On a diophantine inequality with prime numbers of a special type

TL;DR

This paper proves the existence of prime solutions to a specific Diophantine inequality involving prime numbers of a special form, where each prime plus two has a limited number of prime factors, for a range of c values.

Contribution

It establishes the solvability of a Diophantine inequality with primes of a restricted form, extending previous results to a new class of primes with bounded prime factors.

Findings

Solutions exist for the inequality with primes where p+2 has limited prime factors.

The result holds for c in the interval (1, 15/14).

The proof introduces new methods for handling primes with restricted prime factors.

Abstract

We consider the Diophantine inequality \[ \left| p_1^{c} + p_2^{c} + p_3^c- N \right| < (\log N)^{-E} , \] where $1 < c < \frac{15}{14}$, $N$ is a sufficiently large real number and $E>0$ is an arbitrarily large constant. We prove that the above inequality has a solution in primes $p_1$, $p_2$, $p_3$ such that each of the numbers $p_1 + 2, p_2 + 2, p_3 + 2$ has at most $\left[ \frac{369}{180 - 168 c} \right]$ prime factors, counted with the multiplicity.