On an additive problem involving fractional powers with one prime and an almost prime variables

Abstract

For any real number $t$, let $[t]$ denote the integer part of $t$. In this paper it is proved that if $1<c<\frac{247}{238}$, then for sufficiently large integer $N$, the equation \[\left[p^{c}\right]+\left[m^{c}\right]=N\] has a solution in a prime $p$ and an almost prime $m$ with at most $\left[\frac{450}{247-238c}\right]+1$ prime factors. This result constitutes an improvement upon that of Petrov and Tolev.