On a binary additive problem involving fractional powers

We show that, for any given c is an element of (1, 11/10), every sufficiently large integer n can be represented as n = [m(c)] + [p(c)], where m is a positive integer and p is a prime, and [t] is the integer part of the real number t. We also prove that, when c is an element of (1, 1+root 5/2), such representation exists for almost all positive integers n. These respectively improve the results of A. Kumchev [9], and Balanzario, Garaev, and Zuazua [1]. (C) 2019 Elsevier Inc. All rights reserved.