A Tangent Inequality Over Primes

Abstract

We introduce a new Diophantine inequality with prime numbers. Let \(1<c<\frac{10}{9}.\) We show that, for any fixed θ > 1, every sufficiently large positive number N, and a small constant ε > 0, the tangent inequality

$$\left|{p}_{1}^{c} {\mathrm{tan}}^{\theta }\left(\mathrm{log}{p}_{1}\right)+{p}_{2}^{c} {\mathrm{tan}}^{\theta }\left(\mathrm{log}{p}_{2}\right)+{p}_{3}^{c} {\mathrm{tan}}^{\theta }\left(\mathrm{log}{p}_{3}\right)-N\right|<\varepsilon $$

has a solution in prime numbers p₁, p₂, and p₃.