Recursive sufficiency for the Collatz conjecture and computational verification

Mohammad Ansari
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 3, Pages 471–480
DOI: 10.7546/nntdm.2025.31.3.471-480
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Authors and affiliations

Mohammad Ansari
Department of Mathematics, Azad University of Gachsaran
Gachsaran, Iran

Abstract

We define the notion of recursive sufficiency for the Collatz conjecture and we use it to present some results concerning the computational verification of the conjecture. For any integer N\ge 1 and any recursively sufficient set F, it is proved that all integers in the interval [1, N] satisfy the conjecture if and only if F\cap [1, N] satisfies the conjecture. We offer a sequence of sieves for which the corresponding sequence of elimination percentages tends to 100\%, and as a result, for any integer P arbitrarily close to 100, we give a sieve whose elimination percentage is at least P\%. Also, we prove that if N=2(3^n)+1 is the largest known integer for which all integers 1, 2, \ldots , N satisfy the conjecture, then all integers N+1, N+2, \ldots, 2N will satisfy the conjecture as well, and hence, they can be eliminated from the verification process.

Keywords

  • Collatz conjecture
  • Recursive sufficiency
  • Computational verification

2020 Mathematics Subject Classification

  • 11A99
  • 11Y55

References

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Manuscript history

  • Received: 13 December 2024
  • Revised: 24 July 2025
  • Accepted: 28 July 2025
  • Online First: 4 August 2025

Copyright information

Ⓒ 2025 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Ansari, M. (2025). Recursive sufficiency for the Collatz conjecture and computational verification. Notes on Number Theory and Discrete Mathematics, 31(3), 471-480, DOI: 10.7546/nntdm.2025.31.3.471-480.

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