A tangent inequality over primes

S. I.  Dimitrov

doi:10.37863/umzh.v75i7.7184

S. I. Dimitrov Faculty of Applied Mathematics and Informatics, Technical University of Sofia, Bulgaria

DOI: https://doi.org/10.37863/umzh.v75i7.7184

Keywords: Diophantine inequality, Tangent inequality, Prime numbers

Abstract

UDC 511

We introduce a new Diophantine inequality with prime numbers. Let $1<c<\dfrac{10}{9}.$ 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 $p_1,$ $p_2,$ and $p_3.$

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