On an tangent equation by primes

TL;DR

This paper introduces a new prime-based Diophantine equation involving tangent and logarithmic functions, proving that large integers can be represented as sums of three prime-related terms within certain bounds.

Contribution

It establishes the solvability of a novel prime Diophantine equation involving tangent functions and provides an asymptotic count of solutions for large integers.

Findings

Every sufficiently large integer can be expressed as a sum of three prime-based tangent terms.

The paper proves the existence of solutions within specific parameter ranges.

An asymptotic formula for the number of such representations is derived.

Abstract

In this paper we introduce a new diophantine equation with prime numbers. Let $[\, \cdot\,]$ be the floor function. We prove that when $1<c<\frac{23}{21}$ and $\theta>1$ is a fixed, then every sufficiently large positive integer $N$ can be represented in the form \begin{equation*} N=\big[p^c_1\tan^\theta(\log p_1)\big]+ \big[p^c_2\tan^\theta(\log p_2)\big]+ \big[p^c_3\tan^\theta(\log p_3)\big]\,, \end{equation*} where $p_1,\, p_2,\, p_3$ are prime numbers. We also establish an asymptotic formula for the number of such representations.