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On a Diophantine equation involving primes

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Abstract

Let \([\theta ]\) denote the integral part of the real number \(\theta \). In this paper it is proved that for \(1<c< \frac{137}{119}\), the Diophantine equation \( [p^c_1] + [p^c_2] + [p^c_3] = N\) is solvable in prime variables \(p_1, p_2, p_3\) for sufficiently large integer N. The range \(1<c< \frac{137}{119}\) constitutes an extension of \(1< c < \frac{258}{235}\) due to Zhai and Cao.

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Acknowledgements

The author would like to thank the anonymous referee for his/her patience and time in refereeing this paper.

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Authors and Affiliations

  1. School of Mathematical Science, Tongji University, Shanghai, 200092, People’s Republic of China

    Yingchun Cai

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  1. Yingchun Cai
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Correspondence to Yingchun Cai.

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Project supported by The National Science Foundation of China (Grant No. 11771333).

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Cai, Y. On a Diophantine equation involving primes. Ramanujan J 50, 151–162 (2019). https://doi.org/10.1007/s11139-018-0027-6

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  • DOI: https://doi.org/10.1007/s11139-018-0027-6

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