The Collatz conjecture and De Bruijn graphs

TL;DR

This paper explores the structure of Collatz-related graphs, revealing isomorphisms with De Bruijn graphs for certain moduli, and extends these concepts to 2-adic and p-adic integers, uncovering deep algebraic connections.

Contribution

It establishes isomorphisms between Collatz graphs and De Bruijn graphs for powers of two and generalizes these to p-adic contexts, linking Collatz dynamics with algebraic graph structures.

Findings

Collatz graphs mod powers of 2 are isomorphic to binary De Bruijn graphs

The isomorphism extends to 2-adic integers, matching known conjugacy maps

Generalizations of the 3n+1 function relate to p-adic De Bruijn graphs

Abstract

We study variants of the well-known Collatz graph, by considering the action of the 3n+1 function on congruence classes. For moduli equal to powers of 2, these graphs are shown to be isomorphic to binary De Bruijn graphs. Unlike the Collatz graph, these graphs are very structured, and have several interesting properties. We then look at a natural generalization of these finite graphs to the 2-adic integers, and show that the isomorphism between these infinite graphs is exactly the conjugacy map previously studied by Bernstein and Lagarias. Finally, we show that for generalizations of the 3n+1 function, we get similar relations with 2-adic and p-adic De Bruijn graphs.