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Results of Diophantine approximation by unlike powers of primes

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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 ε > 0; we consider the inequality |λ1p1 + λ2p 22 + λ3p 33 + λ4p 44 + λ5p k5 + η | < (max pj)-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.

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Acknowledgements

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).

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Authors and Affiliations

  1. School of Mathematics, Tianjin University, Tianjin, 300350, China

    Gaiyun Gao & Zhixin Liu

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  1. Gaiyun Gao
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  2. Zhixin Liu
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Correspondence to Zhixin Liu.

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Gao, G., Liu, Z. Results of Diophantine approximation by unlike powers of primes. Front. Math. China 13, 797–808 (2018). https://doi.org/10.1007/s11464-018-0713-0

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  • DOI: https://doi.org/10.1007/s11464-018-0713-0

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