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

TL;DR

This paper investigates an equation involving fractional powers with prime and almost prime variables, establishing conditions under which solutions exist for large integers and providing bounds on the number of prime factors.

Contribution

It proves the existence of solutions to the equation with specific bounds on prime factors for the first time in this context.

Findings

Solutions exist for large N when 1 < c < 29/28.

The almost prime variable has at most [52/(29-28c)] + 1 prime factors.

The result extends understanding of equations involving fractional powers and 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.