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 (x, n) 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 \(N(2^r,p)\le 1\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
F. Beukers, On the generalized Ramanujan–Nagell equation II. Acta Arith. 39(1), 113–123 (1981)
M.-H. Le, On the generalized Ramanujan–Nagell equation \(x^2-D=p^n\). Acta Arith. 58(3), 289–298 (1991)
M.-H. Le, On the number of solutions of diophantine equation \(x^2-D=p^n\). J. Math. 34(3), 378–387 (1991)
M.-H. Le, On the number of solutions of the generalized Ramanujan–Nagell equation \(x^2-D=p^n\). Publ. Math. Debr. 45(3–4), 239–254 (1994)
P.-Z. Yuan, On the number of the solutions of \(x^2-D=p^n\). J. Sichuan Univ. Nat. Sci. Ed. 35(3), 311–316 (1998)
M. Bauer, M.A. Bennett, Application of the hypergeometric method to the generalized Ramanujan–Nagell equation. Ramanujan J. 6(2), 209–270 (2002)
J.-M. Yang, The number of solutions of the generalized Ramanujan–Nagell equation \(x^2-D=3^n\). J. Math. 51(2), 351–356 (2008)
S. Siksek, On the Diophantine equation \(x^2=y^p+2^kz^p\). J. Thorie Nombres Bordx. 15, 839–846 (2003)
Y.-E. Zhao, T.-T. Wang, A note on the number of solutions of the generalized Ramanujan–Nagell equation \(x^2-D=p^n\). Czech Math. J. 62(2), 381–389 (2012)
L.-G. Hua, An Introduction to Number Theory (Science Publishing Company, Beijing, 1979)
A.J. Stephens, H.C. Williams, Some computational results on a problem concerning powerful numbers. Math. Comput. 50, 619–632 (1988)
P. Mihǎilescu, Primary cyclotomic units and a proof of Catalan’s conjecture. J. Reine Angew. Math. 572, 167–195 (2004)
M.A. Bennett, C.M. Skinner, Ternary diophantine equations via Galois representations and modular forms. Can. J. Math. 56(1), 23–54 (2004)
H. Cohen, Number Theory, vol. II (Springer, New York, 2007)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, T., Jiang, Y. On the number of positive integer solutions (x, n) of the generalized Ramanujan–Nagell equation \(x^2-2^r=p^n\) . Period Math Hung 75, 150–154 (2017). https://doi.org/10.1007/s10998-016-0173-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10998-016-0173-9