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

Abteilung für Mathematik und ihre Didaktik

Europa-Universität Flensburg

Auf dem Campius 1b

24944 Flensburg

Germany

**Abstract:**

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 3*n*/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· 10^{11}. We prove
that it suffices to show that, for every integer smaller than or equal to
1536 · 2^{60} = 3 · 2^{69},
the respective Collatz sequence enters the trivial cycle.
This reduces the range of numbers to be checked by nearly 60%.

(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.

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