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Parity-Based Level-Set Approach to the Collatz Conjecture

Author

Listed:
  • Selcuk Koyuncu

    (Department of Mathematics, College of Science and Technology, Temple University Japan, Tokyo 154-0004, Japan)

  • Thevasha Sathiyakumar

    (Department of Mathematics and Computer Science, College of Arts, Science, and Education, Coppin State University, Baltimore, MD 21216, USA)

  • Praise Alayode

    (Department of Mathematics and Computer Science, College of Arts, Science, and Education, Coppin State University, Baltimore, MD 21216, USA)

  • Christopher Ellis

    (Department of Mathematics and Computer Science, College of Arts, Science, and Education, Coppin State University, Baltimore, MD 21216, USA)

  • Peyton Thomas

    (Department of Mathematics and Computer Science, College of Arts, Science, and Education, Coppin State University, Baltimore, MD 21216, USA)

Abstract

The Collatz conjecture concerns the iteration of the map f ( n ) = 3 n + 1 for odd n and f ( n ) = n / 2 for even n . In this paper, we study the level sets l x = { n ∈ N ∣ L ( n ) = x } , where L ( n ) denotes the Collatz length. Using the parity representation of Collatz trajectories, we partition each l x according to the number of odd steps and analyze the corresponding means μ x , k . Under a natural scaling assumption, these means satisfy an approximate geometric progression, so that log μ x , k is approximately linear in k . Computations for n ≤ 100 , 000 and 10 ≤ x ≤ 50 show highly stable regression parameters and near-perfect linear fits.

Suggested Citation

  • Selcuk Koyuncu & Thevasha Sathiyakumar & Praise Alayode & Christopher Ellis & Peyton Thomas, 2026. "Parity-Based Level-Set Approach to the Collatz Conjecture," Mathematics, MDPI, vol. 14(10), pages 1-12, May.
    • Handle: RePEc:gam:jmathe:v:14:y:2026:i:10:p:1763-:d:1947637
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