View source for Collatz conjecture

{{unsolved|mathematics|Does the Collatz sequence eventually reach 1 for all positive integer initial values?}}
The '''Collatz conjecture''' is a [[conjecture]] in [[mathematics]] that concerns a [[sequence]] defined as follows: start with any [[positive integer]] {{mvar|n}}. Then each term is obtained from the previous term as follows: if the previous term is even, the next term is one half 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 {{mvar|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.<ref>{{cite web|last1= O'Connor|first1=J.J.|last2=Robertson|first2=E.F.|title=Lothar Collatz|year=2006|url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Collatz.html|publisher=St Andrews University School of Mathematics and Statistics, Scotland}}</ref>  It is also known as the '''{{math|3''n'' + 1}} problem''', the '''{{math|3''n'' + 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'''.<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.&nbsp;4, the name "Syracuse problem" was proposed by Hasse in the 1950s, during a visit to [[Syracuse University]].}} 
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),<ref>{{cite book |last=Pickover |first=Clifford A. |year=2001 |title=Wonders of Numbers |publisher=Oxford University Press |location=Oxford |isbn=0-19-513342-0 |pages=116–118}}</ref><ref name=hn>{{cite web |title=Hailstone Number |url=http://mathworld.wolfram.com/HailstoneNumber.html |work=MathWorld |publisher=Wolfram Research}}</ref> or as '''wondrous numbers'''.<ref>{{cite book |last=Hofstadter |first=Douglas R. |authorlink=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)"/> He also offered US$500 for its solution.<ref>{{cite journal |last=Guy |first=R. K. |year=1983 |title=Don't try to solve these problems |journal=Amer. Math. Monthly |volume=90 |issue= 1|pages=35–41 |jstor=2975688 |doi=10.2307/2975688}} By this Erdos means that there aren't powerful tools for manipulating such objects.</ref> [[Jeffrey Lagarias]] in 2010 claimed that based only on known information about this problem, "this is an extraordinarily difficult problem, completely out of reach of present day mathematics."<ref>{{cite book |editor1-last=Lagarias |editor1-first=Jeffrey C. |date=2010 |title=The ultimate challenge: the 3''x'' + 1 problem |publisher=American Mathematical Society |location=Providence, R.I. |isbn=978-0821849408 |page=4}}</ref>

==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:Collatz-10Million.png|thumb|Iteration time for inputs of 2 to 10<sup>7</sup>.]]

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 (mathematics)|function]] {{mvar|f}} as follows:

: <math> f(n) = \begin{cases} \frac{n}{2} &\text{if } n \equiv 0 \pmod{2}\\[4px] 3n+1 & \text{if } n\equiv 1 \pmod{2} .\end{cases}</math>

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> 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''{{isup|''i''}}(''n'')}}).

The Collatz conjecture is: ''This process will eventually reach the number 1, regardless of which positive integer is chosen initially.''

That smallest {{mvar|i}} such that {{math|''a<sub>i</sub>'' {{=}} 1}} is called the '''total stopping time''' of {{mvar|n}}.<ref name="Lagarias (1985)"/>  The conjecture asserts that every {{mvar|n}} has a well-defined total stopping time. If, for some {{mvar|n}}, such an {{mvar|i}} doesn't exist, we say that {{mvar|n}} has infinite total stopping time and the conjecture is false.

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.

==Examples==
For instance, starting with {{math|''n'' {{=}} 12}}, one gets the sequence 12, 6, 3, 10, 5, 16, 8, 4, 2, 1.

{{math|''n'' {{=}} 19}}, for example, 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.

<!-- 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 large font), climbing to a high of <span style="color:#FF0000"> 9232</span> before descending to 1.

:<big>'''27'''</big>, 82, <big>'''41'''</big>, 124, 62, <big>'''31'''</big>, 94, <big>'''47'''</big>, 142, <big>'''71'''</big>, 214, <big>'''107'''</big>, 322, <big>'''161'''</big>, 484, 242, <big>'''121'''</big>, 364, 182, <big>'''91'''</big>, 274, <big>'''137'''</big>, 412, 206, <big>'''103'''</big>, 310, <big>'''155'''</big>, 466, <big>'''233'''</big>, 700, 350, <big>'''175'''</big>, 526, <big>'''263'''</big>, 790, <big>'''395'''</big>, 1186, <big>'''593'''</big>, 1780, 890, <big>'''445'''</big>, 1336, 668, 334, <big>'''167'''</big>, 502, <big>'''251'''</big>, 754, <big>'''377'''</big>, 1132, 566, <big>'''283'''</big>, 850, <big>'''425'''</big>, 1276, 638, <big>'''319'''</big>, 958, <big>'''479'''</big>, 1438, <big>'''719'''</big>, 2158, <big>'''1079'''</big>, 3238, <big>'''1619'''</big>, 4858, <big>'''2429'''</big>, 7288, 3644, 1822, <big>'''911'''</big>, 2734, <big>'''1367'''</big>, 4102, <big>'''2051'''</big>, 6154, <big>'''3077'''</big>, <big><span style="color:#FF0000"> '''9232'''</span></big>, 4616, 2308, 1154, <big>'''577'''</big>, 1732, 866, <big>'''433'''</big>, 1300, 650, <big>'''325'''</big>, 976, 488, 244, 122, <big>'''61'''</big>, 184, 92, 46, <big>'''23'''</big>, 70, <big>'''35'''</big>, 106, <big>'''53'''</big>, 160, 80, 40, 20, 10, <big>'''5'''</big>, 16, 8, 4, 2, <big>'''1'''</big> {{OEIS|id=A008884}}

[[File:Collatz5.svg|frameless|upright=2|center]]

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, ... {{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}}

Number of steps for {{mvar|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, ... {{OEIS|id=A006577}}

The longest progression for any initial starting number
: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<sup>4</sup> is 6171, which has 261 steps,
:less than 10<sup>5</sup> is {{val|77031}}, which has 350 steps,
: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 Search Programs |journal=Computers & Mathematics with Applications |volume=24 |issue=11 |pages=79–99 |doi=10.1016/0898-1221(92)90034-F}}</ref>
: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 [[Power of two|powers of two]] converge to one quickly because {{math|2<sup>''n''</sup>}} is halved {{mvar|n}} times to reach 1, and is never increased.

==Visualizations==
<gallery widths="180px" heights="250px">
Collatz-graph-all-30-no27.svg|[[Directed graph]] showing the [[Orbit (dynamics)|orbits]] of small numbers under the Collatz map. The Collatz conjecture is equivalent to the statement that all paths eventually lead to 1.
Collatz Orbits of FIrst 1000 Integers.svg|Directed graph showing the orbits of the first 1000 numbers.
CollatzConjectureGraphMaxValues.jpg|The {{mvar|x}} axis represents starting number, the {{mvar|y}} axis represents the highest number reached during the chain to&nbsp;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}})
</gallery>

==Cycles==
In this part, consider the "shortcut" form of the Collatz function
: <math> T(x) = \begin{cases} \frac{x}{2} &\text{if } x \equiv 0 \pmod{2},\\[4px] \frac{3x+1}{2} & \text{if } x\equiv 1 \pmod{2}. \end{cases}</math>
A ''cycle'' is a sequence {{math|(''a''<sub>0</sub>; ''a''<sub>1</sub>; ...; ''a<sub>q</sub>'')}} of distinct positive integers where {{math|''T''(''a''<sub>0</sub>) {{=}} ''a''<sub>1</sub>}}, {{math|''T''(''a''<sub>1</sub>) {{=}} ''a''<sub>2</sub>}}, ..., and {{math|''T''(''a<sub>q</sub>'') {{=}} ''a''<sub>0</sub>}}.

The only known cycle is {{math|(1;2)}} of length 2, called the trivial cycle.

===Cycle length===
The length of a non-trivial cycle is known to be at least {{val|17087915}}.<ref>{{Cite journal|last=Eliahou|first=Shalom|date=1993-08-01|title=The 3''x'' + 1 problem: new lower bounds on nontrivial cycle lengths|url=|journal=Discrete Mathematics|volume=118|issue=1|pages=45–56|doi=10.1016/0012-365X(93)90052-U}}</ref> In fact, Eliahou (1993) proved that the period {{mvar|p}} of any non-trivial cycle is of the form 
:<math>p = 301994 a + 17087915 b + 85137581 c</math> 
where {{mvar|a}}, {{mvar|b}} and {{mvar|c}} are non-negative integers, {{math|''b'' ≥ 1}} and {{math|''ac'' {{=}} 0}}. This result is based on the [[continued fraction]] expansion of {{math|{{sfrac|ln 3|ln 2}}}}.

A similar reasoning that accounts for the recent verification of the conjecture up to {{val|5e18}} leads to the improved lower bound {{val|10439860591}}  (or {{val|17026679261}} without the "shortcut").

==={{mvar|k}}-cycles===
A {{mvar|k}}-cycle is a cycle that can be partitioned into {{math|2''k''}} contiguous subsequences: {{mvar|k}} increasing sequences of odd numbers alternating with {{mvar|k}} decreasing sequences of even numbers. 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 ''1-cycle''.<ref name="Simons & de Weger (2003)"/>

Steiner (1977) proved that there is no 1-cycle other than the trivial {{math|(1;2)}}.<ref name="Steiner (1977)"/> Simons (2004) used Steiner's method to prove that there is no 2-cycle.<ref>{{cite journal |last=Simons |first=John L. |year=2005 |title=On the nonexistence of 2-cycles for the 3''x'' + 1 problem |journal=Math. Comp. |volume=74 |issue= |pages=1565–72 |mr=2137019 |doi=10.1090/s0025-5718-04-01728-4|bibcode=2005MaCom..74.1565S }}</ref> Simons & de Weger (2005) extended this proof up to 68-cycles: there is no {{mvar|k}}-cycle up to {{math|''k'' {{=}} 68}}.<ref name="Simons & de Weger (2003)"/> Beyond 68, this method gives upper bounds for the elements in such a cycle: for example, if there is a 75-cycle, then at least one element of the cycle is less than {{gaps|2385|×|2<sup>50</sup>}}.<ref name="Simons & de Weger (2003)"/> Therefore, as exhaustive computer searches continue, larger cycles may be ruled out. To state the argument more intuitively: we need not look for cycles that have at most 68 trajectories, where each trajectory consists of consecutive ups followed by consecutive downs.

==Supporting arguments==
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===
{{asof|2017}}, the conjecture has been checked by computer for all starting values up to {{gaps|87|×|2<sup>60</sup>}}.<ref name=Roosendaal2>{{cite web |last=Roosendaal |first=Eric |title=On the 3x+1 Problem |url=http://www.ericr.nl/wondrous/ |accessdate=6 June 2018}}</ref> All initial values tested so far eventually end in the repeating cycle {{math|(4;2;1)}}, which has only three terms.  From this lower bound on the starting value, a lower bound can also be obtained for the number of terms a repeating cycle other than {{math|(4;2;1)}} must have.<ref name=Garner>{{cite web |last=Garner |first=Lynn E. |title=On The Collatz 3n + 1 Algorithm |url=http://www.ams.org/journals/proc/1981-082-01/S0002-9939-1981-0603593-2/S0002-9939-1981-0603593-2.pdf |accessdate=27 March 2015}}</ref> When this relationship was established in 1981, the formula gave a lower bound of {{val|35400}} terms.<ref name=Garner/>

This computer evidence is not a proof that the conjecture is true. As shown in the cases of the [[Pólya conjecture]], the [[Mertens conjecture]], and [[Skewes' number]], sometimes a conjecture's only [[counterexamples]] are found when using very large numbers.

===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.  The argument is not a proof because it assumes that Hailstone sequences are assembled from uncorrelated probabilistic events.  (It does rigorously establish that the [[p-adic numbers|2-adic]] extension of the Collatz process has two division steps for every multiplication step for almost all 2-adic starting values.)

And even if the probabilistic reasoning were rigorous, this would still imply only that the conjecture is [[almost surely]] true for any given integer, which does not necessarily imply that it is true for all integers.

[[Terry Tao]] (2019) proved using probability that almost all Collatz orbits attain almost bounded values.<ref>{{cite arXiv |last=Tao |first=Terence |author-link=Terence Tao |eprint=1909.03562 |title=Almost all orbits of the Collatz map attain almost bounded values |class=math.PR |date=2019 }}</ref><ref name="Tao">{{cite web |last1=Tao |first1=Terence |title=Almost all Collatz orbits attain almost bounded values |url=https://terrytao.wordpress.com/2019/09/10/almost-all-collatz-orbits-attain-almost-bounded-values/ |website=What's new |accessdate=11 September 2019 |language=en |date=10 September 2019}}</ref>

===Rigorous bounds===
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. | authorlink2 = Jeffrey Lagarias
 | year = 2003
 | title = Bounds for the 3''x''&nbsp;+&nbsp;1 problem using difference inequalities
 | journal = Acta Arithmetica
 | url = https://www.impan.pl/download/pdf/aa109-3-4
 | doi = 10.4064/aa109-3-4
 | mr = 1980260
 | volume = 109
 | issue = 3
 | pages = 237–258| arxiv = math/0205002
 }}</ref>

==Other formulations of the conjecture==

===In reverse===
[[File:Collatz-tree, depth=20.svg|thumb|upright=2|The first 21 levels of the ''Collatz [[Graph (discrete mathematics)|graph]]'' generated in bottom-up fashion. The graph includes all numbers with an orbit length of 21 or less.]]
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> R(n) = \begin{cases} \{2n\} & \text{if } n\equiv 0,1,2,3,5 \\[4px] \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)}} [[iff]] {{math|3''n'' + 1 ≡ 4 (mod 6)}}. Equivalently, {{math|{{sfrac|''n'' − 1|3}} ≡ 1 (mod 2)}} iff {{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 [[Collatz conjecture#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> R(n) = \begin{cases} \{2n\} & \text{if } n\equiv 0,1 \\[4px] \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)}} iff {{math|{{sfrac|3''n'' + 1|2}} ≡ 2 (mod 3)}}. Equivalently, {{math|{{sfrac|2''n'' − 1|3}} ≡ 1 (mod 2)}} iff {{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}}  with {{math|{{sfrac|''n''′|''H''(''n''′)}}}} where {{math|''n''′ {{=}} 3''n'' + 1}}  and {{math|''H''(''n''′)}} is the highest [[power of 2]] that divides {{math|''n''′}} (with no [[remainder]]).  The resulting function {{mvar|f}} maps from [[odd number]]s to odd numbers.  Now suppose that for some odd number {{mvar|n}}, applying this operation {{mvar|k}} times yields the number 1 (that is, {{math|''f''{{isup|''k''}}(''n'') {{=}} 1}}).  Then in [[Binary number|binary]], the number {{mvar|n}} can be written as the concatenation of [[String (computer science)|strings]] {{math|''w''<sub>''k''</sub> ''w''<sub>''k''−1</sub> … ''w''<sub>1</sub>}} where each {{math|''w''<sub>''h''</sub>}} is a finite and contiguous extract from the representation of {{math|{{sfrac|1|3<sup>''h''</sup>}}}}.<ref name="Colussi2011">{{cite journal |last=Colussi |first=Livio |date=9 September 2011 |title=The convergence classes of Collatz function |journal=Theoretical Computer Science |doi=10.1016/j.tcs.2011.05.056 |volume=412 |issue=39 |pages=5409–5419}}</ref> The representation of {{mvar|n}} therefore holds the [[Repeating decimal|repetends]] of {{math|{{sfrac|1|3<sup>''h''</sup>}}}}, where each repetend is optionally rotated and then replicated up to a finite number of bits.  It is only in binary that this occurs.<ref name="Hew2016">{{cite journal |last=Hew |first=Patrick Chisan |date=7 March 2016 |title=Working in binary protects the repetends of 1/3<sup>''h''</sup>: Comment on Colussi's 'The convergence classes of Collatz function' |journal=Theoretical Computer Science |doi=10.1016/j.tcs.2015.12.033 |volume=618 |pages=135–141}}</ref> Conjecturally, every binary string {{mvar|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&nbsp;{{mvar|s}}).

===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.  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 {{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 "0"s (i.e. repeatedly divide by two until the result is odd).

This prescription is plainly equivalent to computing a Hailstone sequence in base two.

====Example====
The starting number 7 is written in base two as 111.  The resulting Collatz sequence is:

<div style="font-family:'Courier New', 'Lucida Console', 'Courier', Monospace">
            111
           <u>111'''1'''</u>
          1011<s>0</s>
         <u>1011'''1'''</u>
        10001<s>0</s>
       <u>10001'''1'''</u>
       1101<s>00</s>
      <u>1101'''1'''</u>
     101<s>000</s>
    <u>101'''1'''</u>
   1<s>0000</s>
</div>

===As a parity sequence===
For this section, consider the Collatz function in the slightly modified form

: <math> f(n) = \begin{cases} \frac{n}{2} &\text{if } n \equiv 0 \\[4px] \frac{3n +1}{2} & \text{if } n \equiv 1 \end{cases} \pmod{2}.</math>

This can be done because when {{mvar|n}} is odd, {{math|3''n'' + 1}} is always even.

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>}}.  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.<ref name="Lagarias (1985)"/><ref>{{citation
 | 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
}}</ref>

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 (e.g. for {{math|2<sup>5</sup>''a'' + 1}} there are 3 increases as 1 iterates to 2, 1, 2, 1, and finally to 2 so the result is {{math|3<sup>3</sup>''a'' + 2}}; for {{math|2<sup>2</sup>''a'' + 1}} there is only 1 increase as 1 rises to 2 and falls to 1 so the result is {{math|3''a'' + 1}}). When {{mvar|b}} is {{math|2<sup>''k''</sup> − 1}} then there will be {{mvar|k}} rises and the result will be {{math|2 × 3<sup>''k''</sup>''a'' − 1}}. The factor of 3 multiplying {{mvar|a}} is independent of the value of {{mvar|a}}; it depends only on the behavior of {{mvar|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. {{math|4''a'' + 1}} becomes {{math|3''a'' + 1}} after two applications of {{mvar|f}} and {{math|16''a'' + 3}} becomes {{math|9''a'' + 2}} after 4 applications of {{mvar|f}}. Whether those smaller numbers continue to 1, however, depends on the value of {{mvar|a}}.

===As a tag system===
For the Collatz function in the form

<math> f(n) = \begin{cases} \frac{n}{2} &\text{if } n \equiv 0 \\[4px] \frac{3n+1}{2} & \text{if } n \equiv 1. \end{cases} \pmod{2}</math>

Hailstone sequences can be computed by the extremely simple [[Tag system#Example: Computation of Hailstone sequences|2-tag system]] with production rules

:{{math|''a'' → ''bc''}}, {{math|''b'' → ''a''}}, {{math|''c'' → ''aaa''}}.

In this system, the positive integer {{mvar|n}} is represented by a string of {{mvar|na}}, and iteration of the tag operation halts on any word of length less than&nbsp;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]]'' 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 4 known cycles, which all nonzero integers seem to eventually fall into under iteration of {{mvar|f}}.  These cycles are listed here, starting with the well-known cycle for positive&nbsp;{{mvar|n}}:

Odd values are listed in large bold. Each cycle is listed with its member of least absolute value (which is always odd) first.

{| class="wikitable" style="text-align: center;"
! Cycle !! Odd-value cycle length !! Full cycle length
|-
|style="text-align: left;"| <big>'''1'''</big> → 4 → 2 → <big>'''1'''</big> '''...''' || 1 || 3
|-
|style="text-align: left;"| <big>'''−1'''</big> → −2 → <big>'''−1'''</big> '''...''' || 1 || 2
|-
|style="text-align: left;"| <big>'''−5'''</big> → −14 → <big>'''−7'''</big> → −20 → −10 → <big>'''−5'''</big> '''...''' || 2 || 5
|-
|style="text-align: left;"| <big>'''−17'''</big> → −50 → <big>'''−25'''</big> → −74 → <big>'''−37'''</big> → −110 → <big>'''−55'''</big> → −164 → −82 → <big>'''−41'''</big> → −122 → <big>'''−61'''</big> → −182 → <big>'''−91'''</big> → −272 → −136 → −68 → −34 → <big>'''−17'''</big> '''...''' || 7 || 18
|}

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. 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 ===
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.  A closely related fact is that the Collatz map extends to the ring of [[2-adic 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 [[#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}}</ref> Conversely, it is conjectured that every rational with an odd denominator has an eventually cyclic parity sequence (Periodicity Conjecture <ref name="Lagarias (1985)"/>).

If a parity cycle has length {{mvar|n}} and includes odd numbers exactly {{mvar|m}} times at indices {{math|''k''<sub>0</sub> < … < ''k''<sub>''m''−1</sub>}}, then the unique rational which generates immediately and periodically this parity cycle is
{{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
:<math>\frac{3^3 2^0 + 3^2 2^2 + 3^1 2^3 + 3^0 2^6}{2^7 - 3^4} = \frac{151}{47}</math>
as the latter leads to the rational cycle 
:<math>\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} </math>.

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
:<math>\frac{3^3 2^1 + 3^2 2^2 + 3^1 2^5 + 3^0 2^6}{2^7 - 3^4} = \frac{250}{47}</math>.

For a one-to-one correspondence, a parity cycle should be ''irreducible'', i.e., not partitionable into identical sub-cycles. As an illustration of this, the parity cycle {{nowrap|(1 1 0 0 1 1 0 0)}} and its sub-cycle {{nowrap|(1 1 0 0)}} are associated to the same fraction {{sfrac|5|7}} when reduced to lowest terms.

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 of the Collatz function
: <math> T_d(x) = \begin{cases} \frac{x}{2} &\text{if } x \equiv 0 \pmod{2},\\[4px] \frac{3x+d}{2} & \text{if } x\equiv 1 \pmod{2}. \end{cases}</math>

===2-adic extension===
The function 
: <math> T(x) = \begin{cases} \frac{x}{2} &\text{if } x \equiv 0 \pmod{2}\\[4px] \frac{3x+1}{2} & \text{if } x\equiv 1 \pmod{2} \end{cases}</math>
is well-defined on the ring {{math|ℤ<sub>2</sub>}} of [[2-adic integers]], where it is continuous and [[Measure-preserving transformation|measure-preserving]] with respect to the 2-adic measure. Moreover, its dynamics is known to be [[Ergodic theory|ergodic]].<ref name="Lagarias (1985)"/>

Define the ''parity vector'' function {{mvar|Q}} acting on {{math|ℤ<sub>2</sub>}} as
: <math> Q(x) = \sum_{k=0}^{\infty} \left( T^k (x) \mod 2 \right) 2^k </math>.

The function {{mvar|Q}} is a 2-adic [[isometry]].<ref>{{Cite journal|last=Lagarias|first=Jeffrey C.|last2=Bernstein|first2=Daniel J.|date=1996|title=The 3''x'' + 1 Conjugacy Map|journal=Canadian Journal of Mathematics|language=en|volume=48|issue=6|pages=1154–1169|doi=10.4153/CJM-1996-060-x|issn=0008-414X}}</ref> Consequently, every infinite parity sequence occurs for exactly one 2-adic integer, so that almost all trajectories are acyclic in <math>\mathbb{Z}_2</math>.

An equivalent formulation of the Collatz conjecture is that
:<math> Q\left(\mathbb{Z}^{+}\right) \subset \tfrac13 \mathbb{Z}.</math>

===Iterating on real or complex numbers{{anchor|Collatz_fractal}}===
[[File:CobwebCollatz2.PNG|thumb|[[Cobweb plot]] of the orbit 10-5-8-4-2-1-2-1-2-1-etc. in the real extension of the Collatz map (optimized by replacing "{{math|3''n'' + 1}}" with "{{math|{{sfrac|3''n'' + 1|2}}}}")]]

The Collatz map can be viewed as the restriction to the integers of the smooth real and complex map

:<math>f(z)=\frac 1 2 z \cos^2\left(\frac \pi 2 z\right)+(3z+1)\sin^2\left(\frac \pi 2 z\right).</math>

If the standard Collatz map defined above is optimized by replacing the relation {{math|3''n'' + 1}} with the common substitute "shortcut" relation {{math|{{sfrac|3''n'' + 1|2}}}}, it can be viewed as the restriction to the integers of the smooth real and complex map

:<math>f(z)=\frac 1 2 z \cos^2\left(\frac \pi 2 z\right)+\frac {3z+1} {2} \sin^2\left(\frac \pi 2 z\right).</math>

[[File:CollatzFractal.png|thumb|left|Collatz map [[fractal]] in a neighbourhood of the real line]]

The point of view of iteration on the real line was investigated by {{named ref|name=Chamberland (1996)}}. 
He showed that the conjecture does not hold for real numbers since there are infinitely many fixed points, as well as orbits escaping monotonically to infinity. He also showed that there is, at least, another attracting cycle: 1.1925 → 2.1386.

On the complex plane, it was investigated by {{named ref|name=Letherman, Schleicher, and Wood (1999)|separator=.}}
Most points of the plane diverge to infinity, as seen in blue on the illustration below. The boundary between diverging and non-diverging regions show a [[fractal]] pattern called the "Collatz fractal".

{{clear}}

==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.  To jump ahead {{mvar|k}} steps on each iteration (using the {{mvar|f}} function from that section), break up the current number into two parts, {{mvar|b}} (the {{mvar|k}} least significant bits, interpreted as an integer), and {{mvar|a}} (the rest of the bits as an integer).  The result of jumping ahead {{mvar|k}} steps can be found as:

:{{math|''f''{{isup|''k''}}(2<sup>''k''</sup>''a'' + ''b'') {{=}} 3<sup>''c''(''b'')</sup>''a'' + ''d''(''b'')}}.

The {{mvar|c}} (or better {{math|3<sup>''c''</sup>}}) and {{mvar|d}} arrays are precalculated for all possible {{mvar|k}}-bit numbers {{mvar|b}}, where {{math|''d''(''b'')}} is the result of applying the {{mvar|f}} function {{mvar|k}} times to {{mvar|b}}, and {{math|''c''(''b'')}} is the number of odd numbers encountered on the way.<ref>{{citation |last=Scollo |first=Giuseppe |year=2007 |contribution=Looking for Class Records in the 3''x'' + 1 Problem by means of the COMETA Grid Infrastructure |title=Grid Open Days at the University of Palermo |url=http://www.dmi.unict.it/~scollo/seminars/gridpa2007/CR3x+1paper.pdf}}</ref>  For example, if {{mvar|''k'' {{=}} 5}}, one can jump ahead 5 steps on each iteration by separating out the 5 least significant bits of a number and using:

: {{math|''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}
: {{math|''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 {{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&nbsp;{{mvar|n}}.  If, for some given {{mvar|b}} and {{mvar|k}}, the inequality

:{{math|''f''{{isup|''k''}}(2<sup>''k''</sup>''a'' + ''b'')  {{=}} 3<sup>''c''(''b'')</sup>''a'' + ''d''(''b'') < 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/>  For instance, the first counterexample must be odd because {{math|''f''(2''n'') {{=}} ''n''}}, smaller than {{math|2''n''}}; and it must be 3 mod 4 because {{math|''f''{{isup|2}}(4''n'' + 1) {{=}} 3''n'' + 1}}, smaller than {{math|4''n'' + 1}}.  For each starting value {{mvar|a}} which is not a counterexample to the Collatz conjecture, there is a {{mvar|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 {{mvar|k}} increases, the search only needs to check those residues {{mvar|b}} that are not eliminated by lower values of&nbsp;{{mvar|k}}.  Only an exponentially small fraction of the residues survive.{{refn|{{named ref|name=Lagarias (1985)}} Theorem D.}} For example, the only surviving residues mod 32 are 7, 15, 27, and 31.

==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''′}} with {{math|''k''′}} odd and {{math|''a'' ≥ 1}}.  The '''Syracuse function''' is the function {{mvar|f}} from the set {{mvar|I}} of odd integers into itself, for which {{math|''f''(''k'') {{=}} ''k''′}} {{OEIS|id=A075677}}.

Some properties of the Syracuse function are:

* For all {{math|''k'' ∈ ''I''}}, {{math|''f''(4''k'' + 1) {{=}} ''f''(''k'')}}. (Because {{math|3(4''k'' + 1) + 1 {{=}} 12''k'' + 4 {{=}} 4(3''k'' + 1)}}.)
* In more generality: For all {{math|''p'' ≥ 1}} and odd {{mvar|h}}, {{math|''f''{{isup|''p'' − 1}}(2<sup>''p''</sup>''h'' − 1) {{=}} 2 × 3<sup>''p'' − 1</sup>''h'' − 1}}. (Here {{math|''f''{{isup|''p'' − 1}}}} is [[Functional power|function iteration notation]].)
* For all odd {{mvar|h}}, {{math|''f''(2''h'' − 1) ≤ {{sfrac|3''h'' − 1|2}}}}

The Collatz  conjecture is equivalent to the statement that, for all {{mvar|k}} in {{mvar|I}}, there exists an integer {{math|''n'' ≥ 1}} such that {{math|''f''{{isup|''n''}}(''k'') {{=}} 1}}.

== Undecidable generalizations ==
In 1972, [[John Horton Conway]] proved that a natural generalization of the Collatz problem is algorithmically [[undecidable problem|undecidable]].<ref>{{citation |last=Conway |first=John H. |year=1972 |contribution=Unpredictable Iterations |title=Proc. 1972 Number Theory Conf., Univ. Colorado, Boulder |pages=49–52}}</ref>

Specifically, he considered functions of the form

: <math>g(n) = a_i n + b_i,  {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 1?

is undecidable, by representing the [[halting problem]] in this way.

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 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 |chapterurl=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 above problem is, in fact, undecidable and even higher in the [[arithmetical hierarchy]], specifically {{math|Π{{su|b=0|p=2}}}}-complete. This hardness result holds even if one restricts the class of functions {{mvar|g}} by fixing the modulus {{mvar|P}} to 6480.<ref>{{citation |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>

==See also==
{{portal|Mathematics}}
* [[3x + 1 semigroup|{{math|3''x'' + 1}} semigroup]]
* [[Arithmetic dynamics#Other areas in which number theory and dynamics interact|Arithmetic dynamics]]
* [[Modular arithmetic]]
* [[Residue-class-wise affine group]]

==Further reading==
* ''The Ultimate Challenge: the 3''x'' + 1 problem'':
:This volume,<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> 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==
{{Reflist|30em|refs=

===Papers===
<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  |last=Chamberland |title=A continuous extension of the 3''x''&nbsp;+&nbsp;1 problem to the real line |journal=Dynam. Contin. Discrete Impuls Systems |volume=2 |issue=4 |pages=495–509 |year=1996 |doi= |url=}}</ref>
<!--<ref name="Garner (1981)">{{cite journal |last=Garner |first=Lynn E. |year=1981 |title=On the Collatz 3''n''&nbsp;+&nbsp;1 Algorithm |journal=Proceedings of the American Mathematical Society |volume=82 |issue=1 |pages=19–22 |doi=10.2307/2044308 |jstor=2044308|accessdate= |quote= }}</ref>-->
<ref name="Letherman, Schleicher, and Wood (1999)">{{cite journal |first=Simon |last=Letherman |first2=Dierk |last2=Schleicher |first3=Reg |last3=Wood |title=The (3''n''&nbsp;+&nbsp;1)-Problem and Holomorphic Dynamics |journal=Experimental Mathematics |volume=8 |issue=3 |pages=241–252 |year=1999 |doi= 10.1080/10586458.1999.10504402|url=}}</ref>
<!--<ref name="Eliahou (1993)">Eliahou, Shalom, ''The 3x+1 problem: new lower bounds on nontrivial cycle lengths'', Discrete Mathematics 118 (1993) p.&nbsp;45-56; [http://images.math.cnrs.fr/Le-probleme-3n-1-y-a-t-il-des.html ''Le problème 3n+1 : y a-t-il des cycles non triviaux ?''], ''[[Images des mathématiques]]'' (2011) {{Link language|fr}}.</ref>
<ref name="Andrei (1998)">{{cite journal |author1=Andrei, Stefan |author2=Masalagiu, Cristian |doi=10.1007/s002360050117 |title=About the Collatz conjecture |year=1998 |journal=Acta Informatica |volume=35 |issue=2 |pages=167–179}}</ref>
<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 |issue= |pages=7–23 |year=2005 |doi= 10.12775/llp.2005.002|url=https://compmath.files.wordpress.com/2008/08/jpvb_collatz.pdf |format=PDF}}</ref>
<ref name="Belaga (2006)">{{cite book |first=Edward G. |last=Belaga |last2=Mignotte |first2=Maurice |chapter=Walking Cautiously into the Collatz Wilderness: Algorithmically, Number Theoretically, Randomly |chapterurl=http://www-irma.u-strasbg.fr/~belaga/a8*BelagaMathInfo06Presentation060920.ppt |format=PowerPoint |editor= |title=Fourth Colloquium on Mathematics and Computer Science : Algorithms, Trees, Combinatorics and Probabilities, September 18–22, 2006, Institut Élie Cartan, Nancy, France |publisher= |location= |year= |isbn= |pages= |url=}}</ref>
<ref name="Belaga (1998a)">{{cite journal |first=Edward G. |last=Belaga |last2=Mignotte |first2=Maurice |title=Embedding the 3x+1 Conjecture in a 3x+d Context |journal=Experimental Mathematics |volume=7 |issue=2 |pages= |year=1998 |doi= |url=http://www.emis.de/journals/EM/expmath/volumes/7/7.html}}</ref>-->
<ref name="Steiner (1977)">{{cite book |first=R. P. |last=Steiner |chapter=A theorem on the syracuse problem |chapterurl= |editor= |title=Proceedings of the 7th Manitoba Conference on Numerical Mathematics |location= |year=1977 |isbn= |pages=553–9 |mr=535032}}</ref>
<ref name="Simons & de Weger (2003)">{{cite journal |first=J. |last=Simons |first2=B. |last2=de Weger |title=Theoretical and computational bounds for ''m''-cycles of the 3''n''&nbsp;+&nbsp;1 problem |journal=Acta Arithmetica |volume=117 |issue=1 |pages=51–70 |year=2003 |doi=10.4064/aa117-1-3 |url=http://deweger.xs4all.nl/papers/%5B35%5DSidW-3n+1-ActaArith%5B2005%5D.pdf |bibcode=2005AcAri.117...51S }}</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>-->

===Preprints===
<!--<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>
<ref name="De Mol (2008)">{{cite journal | last = De Mol | first = Liesbeth | title = Tag systems and Collatz-like functions | journal = Theoretical Computer Science | volume = 390 | issue = 1 | pages =92–101 | date = January 2008 | url = http://logica.ugent.be/liesbeth/TagColOK.pdf | doi=10.1016/j.tcs.2007.10.020}}</ref>
<ref name="Lagarias (2006)">{{cite arXiv |author=Jeffrey C. Lagarias |eprint=math.NT/0608208 |title=The 3''x''&nbsp;+&nbsp;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 | archivedate = 2009-10-24 | archiveurl = https://web.archive.org/web/20091024183537/http://geocities.com/mattiksinisalo/collatz.doc | url = http://geocities.com/mattiksinisalo/collatz.doc | title = On the minimal cycle lengths of the Collatz sequences | date = June 2003 | publisher = University of Oulu }}</ref>
<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 | title = Some Holomorphic Functions connected with the Collatz Problem }}{{dead link|date=August 2016|bot=medic}}{{cbignore|bot=medic}}</ref>-->

===Books===
<!--<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. | authorlink=Richard K. Guy | title=Unsolved problems in number theory | publisher=[[Springer-Verlag]] |edition=3rd | year=2004 |isbn=0-387-20860-7 | zbl=1058.11001 | chapter="E17: Permutation Sequences" |pages=336–7 |chapter-url=https://books.google.com/books?id=1AP2CEGxTkgC&pg=PA337 |ref=harv}}</ref>
}}

==External links==
* [http://www.numbertheory.org/3x+1/ Keith Matthews' 3''x''&nbsp;+&nbsp;1 page: Review of progress, plus various programs].
* [[Distributed computing]] ([[BOINC]]) [http://boinc.thesonntags.com/collatz/ project] that verifies the Collatz conjecture for larger values.
* An ongoing [[distributed computing]] [http://www.ericr.nl/wondrous/index.html project] by Eric Roosendaal verifies the Collatz conjecture for larger and larger values.
* Another ongoing [[distributed 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 |date= |work=[[Wolfram Demonstrations Project]] |url=http://demonstrations.wolfram.com/CollatzPaths/ }}
* [https://www.popularmechanics.com/science/math/a29033918/math-riddle-collatz-conjecture/ Mathematicians Are So Close to Cracking This 82-Year-Old Riddle]

[[Category:Articles with inconsistent citation formats]]
[[Category:Conjectures]]
[[Category:Arithmetic dynamics]]
[[Category:Integer sequences]]