{=}} ''n''}}, and {{math|''a''<sub>''i''+1</sub> {{=}} ''f''(''a''<sub>''i''</sub>)}}.
Which operation is performed, {{math|{{sfrac|3''n'' + 1|2}}}} or {{math|{{sfrac|''n''|2}}}}, depends on the parity. The parity sequence is the same as the sequence of operations.
Using this form for {{math|''f''(''n'')}}, it can be shown that the parity sequences for two numbers {{mvar|m}} and {{mvar|n}} will agree in the first {{mvar|k}} terms if and only if {{mvar|m}} and {{mvar|n}} are equivalent modulo {{math|2<sup>''k''</sup>}}.
Applying the {{mvar|f}} function {{mvar|k}} times to the number {{math|''n'' {{=}} 2<sup>''k''</sup>''a'' + ''b''}} will give the result {{math|3<sup>''c''</sup>''a'' + ''d''}}, where {{mvar|d}} is the result of applying the {{mvar|f}} function {{mvar|k}} times to {{mvar|b}}, and {{mvar|c}} is how many increases were encountered during that sequence
===As a tag system===
For the Collatz function in the shortcut form
<math> f(n) = \begin{cases} \frac{n}{2} &\text{if } n \equiv 0 \\
Hailstone sequences can be computed by the
:{{math|''a'' → ''bc''}}, {{math|''b'' → ''a''}}, {{math|''c'' → ''aaa''}}.
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In this system, the positive integer {{mvar|n}} is represented by a string of {{mvar|n}} copies of {{mvar|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 {{mvar|a}} as the initial word, eventually halts (see ''[[Tag system#Example: Computation of Collatz sequences|Tag system]]'' for a worked example).
==Extensions to larger domains==
===Iterating on all integers===
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
Odd values are listed in large bold. Each cycle is listed with its member of least absolute value (which is always odd) first.
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|}
The generalized Collatz conjecture is the assertion that every integer, under iteration by {{mvar|f}}, eventually falls into one of the four cycles above or the cycle 0 → 0
=== Iterating on rationals with odd denominators ===
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.
When using the "shortcut" definition of the Collatz map, it is known that any periodic [[#As a parity sequence|parity sequence]] is generated by exactly one rational.<ref>{{Cite journal|last=Lagarias|first=Jeffrey|date=1990|title=The set of rational cycles for the 3x+1 problem|url=https://eudml.org/doc/206298|journal=Acta Arithmetica|volume=56|issue=1|pages=33–53|issn=0065-1036|doi=10.4064/aa-56-1-33-53|doi-access=free}}</ref> Conversely, it is conjectured that every rational with an odd denominator has an eventually cyclic parity sequence (Periodicity Conjecture
If a parity cycle has length {{mvar|n}} and includes odd numbers exactly {{mvar|m}} times at indices {{math|''k''<sub>0</sub> <
{{NumBlk|:|<math>\frac{3^{m-1} 2^{k_0} + \cdots + 3^0 2^{k_{m-1}}}{2^n - 3^m}.</math>|{{EquationRef|1}}}}
For example, the parity cycle {{nowrap|(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
as the latter leads to the rational cycle
Any cyclic permutation of {{nowrap|(1 0 1 1 0 0 1)}} is associated to one of the above fractions. For instance, the cycle {{nowrap|(0 1 1 0 0 1 1)}} is produced by the fraction
For a one-to-one correspondence, a parity cycle should be ''irreducible'',
In this context, assuming the validity of the Collatz conjecture implies that {{nowrap|(1 0)}} and {{nowrap|(0 1)}} are the only parity cycles generated by positive whole numbers (1 and 2, respectively).
If the odd denominator {{mvar|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 "{{math|3''n'' + ''d''}} " generalization<ref name="Belaga (1998a)"/> of the Collatz function
\frac{x}{2} &\text{if } x \equiv 0 \pmod{2},\\ \frac{3x+d}{2} & \text{if } x\equiv 1 \pmod{2}. \end{cases}</math> ===2-adic extension===
The function
is well-defined on the ring
Define the ''parity vector'' function {{mvar|Q}} acting on
The function {{mvar|Q}} is a 2-adic [[isometry]].<ref>{{Cite journal|last1=
An equivalent formulation of the Collatz conjecture is that
===Iterating on real or complex numbers{{anchor|Collatz_fractal}}===
[[File:
The Collatz map can be extended to the [[real line]] by choosing any function which evaluates to <math>x/2</math> when <math>x</math> is an even integer, and to either <math>3x + 1</math> or <math>(3x + 1)/2</math> (for the "shortcut" version) when <math>x</math> is an odd integer. This is called an [[interpolating]] function. A simple way to do this is to pick two functions <math>g_1</math> and <math>g_2</math>, where:
:<math>g_1(n) = \begin{cases}1, &n\text{ is even,}\\ 0, &n\text{ is odd,}\end{cases}</math>
:<math>g_2(n) = \begin{cases}0, &n\text{ is even,}\\1, &n\text{ is odd,}\end{cases}</math>
and use them as switches for our desired values:
:<math>f(x) \triangleq \frac{x}{2}\cdot g_1(x) \,+\, \frac{3x + 1}{2}\cdot g_2(x)</math>.
One such choice is <math>g_1(x) \triangleq \cos^2\left(\tfrac{\pi}{2} x\right)</math> and <math>g_2(x) \triangleq \sin^2\left(\tfrac{\pi}{2} x\right)</math>. The [[iterations]] of this map lead to a [[dynamical system]], further investigated by Marc Chamberland.<ref name="Chamberland (1996)"/> He showed that the conjecture does not hold for positive real numbers since there are infinitely many [[Fixed point (mathematics)|fixed points]], as well as [[Orbit (dynamics)|orbits]] escaping [[monotonic function|monotonically]] to infinity. The function <math>f</math> has two [[attractor|attracting]] cycles of period <math>2</math>: <math>(1;\,2)</math> and <math>(1.1925...;\,2.1386...)</math>. Moreover, the set of unbounded orbits is conjectured to be of [[Lebesgue measure|measure]] <math>0</math>.
Letherman, Schleicher, and Wood extended the study to the [[complex plane]].<ref name="Letherman, Schleicher, and Wood (1999)"/> They used Chamberland's function for [[Trigonometric_functions#In_the_complex_plane|complex sine and cosine]] and added the extra term <math>\tfrac{1}{\pi}\left(\tfrac12 - \cos(\pi z)\right)\sin(\pi z)\,+</math>
<math>h(z)\sin^2(\pi z)</math>, where <math>h(z)</math> is any [[entire function]]. Since this expression evaluates to zero for real integers, the extended function
:<math>\begin{align}f(z) \triangleq \;&\frac{z}{2}\cos^2\left(\frac{\pi}{2} z\right) + \frac{3z + 1}{2}\sin^2\left(\frac{\pi}{2} z\right) \, + \\
&\frac{1}{\pi}\left(\frac12 - \cos(\pi z)\right)\sin(\pi z) + h(z)\sin^2(\pi z)\end{align}</math>
is an interpolation of the Collatz map to the complex plane. The reason for adding the extra term is to make all integers [[Critical point (mathematics)|critical points]] of <math>f</math>. With this, they show that no integer is in a [[Classification_of_Fatou_components#Baker_domain|Baker ___domain]], which implies that any integer is either eventually periodic or belongs to a [[wandering set|wandering ___domain]]. They conjectured that the latter is not the case, which would make all integer orbits finite.
[[File:Collatz Fractal.jpg|thumb|left|A Collatz [[fractal]] centered at the origin, with real parts from –5 to 5.]]
Most of the points have orbits that diverge to infinity. Coloring these points based on how fast they diverge produces the image on the left, for <math>h(z) \triangleq 0</math>. The inner black regions and the outer region are the [[Classification of Fatou components|Fatou components]], and the boundary between them is the [[Julia set]] of <math>f</math>, which forms a [[fractal]] pattern, sometimes called a "Collatz fractal".
[[File:Exponential Collatz Fractal.jpg|thumb|right|Julia set of the exponential interpolation.]]
There are many other ways to define a complex interpolating function, such as using the [[Exponential_function#Complex_plane|complex exponential]] instead of sine and cosine:
:<math>f(z) \triangleq \frac{z}{2} + \frac14(2z + 1)\left(1 - e^{i\pi z}\right)</math>,
which exhibit different dynamics. In this case, for instance, if <math>\operatorname{Im}(z) \gg 1</math>, then <math>f(z) \approx z + \tfrac14</math>. The corresponding Julia set, shown on the right, consists of uncountably many curves, called ''hairs'', or ''rays''.
{{clear}}
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==Optimizations==
===Time–space tradeoff===
The section ''[[#As a parity sequence|As a parity sequence]]'' above gives a way to speed up simulation of the sequence.
:{{math|''f''{{isup|''k''}}(2<sup>''k''</sup>''a'' + ''b'') {{=}} 3<sup>''c''(''b'', ''k'')</sup>''a'' + ''d''(''b'', ''k'')}}.
The values of {{mvar|c}} (or better {{math|3<sup>''c''</sup>}}) and {{mvar|d}} can be precalculated for all possible {{mvar|k}}-bit numbers {{mvar|b}}, where {{math|''d''(''b'', ''k'')}} is the result of applying the {{mvar|f}} function {{mvar|k}} times to {{mvar|b}}, and {{math|''c''(''b'', ''k'')}} is the number of odd numbers encountered on the way.<ref>{{cite web |last=Scollo |first=Giuseppe |year=2007 |title=Looking for class records in the 3''x'' + 1 problem by means of the COMETA grid infrastructure |work=Grid Open Days at the University of Palermo |url=http://www.dmi.unict.it/~scollo/seminars/gridpa2007/CR3x+1paper.pdf}}</ref> For example, if {{math|''k'' {{=}} 5}}, one can jump ahead 5 steps on each iteration by separating out the 5 least significant bits of a number and using
: {{mvar|c}}(0...31, 5) = { 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 },
: {{mvar|d}}(0...31, 5) = { 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 {{math|2<sup>''k''</sup>}} [[precomputation]] and storage to speed up the resulting calculation by a factor of {{mvar|k}}, a [[space–time tradeoff]].
===Modular restrictions===
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 {{mvar|n}}.
:{{math|''f''{{isup|''k''}}(2<sup>''k''</sup>''a'' + ''b'')
holds for all {{mvar|a}}, then the first counterexample, if it exists, cannot be {{mvar|b}} modulo {{math|2<sup>''k''</sup>}}.<ref name="Garner (1981)"/>
Integers divisible by 3 cannot form a cycle, so these integers do not need to be checked as counter examples.<ref name=Clay>{{cite web |last=Clay |first=Oliver Keatinge|title=The Long Search for Collatz Counterexamples |url=https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=2052&context=jhm|pages=208|access-date=26 July 2024}}</ref>
==Syracuse function==
If {{mvar|k}} is an odd integer, then {{math|3''k'' + 1}} is even, so {{math|3''k'' + 1 {{=}} 2<sup>''a''</sup>''k''
Some properties of the Syracuse function are:
Line 331:
== Undecidable generalizations ==
In 1972, [[John Horton Conway]] proved that a natural generalization of the Collatz problem is algorithmically [[undecidable problem|undecidable]].<ref>{{
Specifically, he considered functions of the form
<math display="block"> {g(n) = a_i n + b_i} \text{ when } {n\equiv i \pmod P},</math>
where {{math|''a''<sub>0</sub>, ''b''<sub>0</sub>, ..., ''a''<sub>''P'' − 1</sub>, ''b''<sub>''P'' − 1</sub>}} are rational numbers which are so chosen that {{math|''g''(''n'')}} is always an integer. The standard Collatz function is given by {{math|''P'' {{=}} 2}}, {{math|''a''<sub>0</sub> {{=}} {{sfrac|1|2}}}}, {{math|''b''<sub>0</sub> {{=}} 0}}, {{math|''a''<sub>1</sub> {{=}} 3}}, {{math|''b''<sub>1</sub> {{=}} 1}}. Conway proved that the problem
: Given {{mvar|g}} and {{mvar|n}}, does the sequence of iterates {{math|''g<sup>k</sup>''(''n'')}} reach {{math|1}}?
is undecidable, by representing the [[halting problem]] in this way.
Line 347 ⟶ 343:
Closer to the Collatz problem is the following ''universally quantified'' problem:
: Given {{mvar|g}}, does the sequence of iterates {{math|''g<sup>k</sup>''(''n'')}} reach {{math|1}}, for all {{math|''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<ref name=KurtzSimon>{{cite book |last1=Kurtz |first1=Stuart A. |last2=Simon |first2=Janos |editor1-last=Cai |editor1-first=J.-Y. |editor2-last=Cooper |editor2-first=S. B. |editor3-last=Zhu |editor3-first=H. |year=2007 |chapter-url=https://books.google.com/books?id=mhrOkx-xyJIC&pg=PA542 |chapter=The undecidability of the generalized Collatz problem |title=Proceedings of the 4th International Conference on Theory and Applications of Models of Computation, TAMC 2007, held in Shanghai, China in May 2007 |isbn=978-3-540-72503-9 |doi=10.1007/978-3-540-72504-6_49 |pages=542–553}} As [http://www.cs.uchicago.edu/~simon/RES/collatz.pdf PDF]</ref> proved that the universally quantified problem is, in fact, undecidable and even higher in the [[arithmetical hierarchy]]; specifically, it is {{math|Π{{su|b=2|p=0}}}}-complete. This hardness result holds even if one restricts the class of functions {{mvar|g}} by fixing the modulus {{mvar|P}} to 6480.<ref>{{cite journal |last=Ben-Amram |first=Amir M. |year=2015 |title=Mortality of iterated piecewise affine functions over the integers: Decidability and complexity |journal=Computability |doi=10.3233/COM-150032 |volume=1 |issue=1 |pages=19–56}}</ref>
Iterations of {{mvar|g}} in a simplified version of this form, with all <math>b_i</math> equal to zero, are formalized in an [[esoteric programming language]] called [[FRACTRAN]].
==In computational complexity==
The Collatz and related conjectures are often used when studying computational complexity.<ref>{{cite journal |author=Michel, Pascal |title=Busy beaver competition and Collatz-like problems |journal=Archive for Mathematical Logic |volume=32 |issue=5 |year=1993 |pages=351–367|doi=10.1007/BF01409968 }}</ref><ref>{{cite web |url=https://arxiv.org/html/2107.12475v2 |title=Hardness of busy beaver value BB(15)}}</ref> The connection is made through the [[busy beaver]] function, where BB(n) is the maximum number of steps taken by any n state [[Turing machine]] that halts. There is a 15 state Turing machine that halts if and only if a conjecture by [[Paul Erdős]] (closely related to the Collatz conjecture) is false. Hence if BB(15) was known, and this machine did not stop in that number of steps, it would be known to run forever and hence no counterexamples exist (which proves the conjecture true). This is a completely impractical way to settle the conjecture; instead it is used to suggest that BB(15) will be very hard to compute, at least as difficult as settling this Collatz-like conjecture.
In 2024, a six-state machine was found for which determining whether it halts involves solving a Collatz-like problem called the antihydra problem. As proofs of even simple conjectures of this nature are not currently known, this suggests that BB(6) will be very hard to compute.<ref>{{cite magazine|magazine=Quanta|title=With Fifth Busy Beaver, Researchers Approach Computation’s Limits|date=July 2, 2024|first=Ben|last=Brubaker|url=https://www.quantamagazine.org/amateur-mathematicians-find-fifth-busy-beaver-turing-machine-20240702/|access-date=2025-08-24}}</ref><ref>{{cite OEIS|A386792|Antihydra, a BB(6) Turing machine (values of a)}}</ref>
==See also==
{{portal|Mathematics}}
{{commons category}}
* [[3x + 1 semigroup|{{math|3''x'' + 1}} semigroup]]
* [[Arithmetic dynamics#Other areas in which number theory and dynamics interact|Arithmetic dynamics]]
* [[Juggler sequence]]
* [[Modular arithmetic]]
* [[Residue-class-wise affine group]]
==
{{notelist}}
==References==
{{Reflist|30em|refs=
<ref name="Lagarias (1985)">{{cite journal |first=Jeffrey C. |last=Lagarias |title=The 3''x'' + 1 problem and its generalizations |journal=[[The American Mathematical Monthly]] |volume=92 |issue=1 |pages=3–23 |year=1985 |jstor=2322189|doi=10.1080/00029890.1985.11971528 }}</ref>
<!--<ref name="Lagarias (2001)">{{SpringerEOM | urlname=S/s110330 | title=Syracuse problem | author=Jeffrey C. Lagarias}}.</ref>-->
<ref name="Chamberland (1996)">{{cite journal |first=Marc
<ref name="
<ref name="Letherman, Schleicher, and Wood (1999)">{{cite journal |first1=Simon |last1=Letherman |first2=Dierk |last2=Schleicher |first3=Reg |last3=Wood |title=The (3''n'' + 1)-problem and holomorphic dynamics |journal=Experimental Mathematics |volume=8 |issue=3 |pages=241–252 |year=1999 |doi= 10.1080/10586458.1999.10504402}}</ref>
<ref name="
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<!--<ref name="Van Bendegem (2005)">{{cite journal |first=Jean Paul |last=Van Bendegem |title=The Collatz Conjecture: A Case Study in Mathematical Problem Solving |journal=Logic and Logical Philosophy |volume=14 |pages=7–23 |year=2005 |doi= 10.12775/llp.2005.002|url=https://compmath.files.wordpress.com/2008/08/jpvb_collatz.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://compmath.files.wordpress.com/2008/08/jpvb_collatz.pdf |archive-date=2022-10-09 |url-status=live |format=PDF}}</ref>-->
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<ref name="Steiner (1977)">{{cite book |first=R. P. |last=Steiner |chapter=A theorem on the syracuse problem |title=Proceedings of the 7th Manitoba Conference on Numerical Mathematics |year=1977 |pages=553–9 |mr=535032}}</ref>
<ref name="Simons & de Weger (2005)">{{cite journal |first1=J. |last1=Simons |first2=B. |last2=de Weger |title=Theoretical and computational bounds for ''m''-cycles of the 3''n'' + 1 problem |journal=Acta Arithmetica |volume=117 |issue=1 |pages=51–70 |year=2005 |doi=10.4064/aa117-1-3 |url=http://deweger.xs4all.nl/papers/[35]SidW-3n+1-ActaArith[2005].pdf |bibcode=2005AcAri.117...51S |doi-access=free |access-date=2023-03-28 |archive-date=2022-03-18 |archive-url=https://web.archive.org/web/20220318094356/http://deweger.xs4all.nl/papers/[35]SidW-3n+1-ActaArith[2005].pdf |url-status=bot: unknown }}</ref>
<ref name="Terras (1976)">{{cite journal | last = Terras | first = Riho | year = 1976 | title = A stopping time problem on the positive integers | journal = Acta Arithmetica | mr = 0568274 | volume = 30 | issue = 3 | pages = 241–252 | url = http://matwbn.icm.edu.pl/ksiazki/aa/aa30/aa3034.pdf | doi=10.4064/aa-30-3-241-252| doi-access = free }}</ref>
<!--<ref name="Sinyor (2010)">Sinyor, J.; [http://downloads.hindawi.com/journals/ijmms/2010/458563.pdf "The 3x+1 Problem as a String Rewriting System"], ''International Journal of Mathematics and Mathematical Sciences'', volume 2010 (2010), Article ID 458563, 6 pages.</ref>-->
<!--<ref name="Belaga (1998b)">{{cite paper | author1-link = Edward Belaga | author1-last = Belaga | author1-first = Edward G. | citeseerx = 10.1.1.54.483 | title = Reflecting on the 3x+1 Mystery | publisher = [[University of Strasbourg]] | date = 1998 }}</ref>
<ref name="Bruschi (2008)">{{cite arXiv |author=Bruschi, Mario |eprint=0810.5169 |title=A generalization of the Collatz problem and conjecture |class=math.NT |year=2008}}</ref>
Line 389 ⟶ 390:
<ref name="Lagarias (2006)">{{cite arXiv |author=Jeffrey C. Lagarias |eprint=math.NT/0608208 |title=The 3''x'' + 1 problem: An annotated bibliography, II (2000–) |class=math.NT |year=2006}}</ref>
<ref name="Ohira">{{cite paper | last1 = Ohira | first1 = Reiko | last2 = Yamashita | first2 = Michinori | url = http://risweb2.ris.ac.jp/faculty/earth_env/yamasita/open/p-col.pdf | title = A generalization of the Collatz problem | language = ja }}</ref>
<ref name="Sinisalo (2003)">{{cite paper | first = Matti K. | last = Sinisalo |
<ref name="Stadfeld">{{cite paper | first = Paul | last = Stadfeld | url = http://home.versatel.nl/galien8/blueprint/blueprint.html | title = Blueprint for Failure: How to Construct a Counterexample to the Collatz Conjecture }}</ref>
<ref name="Urata">{{cite paper | last = Urata | first = Toshio | url = http://auemath.aichi-edu.ac.jp/~turata/Fall.files/CTZVI.pdf | archive-url = https://web.archive.org/web/20041128171946/http://auemath.aichi-edu.ac.jp/~turata/Fall.files/CTZVI.pdf | url-status = dead | archive-date = 2004-11-28 | title = Some Holomorphic Functions connected with the Collatz Problem }}</ref>-->
<!--<ref name="Everest (2003)">{{cite book | last1=Everest | first1=Graham | last2=van der Poorten | first2=Alf | author2-link=Alfred van der Poorten | last3=Shparlinski | first3=Igor | last4=Ward | first4=Thomas | title=Recurrence sequences | series=Mathematical Surveys and Monographs | volume=104 | ___location=[[Providence, RI|Providence]], Rhode Island, USA | publisher=[[American Mathematical Society]] | year=2003 | isbn=0-8218-3387-1 | zbl=1033.11006 | at=Chapter 3.4 }}</ref>-->
<ref name="Guy (2004)">{{cite book |last=Guy | first=Richard K. |
<ref name="Lagarias (2010)">{{cite book |editor1-last=Lagarias |editor1-first=Jeffrey C. |editor1-link=Jeffrey Lagarias |year=2010 |title=The Ultimate Challenge: The 3''x'' + 1 Problem |publisher=[[American Mathematical Society]] |isbn=978-0-8218-4940-8 |zbl=1253.11003}}</ref>
}}
==External links==
*
* An ongoing [[
*
*
* Another ongoing volunteer computing [http://sweet.ua.pt/tos/3x+1.html 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).
* {{MathWorld | urlname=CollatzProblem | title=Collatz Problem}}
* {{PlanetMath | urlname=CollatzProblem | title=Collatz Problem}}
* {{cite web |first=Jesse |last=Nochella |title=Collatz Paths
* {{cite AV media |medium=short video |people=[[David Eisenbud|Eisenbud, D.]] |title=Uncrackable? The Collatz conjecture |series=Numberphile |date=8 August 2016 |via=YouTube |url=https://www.youtube.com/watch?v=5mFpVDpKX70 |archive-url=https://ghostarchive.org/varchive/youtube/20211211/5mFpVDpKX70| archive-date=2021-12-11 |url-status=live}}{{cbignore}}
* {{cite AV media |people=[[David Eisenbud|Eisenbud, D.]] |title=Uncrackable? Collatz conjecture |medium=extra footage |series=Numberphile |date=August 9, 2016 |via=YouTube |url=https://www.youtube.com/watch?v=O2_h3z1YgEU |archive-url=https://ghostarchive.org/varchive/youtube/20211211/O2_h3z1YgEU| archive-date=2021-12-11 |url-status=live}}{{cbignore}}
* {{cite AV media |medium=short video |people=[[Alex Kontorovich]] (featuring) |title=The simplest math problem no one can solve |date=30 July 2021 |series=Veritasium |via=YouTube |url=https://www.youtube.com/watch?v=094y1Z2wpJg}}
* [https://www.technologyreview.com/2021/07/02/1027475/computers-ready-solve-this-notorious-math-problem/ Are computers ready to solve this notoriously unwieldy math problem?]
[[Category:Conjectures]]
[[Category:Arithmetic dynamics]]
[[Category:Integer sequences]]
[[Category:Unsolved problems in number theory]]
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