The Collatz problem in Fp[x] and Fp[[x]]

Angelot Behajaina, Elad Paran

Research output: Contribution to journalArticlepeer-review


We study the analogue of the Collatz map in the polynomial ring Fp[x], for any prime number p, and the corresponding dynamical system. We show that every f∈Fp[x] is eventually periodic in this system, in a quadratic number of iterations in deg(f), and describe explicitly all corresponding cycles. This extends a result of Hicks, Mullen, Yucas and Zavislak, who studied the case p=2. We also study the Collatz map in the formal power series ring Fp[[x]], observe that in Fp[[x]] all but countably many power series generate divergent trajectories via iterations of this map, and characterize those power series that are eventually periodic.

Original languageEnglish
Article number102265
Pages (from-to)102265
Number of pages1
JournalFinite Fields and Their Applications
StatePublished - Oct 2023

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  • Collatz problem
  • Dynamics
  • Finite fields
  • Polynomials


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