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 ...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 ε 〉 0, we consider the inequality ｜λ1p1 ＋ λ2p2^2 ＋λ3p3^3＋λ4p4^4 ＋ λ5p5^k ＋ η｜ 〈 （maxpj）^-σ（k）＋ε 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.展开更多
基金The authors would like to express their thanks to the referees for many useful suggestions and comments on the manuscript. This work was supported by the National Natural Science Foundation of China （Grant No. 11301372） and the Specialized Research Fund for the Doctoral Program of Higher Education （No. 20130032120073）.
文摘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 ε 〉 0, we consider the inequality ｜λ1p1 ＋ λ2p2^2 ＋λ3p3^3＋λ4p4^4 ＋ λ5p5^k ＋ η｜ 〈 （maxpj）^-σ（k）＋ε 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.
