三个二次Collatz型映射的轨道分类
Classification of the Orbits of Three Quadratic Collatz-type Maps
- Purdue University（普渡大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文对非负整数上的三个二次Collatz型映射的轨道进行分类，结合Szalay的丢番图方程解分类确定有界轨道初始值，补充了Sedaghat相关分类的新证明及另一映射的无界轨道判据，另外两个映射的分类为新结果。
AI中文摘要：
我们研究非负整数上的三个二次Collatz型映射。每个映射均将偶数折半，且分别将奇数n映射为n(n-1)/2、n(n+1)/2和(n²-1)/4。沿轨道，奇数项会增长至形如2ᵐ+1或2ᵐ-1的数（取决于映射类型），当奇数项大于1时，轨道有界当且仅当出现该类数。到达此类数的一步给出指数丢番图方程的解，配方后该方程变为2ᵃ±2ᵇ+1=z²。结合Szalay对该方程解的分类，我们对这三个映射的所有轨道进行分类。每个轨道要么最终周期，要么趋于无穷，我们精确确定了有界轨道的初始值。对于奇数分支为n(n-1)/2的映射，Sedaghat已给出分类，我们提供了新的完整证明；针对该映射，我们还仅基于整除性给出了无界轨道的初等判据。据我们所知，另外两个映射的分类是全新的。
英文摘要：
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.
