Journal of Integer Sequences, Vol. 26 (2023), Article 23.3.5

There are no Collatz m-Cycles with m ≤ 91

Christian Hercher
Abteilung für Mathematik und ihre Didaktik
Europa-Universität Flensburg
Auf dem Campius 1b
24944 Flensburg


The Collatz conjecture (or "Syracuse problem") considers recursively-defined sequences of positive integers where n is succeeded by n/2, if n is even, or 3n/2 if n is odd. The conjecture states that for all starting values n the sequence eventually reaches the trivial cycle 1, 2, 1, 2, ... . We are interested in the existence of nontrivial cycles.

Let m be the number of local minima in such a nontrivial cycle. Simons and de Weger proved that m ≥ 76. With newer bounds on the range of starting values for which the Collatz conjecture has been checked, one gets m ≥ 83. In this paper, we prove m ≥ 92.

The last part of this paper considers what must be proven in order to raise the number of odd members a nontrivial cycle has to have to the next bound—that is, to at least K ≥ 1.375· 1011. We prove that it suffices to show that, for every integer smaller than or equal to 1536 · 260 = 3 · 269, the respective Collatz sequence enters the trivial cycle. This reduces the range of numbers to be checked by nearly 60%.

Full version:  pdf,    dvi,    ps,    latex,     C++ program 1, Corollary 29,     C++ program 2, Corollary 19,     SageMath notebook

(Concerned with sequences A006577 A025586.)

Received January 23 2022; revised version received January 24 2022, December 21 2022; December 28 2022; March 12 2023. Published in Journal of Integer Sequences, March 15 2023.

Return to Journal of Integer Sequences home page