# Diophantine approximation by special primes

## TL;DR

This paper proves the existence of infinitely many prime triples with specific linear relations and bounded prime factors in their shifted forms, advancing understanding of Diophantine approximation with primes.

## Contribution

It establishes new results on prime triples approximating linear forms with constraints on the number of prime factors of shifted primes.

## Key findings

- Infinitely many prime triples satisfy the inequality with a specific bound.
- Each prime in the triple has at most 28 prime factors when increased by 2.
- The results depend on certain necessary conditions on the coefficients.

## Abstract

We show that whenever $\delta>0$, $\eta$ is real and constants $\lambda_i$ satisfy some necessary conditions, there are infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying the inequality $|\lambda_1p_1 + \lambda_2p_2 + \lambda_3p_3+\eta|<(\max p_j)^{-1/12+\delta}$ and such that, for each $i\in\{1,2,3\}$, $p_i+2$ has at most $28$ prime factors.