A tangent inequality over primes

TL;DR

This paper proves that for large numbers, the tangent inequality involving prime powers and tangent functions has solutions in primes, extending understanding of prime-related diophantine inequalities.

Contribution

It introduces a new diophantine inequality involving primes and tangent functions, establishing the existence of solutions under certain conditions.

Findings

Solutions exist for sufficiently large N

The inequality holds for 1<c<10/9

Results apply to primes with a tangent function component

Abstract

In this paper we introduce a new diophantine inequality with prime numbers. Let $1<c<\frac{10}{9}$. We show that for any fixed $\theta>1$, every sufficiently large positive number $N$ and a small constant $\varepsilon>0$, the tangent inequality \begin{equation*} \big|p^c_1\tan^\theta(\log p_1)+ p^c_2\tan^\theta(\log p_2)+ p^c_3\tan^\theta(\log p_3) -N\big|<\varepsilon \end{equation*} has a solution in prime numbers $p_1,\,p_2,\,p_3$.