On an equation involving fractional powers with one prime and one almost prime variables

Abstract

In this paper we consider the equation $[p^{c}] + [m^{c}] = N$, where $N$ is a sufficiently large integer, and prove that if $1 < c < \frac{29}{28}$, then it has a solution in a prime $p$ and an almost prime $m$ with at most $\left[ \frac{52}{29 - 28 c}\right] + 1 $ prime factors.