The Real 3x+1 Problem

In this work, we introduce another extension U of the 3n+1 function to the real line. We propose a conjecture about the U-trajectories that generalizes the famous 3n+1 (or Collatz) conjecture. We then prove our main result about the iterates of U (which is directly related to both of these conjectures). We also introduce the "flipped 3x+1" function \widetilde U and prove an analogous result for its trajectories. In the final section, we pose some interesting questions about the iterates of U (and \widetilde U), prove a couple of simple results about the iterates of U and \widetilde U, introduce other related functions and propose yet more conjectures and questions about their iterates. It's our hope that the results, conjectures and questions presented here will be not only relevant to the 3n+1 conjecture itself, but also of interest in their own right.Comment: 12 pages. Accepted for publication in Acta Arithmetica. Added more references. The published version is slightly differen

The Collatz conjecture and De Bruijn graphs

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.Comment: 9 pages, 8 figure