A quaternary diophantine inequality by prime numbers of a special type

TL;DR

This paper proves that for certain exponents, the inequality involving four prime powers can be approximately solved with primes where each prime plus two has limited prime factors.

Contribution

It establishes the existence of solutions to a quaternary Diophantine inequality with primes of a special type, extending previous results to a new range of exponents.

Findings

Solutions exist for large N within the specified inequality

Primes involved have at most 32 prime factors when increased by 2

The result applies for 1 < c < 832/825

Abstract

Let $1<c<832/825$. For large real numbers $N>0$ and a small constant $\vartheta>0$, the inequality \begin{equation*} |p_1^c+p_2^c+p_3^c+p_4^c-N|<\vartheta \end{equation*} has a solution in prime numbers $p_1,\,p_2,\,p_3,\,p_4$ such that, for each $i\in\{1,2,3,4\}$, $p_i+2$ has at most $32$ prime factors.