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 $\left[\frac{52}{29-28c}\right]+1$ prime factors.