On an tangent equation by primes

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.