Editing Collatz conjecture (section)

==Cycles==
In this part, consider 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>
A [[Periodic sequence|cycle]] is a sequence {{math|(''a''<sub>0</sub>, ''a''<sub>1</sub>, ..., ''a<sub>q</sub>'')}} of distinct positive integers where {{math|''f''(''a''<sub>0</sub>) {{=}} ''a''<sub>1</sub>}}, {{math|''f''(''a''<sub>1</sub>) {{=}} ''a''<sub>2</sub>}}, ..., and {{math|''f''(''a<sub>q</sub>'') {{=}} ''a''<sub>0</sub>}}.

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

===Cycle length===
The length of a non-trivial cycle is known to be at least {{val|114208327604}} (or {{val|186265759595}} without shortcut). If it can be shown that for all positive integers less than <math>3 \times 2^{69}</math> the Collatz sequences reach 1, then this bound would raise to {{val|217976794617}} ({{val|355504839929}} without shortcut).<ref name="Hercher (2023)"/><ref name="Eliahou (1993)"/> In fact, Eliahou (1993) proved that the period {{mvar|p}} of any non-trivial cycle is of the form
<math display="block">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|1=''ac'' = 0}}. This result is based on the [[simple continued fraction]] expansion of {{math|{{sfrac|ln 3|ln 2}}}}.<ref name="Eliahou (1993)"/>

==={{mvar|k}}-cycles===
A {{mvar|k}}-cycle is a cycle that can be partitioned into {{math|''k''}} contiguous subsequences, each consisting of an increasing sequence of odd numbers, followed by a decreasing sequence of even numbers.<ref name="Simons & de Weger (2005)"/> 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''.

Steiner (1977) proved that there is no 1-cycle other than the trivial {{math|(1; 2)}}.<ref name="Steiner (1977)"/> Simons (2005) 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 |pages=1565–72 |mr=2137019 |doi=10.1090/s0025-5718-04-01728-4|bibcode=2005MaCom..74.1565S |doi-access=free }}</ref> Simons and 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 (2005)"/> Hercher extended the method further and proved that there exists no ''k''-cycle with {{math|''k'' ≤ 91}}.<ref name="Hercher (2023)"/> As exhaustive computer searches continue, larger {{math|''k''}} values may be ruled out. To state the argument more intuitively; we do not have to search for cycles that have less than 92 subsequences, where each subsequence consists of consecutive ups followed by consecutive downs.{{clarify|date=September 2024}}