Prior Work

About the Author

I'm a software engineer, not an academic mathematician. These are self-published explorations — not peer-reviewed papers. They represent several years of off-and-on investigation into the Collatz conjecture, each building on the last. This site is where that work continues.

I share them here for transparency: so you can see how the ideas developed, and judge the work on its merits.

Writings

The Collatz Conjecture, Pythagorean Triples, and the Riemann Hypothesis

Unveiling a Novel Connection Through Dropping Times (2024)

The most recent of the three, and the source of the dropping-set language used throughout this site. It sorts the integers into dropping sets by dropping time (the classification by stopping time goes back to Terras, 1976), counts the odd steps in each (orbital oddity), and observes that the first few sets are arithmetic progressions. Neither of the two connections in its title is special to Collatz. Its Pythagorean triples come from putting a number $n$ and its dropping destination $d$ into Euclid's formula, $(n^2 - d^2,\; 2nd,\; n^2 + d^2)$, which gives a triple for any pair of integers $n > d > 0$. Its "critical line" is a real part of $\tfrac12$ that turns out to be the identity $\mathrm{Re}\,\frac{(n-d) + di}{n + ni} = \tfrac12$, true for any two numbers, so it shows no link to the Riemann hypothesis.

Read on viXra (2403.0077)

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The Geometric Collatz Correspondence (2023)

The earlier of the two papers. It sorts the integers into stopping classes by stopping time (the same sets as the dropping sets above, under an earlier name), gives each number a stopping signature, and plots each number as a point in the plane. By its own account its statements are conjectures without rigorous proofs, most of them tested on the first million integers. The Terminology Map lines its vocabulary up with the later paper's.

Read on viXra (2309.0109)

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The Collatz Conjecture: A New Perspective on an Old Problem (2022)

Proportional Power Ratios in Base 6

A more accessible exploration, published on Medium, and the earliest of the three. This is where I first plotted Collatz orbits against powers of 6: the Proportional Power Ratio $P(x, 6)$, the position of $x$ between consecutive powers of 6, turns an orbit into a strikingly regular pattern, with a 44-step near-repeat that I could not explain at the time. The explanation is that in base-6 logarithmic coordinates every Collatz step is close to a rotation by $\log_6 3$; Shakibaei Asli (2026) made this precise as an explicit near-conjugacy with a bounded error. The 44 in the article's polar plots is partly a plotting effect (the angle there is the step index in radians, and 44 radians is almost exactly 7 full turns) and partly arithmetic ($44 \log_6 3 \approx 26.98$, so the radius nearly repeats as well).

Read on Medium

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How They Connect to This Site

Writing	Key Idea	Where it shows up here
Paper 1 (Pythagorean Triples)	The dropping-set vocabulary	Foundations and Structure pages
Paper 2 (Geometric Correspondence)	Stopping classes, the same sets under an earlier name	Terminology Map
Medium article (PPR / Base-6)	Base-6 plots of orbits and their 44-step near-repeat	The Hidden Rotation chapter
This site (2026)	Countdown hierarchy, finite-fuel heuristic	The Tour and Connections pages

Key References by Others

• Shakibaei Asli, B. (2026). An explicit near-conjugacy between the Collatz map and a circle rotation. arXiv:2601.04289
• Terras, R. (1976). A stopping time problem on the positive integers. Acta Arithmetica, 30(3), 241-252.
• Everett, C. J. (1977). Iteration of the number-theoretic function f(2n) = n, f(2n+1) = 3n+2. Advances in Mathematics, 25(1), 42-45.
• Steiner, R. P. (1977). A theorem on the Syracuse problem. Proceedings of the 7th Manitoba Conference on Numerical Mathematics, 553-559.
• Böhm, C. and Sontacchi, G. (1978). On the existence of cycles of given length in integer sequences like x_{n+1} = x_n/2 if x_n even, and x_{n+1} = 3x_n + 1 otherwise. Atti della Accademia Nazionale dei Lincei, 64, 260-264.
• Lagarias, J. C. (1985). The 3x+1 problem and its generalizations. American Mathematical Monthly, 92(1), 3-23.
• Eliahou, S. (1993). The 3x+1 problem: new lower bounds on nontrivial cycle lengths. Discrete Mathematics, 118, 45-56.
• Simons, J. and de Weger, B. (2005). Theoretical and computational bounds for m-cycles of the 3n+1 problem. Acta Arithmetica, 117, 51-70.
• Hercher, C. (2023). There are no Collatz m-cycles with m ≤ 91. Journal of Integer Sequences, 26, Article 23.3.5.
• Rozier, O. and Terracol, C. (2026). Paradoxical behavior in Collatz sequences. Discrete Mathematics, 349, 115167.
• Winkler, M. (2026). Admissible qx+1 sequences, semiconvergents, and rational Catalan numbers. arXiv:2609.22303
• OEIS A060445 (dropping times of the odd numbers), A100982 (counts of admissible parity words) and A122437 (allowable dropping times).
• Tao, T. (2019). Almost all orbits of the Collatz map attain almost bounded values. arXiv:1909.03562; published in Forum of Mathematics, Pi, 10 (2022), e12.