Editing Collatz conjecture (section)

{{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.&nbsp;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)"/> [[Jeffrey Lagarias]] stated in 2010 that the Collatz conjecture "is an extraordinarily difficult problem, completely out of reach of present day mathematics".<ref name="Lagarias (2010)"/> However, though the Collatz conjecture itself remains open, efforts to solve the problem have led to new techniques and many partial results.<ref name="Lagarias (2010)"/><ref name="Tao">{{Cite journal |last=Tao |first=Terence |date=2022 |title=Almost all orbits of the Collatz map attain almost bounded values |journal=Forum of Mathematics, Pi |language=en |volume=10 |pages=e12 |doi=10.1017/fmp.2022.8 |issn=2050-5086|doi-access=free |arxiv=1909.03562 }}</ref>