论文标题
Collatz多项式：介绍其零上的界限
Collatz polynomials: an introduction with bounds on their zeros
论文作者
论文摘要
Collatz的猜想（也称为3x+1问题）提出，以下算法将在一定数量的迭代后始终产生数字1：给定数字1：乘以三个，如果数字为奇数，则添加一个，如果数字为奇数，将结果数量减半，然后重复。在本文中，对于Collatz猜想所规定的每个$ n $，我们将$ n^{th} $ collatz多项式定义为具有恒定项$ n $和$ k^{th} $ term（对于$ k> 1 $）$ k^{th} $ n $ n $ ncullatz $ n $ n $ N $ ncullatz $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ ncullatz $ n collatz $ ncullatz $ ncullatz $。特别是，我们绑定了这些多项式根源的模量，证明了它们何时具有理性整数根，并提出了进一步的研究和研究途径。
The Collatz Conjecture (also known as the 3x+1 Problem) proposes that the following algorithm will, after a certain number of iterations, always yield the number 1: given a natural number, multiply by three and add one if the number is odd, halve the resulting number, then repeat. In this article, for each $N$ for which the Collatz Conjecture holds we define the $N^{th}$ Collatz polynomial to be the monic polynomial with constant term $N$ and $k^{th}$ term (for $k > 1$) the $k^{th}$ iterate of $N$ under the Collatz function. In particular, we bound the moduli of the roots of these polynomials, prove theorems on when they have rational integer roots, and suggest further applications and avenues of research.
