Abstract
Let \(p\) be an odd prime. In this paper we study the integer solutions \((x,y,n,a,b)\) of the equation \(x^{2}+2^{a}p^{b}=y^{n}, x\ge 1,y>1,\gcd (x,y)=1,a\ge 0,b\ge 0,n\ge 3\).
Similar content being viewed by others
References
S.A. Arif, F.S. Abu Muriefah, on the Diophantine equation \(x^{2}+q^{2k+1}=y^{n}\). J. Number Theory 95, 95–100 (2002)
M.A. Bennett, C.M. Skinner, Ternary Diophantine equation via Galois representations and modular forms. Can. J. Math. 56, 23–54 (2004)
A. Bérczes, I. Pink, On the Diophantine equation \(x^{2}+p^{2k}=y^{n}\). Arch. Math. 91, 505–517 (2008)
Y. Bilu, G. Hanrot, P.M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers (with Appendix by Mignotte). J. Reine Angew. Math. 539, 75–122 (2001)
I.N. Cangül, M. Demirci, F. Luca, À. Pintér, G. Soydan, On the Diophantine equation \(x^2 + 2^{a} 11^{b} =y^{n}\). Fibonacci Quart. 48(1), 39–46 (2010)
R.D. Carmichael, On the numerical factors of the arithmetic forms \(\alpha ^n-\beta ^n\). Ann. Math. (2) 15, 30–70 (1913)
J.H.E. Cohn, Cohn, the Diophantine equation \(x^2+2^k=y^n\). Arch. Math. Basel 59(4), 341–344 (1992)
A. Dabrowski, On the Lebesgue-Nagell equation. Colloq. Math. 125(2), 245–253 (2011)
M. Le, Some exponential Diophantine equations I: the equation \(D_{1}x^{2}-Dy^{2}=\lambda k^z\). J. Number Theory 55(2), 209–221 (1995)
M. Le, On Cohn’s conjecture concerning the Diophantine equation \(x^{2}+2^{m}=y^{n}\). Arch. Math. (Basel) 78(1), 26–35 (2002)
L.A. Lebesgue, Sur l’impossibilité, en nombres entiers, de l’équation \(x^{m}=y^{2}+1\). Nouv. Ann. Math. (1) 9, 178–181 (1850)
W. Ljunggren, Einige Sätze über unbestimmte Gleichungen von der Form \(Ax^4+Bx^2+C=Dy^2\). Det Norske Vid. Akad. Skr 9, 53 pp (1942)
W. Ljunggren, Über die Gleichungen \(1+Dx^2=2y^n\) und \(1+Dx^2=4y^n\). Norsk Vid. Selsk. Forh. 15(30) 115–118 (1943)
W. Ljunggren, W. Ljunggren, Ein Satz über die Diophantische Gleichung \(Ax^2-By^4=C(C=1,2,4)\). Tolfte Skand. Mat. Lund. 10(2)188–194 (1954)
F. Luca, On the equation \(x^2 + 2^{a} 3^{b} =y^{n}\). Int. J. Math. Math. Sci. 29(3), 239–244 (2002)
F. Luca, A. Togbé, On the equation \(x^2 + 2^{a} 5^{b} =y^{n}\). Int. J. Number Theory 4(6), 973–979 (2008)
F. Luca, A. Togbé, On the equation \(x^2 + 2^{\alpha } 13^{\beta } =y^{n}\). Colloq. Math. 116(1), 139–146 (2009)
P. Mihǎilescu, Primary cyclotomic units and a proof of Catalan’s conjecture. J. Reine Angew. Math. 572, 167–195 (2004)
T. Nagell, Sur l’impossibilité de quelques équations à deux indéterminés. Norske Mat. Forenings Skr. Sér. I 13(1), 65–82 (1923)
T. Nagell, Über die rationaler Punkte auf einigen kubischen Kurven. Tǒhoku Math. J. 24(1), 48–53 (1924)
G. Soydan, M. Ulas, H. Zhu, On the Diophantine equation \(x^2+2^{a}19^{b}=y^n\). Indian J. Pure Appl. Math. 43(3), 251–261 (2012)
P.M. Voutier, Primitive divisors of Lucas and Lehmer sequences. Math. Comput. 64, 869–888 (1995)
D.T. Walker, On the Diophantine equation \(mx^2-ny^2=\pm 1\). Am. Math. Mon. 74(5), 504–513 (1967)
H. Zhu, A note on the Diophantine equation \(x^{2}+q^{m}=y^{3}\). Acta Arith. 146(2), 195–202 (2011)
X. Pan, The exponential Lebesgue-Nagell equation \(x^2+p^{2m}=y^n\). Period. Math. Hung. 67(2), 231–242 (2013)
Acknowledgments
The authors would like to thank the referee for the suggestions to improve this paper. The first author was partly supported by the Fundamental Research Funds for the Central Universities (No. 2012121004) and the Science Fund of Fujian Province (No. 2012J050009, 2013J05019). The second author was supported by NSFC (No. 10971184). The third author was supported by the research fund of Uludağ University Project (No. F-2013/87). The work on this paper was completed during a very enjoyable visit of the fourth author at The Institute of Mathematics of Debrecen. He thanks this institution and Professor Ákos Pintér for the hospitality. He was also supported in part by Purdue University North Central.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhu, H., Le, M., Soydan, G. et al. On the exponential Diophantine equation \(x^{2}+2^{a}p^{b}=y^{n}\) . Period Math Hung 70, 233–247 (2015). https://doi.org/10.1007/s10998-014-0073-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10998-014-0073-9