Results of Diophantine approximation by unlike powers of primes

Abstract

Let k be an integer with k\ge \mathrm{6}: Suppose that {{\displaystyle \lambda }}_{\mathrm{1}},{{\displaystyle \lambda }}_{\mathrm{2}},\mathrm{...},{{\displaystyle \lambda }}_{\mathrm{5}} be nonzero real numbers not all of the same sign, satisfying that {{\displaystyle \lambda }}_{\mathrm{1}}/{{\displaystyle \lambda }}_{\mathrm{2}} is irrational, and suppose that \eta is a real number. In this paper, for any \epsilon \rangle \mathrm{0}; we consider the inequality \left|{{\displaystyle \lambda }}_{\mathrm{1}}{{\displaystyle p}}_{\mathrm{1}}+{{\displaystyle \lambda }}_{\mathrm{2}}{{\displaystyle p}}_{\mathrm{2}}^{\mathrm{2}}+{{\displaystyle \lambda }}_{\mathrm{3}}{{\displaystyle p}}_{\mathrm{3}}^{\mathrm{3}}+{{\displaystyle \lambda }}_{\mathrm{4}}{{\displaystyle p}}_{\mathrm{4}}^{\mathrm{4}}+{{\displaystyle \lambda }}_{\mathrm{5}}{{\displaystyle p}}_{\mathrm{5}}^{\mathrm{5}}+\eta \right|\langle {({{\displaystyle \max p}}_j)}^{-\sigma (k)+\epsilon } has innitely many solutions in prime variables

p_{\mathrm{1}},p_{\mathrm{2}},\mathrm{...},p_{\mathrm{5}},

where \sigma (k) depends on k: Our result gives an improvement of the recent result. Furthermore, using the similar method in this paper, we can rene some results on Diophantine approximation by unlike powers of primes, and get the related problem.

Key words： Waring-Goldbach problem; Diophantine inequality

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