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Periodica Mathematica Hungarica

, Volume 75, Issue 2, pp 150–154 | Cite as

On the number of positive integer solutions (xn) of the generalized Ramanujan–Nagell equation \(x^2-2^r=p^n\)

  • Tingting WangEmail author
  • Yingzhao Jiang
Article
  • 186 Downloads

Abstract

Let p be a fixed odd prime, and let r be a fixed positive integer. Further let \(N(2^r,p)\) denote the number of positive integer solutions (xn) of the generalized Ramanujan–Nagell equation \(x^2-2^r=p^n\). In this paper, we use the elementary method and properties of Pell’s equation to give a sharp upper bound estimate for \(N(2^r,p)\). That is, we prove that \(N(2^r,p)\le 1\).

Keywords

Generalized Ramanujan–Nagell equation Number of solutions Upper bound estimate 

Mathematics Subject Classification

11D61 

Notes

Acknowledgements

The authors express their gratitude to the referee for his very helpful and detailed comments, which have significantly improved the presentation of this paper. This work is supported by the N. S. F. (11501452) and the P. S. F. (2014JQ2-1005) of P. R. China.

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2016

Authors and Affiliations

  1. 1.College of ScienceNorthwest A&F UniversityYanglingPeople’s Republic of China

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