On the stopping time of the Collatz map in F2[x]

Gil Alon; Angelot Behajaina; Elad Paran

doi:https://doi.org/10.1016/j.ffa.2024.102473

On the stopping time of the Collatz map in F₂[x]

Gil Alon, Angelot Behajaina, Elad Paran

Mathematics discipline

Open University of Israel

Research output: Contribution to journal › Article › peer-review

Abstract

We study the stopping time of the Collatz map for a polynomial f∈F₂[x], and bound it by O(deg(f)^1.5), improving upon the quadratic bound proven by Hicks, Mullen, Yucas and Zavislak. We also prove the existence of arithmetic sequences of unbounded length in the stopping times of certain sequences of polynomials, a phenomenon observed in the classical Collatz map.

Original language	English
Article number	102473
Journal	Finite Fields and their Applications
Volume	99
DOIs	https://doi.org/10.1016/j.ffa.2024.102473
State	Published - Oct 2024

Keywords

Collatz map
Finite fields
Polynomials

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Algebra and Number Theory
General Engineering
Applied Mathematics

Access to Document

https://doi.org/10.1016/j.ffa.2024.102473

Cite this

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title = "On the stopping time of the Collatz map in F2[x]",

abstract = "We study the stopping time of the Collatz map for a polynomial f∈F2[x], and bound it by O(deg(f)1.5), improving upon the quadratic bound proven by Hicks, Mullen, Yucas and Zavislak. We also prove the existence of arithmetic sequences of unbounded length in the stopping times of certain sequences of polynomials, a phenomenon observed in the classical Collatz map.",

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AU - Behajaina, Angelot

AU - Paran, Elad

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N2 - We study the stopping time of the Collatz map for a polynomial f∈F2[x], and bound it by O(deg(f)1.5), improving upon the quadratic bound proven by Hicks, Mullen, Yucas and Zavislak. We also prove the existence of arithmetic sequences of unbounded length in the stopping times of certain sequences of polynomials, a phenomenon observed in the classical Collatz map.

AB - We study the stopping time of the Collatz map for a polynomial f∈F2[x], and bound it by O(deg(f)1.5), improving upon the quadratic bound proven by Hicks, Mullen, Yucas and Zavislak. We also prove the existence of arithmetic sequences of unbounded length in the stopping times of certain sequences of polynomials, a phenomenon observed in the classical Collatz map.

KW - Collatz map

KW - Finite fields

KW - Polynomials

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