Parity-Based Level-Set Approach to the Collatz Conjecture

Koyuncu, Selcuk; Sathiyakumar, Thevasha; Alayode, Praise; Ellis, Christopher; Thomas, Peyton

doi:10.3390/math14101763

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Open AccessArticle

Parity-Based Level-Set Approach to the Collatz Conjecture

by

Selcuk Koyuncu

^1,*

,

Thevasha Sathiyakumar

²,

Praise Alayode

²,

Christopher Ellis

² and

Peyton Thomas

²

¹

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

²

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

^*

Author to whom correspondence should be addressed.

Mathematics 2026, 14(10), 1763; https://doi.org/10.3390/math14101763

Submission received: 24 March 2026 / Revised: 5 May 2026 / Accepted: 15 May 2026 / Published: 20 May 2026

(This article belongs to the Section E: Applied Mathematics)

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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 \in 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 \leq 100, 000

and

10 \leq x \leq 50

show highly stable regression parameters and near-perfect linear fits.

Keywords: Collatz length; level sets; parity code; mean; regression; clustering

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MDPI and ACS Style

Koyuncu, S.; Sathiyakumar, T.; Alayode, P.; Ellis, C.; Thomas, P. Parity-Based Level-Set Approach to the Collatz Conjecture. Mathematics 2026, 14, 1763. https://doi.org/10.3390/math14101763

AMA Style

Koyuncu S, Sathiyakumar T, Alayode P, Ellis C, Thomas P. Parity-Based Level-Set Approach to the Collatz Conjecture. Mathematics. 2026; 14(10):1763. https://doi.org/10.3390/math14101763

Chicago/Turabian Style

Koyuncu, Selcuk, Thevasha Sathiyakumar, Praise Alayode, Christopher Ellis, and Peyton Thomas. 2026. "Parity-Based Level-Set Approach to the Collatz Conjecture" Mathematics 14, no. 10: 1763. https://doi.org/10.3390/math14101763

APA Style

Koyuncu, S., Sathiyakumar, T., Alayode, P., Ellis, C., & Thomas, P. (2026). Parity-Based Level-Set Approach to the Collatz Conjecture. Mathematics, 14(10), 1763. https://doi.org/10.3390/math14101763

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Parity-Based Level-Set Approach to the Collatz Conjecture

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