A refinement of the 3x + 1 conjecture

Roger Zarnowski
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 3, Pages 234—244
DOI: 10.7546/nntdm.2020.26.3.234-244
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Authors and affiliations

Roger Zarnowski
Department of Mathematics and Statistics, Northern Kentucky University
Nunn Drive, MEP 401, Highland Heights, KY 41099, USA


The 3x + 1 conjecture pertains to iteration of the function T defined by T(x) = x/2 if x is even and T(x) = (3x + 1)/2 if x is odd. The conjecture asserts that the trajectory of every positive integer eventually reaches the cycle (2, 1). We show that the essential dynamics of T-trajectories can be more clearly understood by restricting attention to numbers congruent to 2 (mod 3). This approach leads to an equivalent conjecture for an underlying function TR whose iterates eliminate many extraneous features of T-trajectories. We show that the function TR that governs the refined conjecture has particularly simple mapping properties in terms of partitions of the set of integers, properties that have no parallel in the classical formulation of the conjecture. We then use those properties to obtain a new characterization of T-trajectories and we show that the dynamics of the 3x + 1 problem can be reduced to an iteration involving only numbers congruent to 2 or 8 (mod 9).


  • 3x + 1 problem
  • Collatz conjecture

2010 Mathematics Subject Classification

  • 11B83


  1. J. C. Lagarias, Editor, The Ultimate Challenge: The 3x + 1 Problem, Amer. Math. Soc., 2010.

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Cite this paper

Zarnowski, R. (2020). A refinement of the 3x + 1 conjecture. Notes on Number Theory and Discrete Mathematics, 26 (3), 234-244, doi: 10.7546/nntdm.2020.26.3.234-244.

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