There are no Collatz-m-Cycles with $m\leq 91$

TL;DR

This paper proves that there are no nontrivial Collatz cycles with fewer than 92 local minima and reduces the range needed to verify the conjecture to a significantly smaller set of numbers.

Contribution

It establishes a new lower bound of 92 for local minima in nontrivial cycles and narrows the verification range for the Collatz conjecture.

Findings

No nontrivial cycles with fewer than 92 local minima.

Reduction of the verification range by nearly 60%.

Provides a criterion involving sequences up to 3·2^{69} for future proofs.

Abstract

The Collatz conjecture (or ``Syracuse problem'') considers recursively-defined sequences of positive integers where $n$ is succeeded by $\tfrac{n}{2}$, if $n$ is even, or $\tfrac{3n+1}{2}$, if $n$ is odd. The conjecture states that for all starting values $n$ the sequence eventually reaches the trivial cycle $1, 2, 1, 2, \ldots$ . 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 $m \geq 76$. With newer bounds on the range of starting values for which the Collatz conjecture has been checked, one gets $m \geq 83$. In this paper, we prove $m \geq 92$. 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 $K\geq1.375\cdot 10^{11}$. We prove that it suffices to show that, for every integer smaller than or equal to $1536\cdot2^{60}=3\cdot2^{69}$, the respective Collatz sequence enters the trivial cycle. This reduces the range of numbers to be checked by nearly $60$\%.