Classification of the Orbits of Three Quadratic Collatz-type Maps
Xuda Ye
math.NT
Oct 7, 2026 · v1
TL;DR
The classification theorems and Szalay's Diophantine lemma are verified in Lean 4 with Mathlib, taking Beukers' results as axioms; code is on GitHub.
Abstract
We study three quadratic Collatz-type maps on the non-negative integers. Each of them halves an even number, and they send an odd $n$ to $n(n-1)/2$, to $n(n+1)/2$ and to $(n^2 - 1)/4$, respectively. Along an orbit, the odd terms grow until they reach a number of the form $2^m + 1$ or $2^m - 1$, depending on the map, and an orbit with an odd term greater than $1$ is bounded exactly when this happens. The step that reaches such a number gives a solution of an exponential Diophantine equation, which becomes the equation $2^a \pm 2^b + 1 = z^2$ after completing the square. With Szalay's classification of the solutions of this equation, we classify all orbits of the maps. Every orbit either is eventually periodic or tends to infinity, and we determine exactly the initial values with bounded orbits. For the map with odd branch $n(n-1)/2$, Sedaghat stated the classification, and we give a new and complete proof. For this map, we also give an elementary criterion for unbounded orbits, based on divisibility alone. To our knowledge, the classifications for the other two maps are new.
Problem
Three quadratic Collatz-type maps on the non-negative integers halve even numbers and send odd n to n(n-1)/2, n(n+1)/2 or (n^2-1)/4. The question is which initial values have bounded orbits, where those orbits end, and whether all other orbits tend to infinity.
Approach
Odd terms along an orbit grow until they reach a terminal number of the form 2^m ± 1. Reaching one yields a solution of an exponential Diophantine equation, which becomes 2^a ± 2^b + 1 = z^2 after completing the square. Szalay's classification of that equation's solutions then gives the full classification. All results, including Szalay's lemma (with Beukers' results as axioms), are verified in Lean 4 with Mathlib, with code written with assistance from Claude.
Results
Every orbit is either eventually periodic or tends to infinity, and the initial values with bounded orbits are determined exactly for all three maps. The exceptional odd parts are none for Q_-, 5 for Q_+, and 11, 23 and 181 for S. The paper also gives a new complete proof of Sedaghat's classification for Q_- and an elementary divisibility criterion for unbounded Q_- orbits.
| map | odd branch | terminal numbers | exceptional odd parts |
|---|
| Q_- | n(n-1)/2 | 2^m+1 | none |
| Q_+ | n(n+1)/2 | 2^m-1 | 5 |
| S | (n^2-1)/4 | 2^m±1 | 11, 23, 181 |
Summary of the three maps
