關於k進表示法的一個問题

<正> §1.設k>1是一個固定的正整數,則每一個正整數x都可以唯一地表成 x=a_1k~n1+a_2k~n2+…+a_1k~nt,其中n_1>n_2>…>n_t≥0都是整數;a_1,…,a_t也都是正整數且≤k-1.我們令,並令.在k=2的情况,文1]的作者們證明了

A PROBLEM ON THE k-ADIC REPRESENTATION OF POSITIVE INTEGERS

Let k≥1 be a fixed integer, then any positive integer x can be uniquely represented by the following form x = a_1 k~(n1) + a_2 k~(n2) + … + a_1 k~(n1), where n_1> n_2 > … > n_t ≥ 0 are integers, and a_1, …, a_t are also positive integers not greater than k-1. Define a(x)Theorem 1. For any k≥2, we have Moreover, the result is the best possible.Let m be a fixed integer, then the equation a(y)=m has infinite many solutions. Let B_m(x) be the number of solutions not greater than x, we haveTheorem 2.