3N+1猜想的投资学模型

Investment models for the 3N+1 conjecture

3N+1猜想是有着70多年历史的数学问题,已入选"10000个科学难题"(数学卷).3N+1猜想的描述非常简单:对任意自然数n,若n为偶数,则除以2;若n为奇数,则乘3加1,经反复迭代最终总得到1.将给定自然数n看作第0期的初始资本,而将第t次迭代的结果看作第t期末的资本,从而建立3N+1函数迭代过程中函数值变化的投资模型.从投资学的角度看,3N+1猜想的本质在于3N+1函数迭代过程中长期资本是否相对于初始资本n衰减,而其根本困难在于确定3N+1函数迭代过程中取值为奇数的概率.在研究3N+1函数迭代过程的动态行为的基础上,进一步运用投资模型证明3N+1猜想成立的必要条件为3N+1函数迭代轨迹中奇数的概率小于ln2/ln3,此条件在一定程度上也是充分的.该投资模型也适用于3N+1猜想的各种推广.

The 3N＋l conjecture is a long-standing open problem in number theory concerning iteration of the 3N＋l function, which combines simplicity of statement with apparent intractability. The central difficulty of the 3N＋I conjecture lies in understanding in detail the dynamic properties of iterates of the 3N＋ 1 function. By viewing the given number n as the initial capital at time 0 and viewing the tth 3N＋I function iteration ofn as capital at time t, this paper establishes investment models to determine the asymptotic behavior of the 3N＋I function iteration. Conditions for the 3N＋l conjecture to be true are given. The investment model applies to generalizations of the 3N＋l conjecture.