Editing Collatz conjecture (section)

==={{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}}