计算Ulam映射高阶关联函数的数论方法

A number theoretic method for high order correlation functions of the Ulam map

为计算Ulam映射的高阶关联函数,本文提出了一种数论方法.该方法先将关联函数的计算转化为一类变系数指数型丢番图方程的求解问题,然后将该方程约化为具有严格单调指数的丢番图方程,最后以降阶法求得方程的解.作为应用,本文计算了Ulam映射的前5阶关联函数.

In this paper, a number theoretic method is introduced to calculate the high order correlation functions of the Ulam map. In this method, the calculation is firstly transformed into solving a class of exponential Diophantine equations with variable coefficients. Then these equations are simplified to the Diophantine equations with strictly monotonic exponentials. Finally, the equations are solved by means of a order reduction method.As an application, the first five order correlation functions of the Ulam map are calculated.