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.