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Published February 3, 2026 | Version 2.0
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Collatz‑3: A Modular Variant of the Collatz Process

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Description

Collatz‑3 is a playful yet mathematically structured variant of the classical Collatz map, redesigned around modulo‑3 arithmetic.
Each integer evolves according to a simple rule based on its remainder when divided by 3, producing a mixture of rapid descent, temporary growth, and oscillatory behavior. Despite the simplicity of the rule, the system generates a rich landscape of trajectories and stopping times.

This work introduces the Collatz‑3 transformation, defines the notion of 3‑Resistance (the number of steps required to reach 1), and analyzes the resistance profile for integers from 1 to 100. The results reveal characteristic valleys, ridges, and isolated peaks within the 3‑adic structure, with 28 emerging as the strongest resistor—the “Boss” of the range.

The document includes:

  • a formal definition of the Collatz‑3 map
  • examples of trajectories
  • a complete stopping‑time dataset for 1–100
  • visualizations such as line charts and resistance profiles
  • a Python implementation for further exploration

This project aims to present Collatz‑3 as an accessible form of playable mathematics, encouraging experimentation, visualization, and curiosity-driven discovery.

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Additional details

Dates

Created
2026-02-03
The date when the Collatz‑3 report was completed.
Issued
2026-02-04
The date when this record and its files were officially published on Zenodo.

References

  • Collatz, L. (1937). On the sequence defined by n → n/2 (n even), 3n+1 (n odd).
  • Lagarias, J. C. (2010). The 3x+1 problem: An annotated bibliography, 1963–1999. arXiv:math/0309224.
  • Lagarias, J. C. (2023). The 3x+1 Problem and Its Generalizations. Notices of the American Mathematical Society.