Collatz conjecture: Difference between revisions

Revision as of 20:30, 24 August 2025

Revision as of 14:23, 25 August 2025

Line 134:

===Cycle length===

The lengthAs of a2025, non-trivialthe cycle isbest known tobound beon atcycle leastlength {{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 tois {{val|217976794617}} ({{val|355504839929}} without shortcut).<ref name="Hercher (2023)"Barina/><ref name="EliahouIn (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)"/>