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

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 $60$\%.