Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Ye, Xuda
Institution: Purdue University West Lafayette
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221181414