A Fibonacci theorem for Collatz trajectories

Abstract

We prove that for each $m\ge1$, exactly $F(m+1)$ odd integers in $\{1,\ldots,2^m\}$ have the property that their orbit under the accelerated Collatz map $T$ avoids residue class $4\pmod{6}$ during steps $2,\ldots,m$, where $F(m+1)$ is the $(m+1)$-th Fibonacci number. The proportion decays at rate $(\varphi/2)^m$. The proof uses the directed graph $G$ of Collatz transitions modulo 6 and its unique absorbing strongly connected component $V_3=\{1,2,4,5\}$. An explicit bijection $\Psi_m$ is constructed encoding integers in $\{1,\ldots,6\cdot2^m\}$ as directed paths of $m$ edges in $G$. It is also shown that every positive cycle of $T$ must visit residue class $2\pmod{6}$, which by a flow conservation identity on $G$ accounts for more than $18\%$ of the steps in any cycle.

Artistic interpretation of graph G

The Collatz conjecture was born in 1937 — the same year as Tolkien's The Hobbit. This coincidence inspired the illustration below, where the nodes of graph G mod 6 become territories on a fantasy map and the edges become bridges connecting them.

Graph G mod 6 as a Tolkien-style fantasy map, with islands, bridges and volcanoes — Graph G mod 6, Tolkien style — illustration by Gema Jiménez Camacho, 2026

Main results

Theorem 1

Fibonacci theorem

Exactly $F(m+1)$ odd integers in $\{1,\ldots,2^m\}$ have trajectories avoiding residue $4\pmod{6}$ at steps $2,\ldots,m$. The proportion decays as $(\varphi/2)^m \to 0$.

Theorem 2

Bijection $\Psi_m$

Explicit construction of a bijection between $\{1,\ldots,6\cdot2^m\}$ and directed paths of $m$ edges in the graph $G$ mod 6 with $m+1$ nodes.

Corollary

Cycles and residue 2

Every positive cycle of $T$ must visit $2\pmod{6}$. By the flow conservation identity on $G$, this accounts for more than 18 % of the steps in any cycle.

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How to cite

@misc{reyesjimenez2026fibonacci,
  title         = {A Fibonacci theorem for {Collatz} trajectories via modular graph structure},
  author        = {Reyes Jim\'enez, Manuel-Alejandro},
  year          = {2026},
  eprint        = {2606.02621},
  archivePrefix = {arXiv},
  primaryClass  = {math.NT},
  doi           = {10.48550/arXiv.2606.02621}
}

A Fibonacci theorem for Collatz trajectories via modular graph structure