Results of Diophantine approximation by unlike powers of primes

Abstract

Let k be an integer with k ≥ 6: Suppose that λ1, λ2,..., λ5 be nonzero real numbers not all of the same sign, satisfying that λ1/λ2 is irrational, and suppose that η is a real number. In this paper, for any e > 0; we consider the inequality |λ1p1 + λ2p 2 2 + λ3p 3 3 + λ4p 4 4 + λ5p 5 k + η | < (max pj)-σ(k)+e has infinitely many solutions in prime variables p1, p2,...,p5, where σ(k) depends on k. Our result gives an improvement of the recent result. Furthermore, using the similar method in this paper, we can refine some results on Diophantine approximation by unlike powers of primes, and get the related problem.