Prime numbers of the form $\mathbf{[n^c tan^\theta(log n)]}$

TL;DR

This paper proves the existence of infinitely many prime numbers of a specific form involving powers and tangent functions, expanding understanding of prime distribution within complex mathematical expressions.

Contribution

It establishes the infinitude of primes of the form [n^c tan^θ(log n)] for certain c and θ, a novel result in prime number theory.

Findings

Infinitely many primes of the specified form exist for 1<c<12/11 and θ>1.

The result extends prime distribution knowledge to complex functions involving tangent.

Provides new insights into prime occurrence in non-linear, transcendental expressions.

Abstract

Let $[\, \cdot\,]$ be the floor function. In the present paper we prove that when $1<c<\frac{12}{11}$ and $\theta>1$ is a fixed, then there exist infinitely many prime numbers of the form $[n^c \tan^\theta(\log n)]$.