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Collatz Attractors Are Space-Filling

Author

Listed:
  • Idriss J. Aberkane

    (UNESCO-UniTwin Complex Systems Digital Campus, ECCE e-Lab, Strasbourg Université, CEDEX, 67081 Strasbourg, France
    Scanderia Online Education, 66 Avenue des Champs-Elysées, 75008 Paris, France)

Abstract

The algebraic topology of Collatz attractors (or “Collatz Feathers”) remains very poorly understood. In particular, when pushed to infinity, is their set of branches discrete or continuous? Here, we introduce a fundamental theorem proving that the latter is true. For any odd x , we first define A x n as the set of all odd numbers with S y r ( x ) in their Collatz orbit and up to n more digits than x in base 2. We then prove lim n → ∞ | A x n | 2 n + c ≥ 1 with c > − 4 for all x and, in particular, c = 0 for x = 1 , which is a result strictly stronger than that of Tao 2019.

Suggested Citation

  • Idriss J. Aberkane, 2022. "Collatz Attractors Are Space-Filling," Mathematics, MDPI, vol. 10(11), pages 1-9, May.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:11:p:1835-:d:824930
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    References listed on IDEAS

    as
    1. Fabian Bocart, 2018. "Inflation Propensity of Collatz Orbits: A New Proof-of-Work for Blockchain Applications," JRFM, MDPI, vol. 11(4), pages 1-18, November.
    2. Alexander Rahn & Eldar Sultanow & Max Henkel & Sourangshu Ghosh & Idriss J. Aberkane, 2021. "An Algorithm for Linearizing the Collatz Convergence," Mathematics, MDPI, vol. 9(16), pages 1-32, August.
    Full references (including those not matched with items on IDEAS)

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    2. Alexander Rahn & Eldar Sultanow & Max Henkel & Sourangshu Ghosh & Idriss J. Aberkane, 2021. "An Algorithm for Linearizing the Collatz Convergence," Mathematics, MDPI, vol. 9(16), pages 1-32, August.

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