Volume 7 Number 2 (Apr. 2017)
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IJAPM 2017 Vol.7(2): 112-117 ISSN: 2010-362X
doi: 10.17706/ijapm.2017.7.2.112-117

On the Generalized LEBESGUE-Ramanujan-Nagell Equation

Ran Yinxia
Abstract—Let p is a prime, we studied the the generalized Lebesgue-Ramanujan-Nagell equation. By using the elementary method and algebraic number theory, we obtain one necessary condition which the equation has integer solutions and some sufficient conditions which the equation has no integer solution. 1). Let x be an odd number, one necessary condition which the equation has integer solutions is that 2n(p-1)-1/p contains some square factors. 2). Let x be an even number, when n=pk(k≥1), all integer solutions for the equation are (x, y)=(0,4k); when n=pk+(p-1)/2(k≥0), all integer solutions are (±2pk+(p-1)/2,22k+1); when n≡1,2,3,…,(p-3)/2, (p+1)/2,…,p-1(modp), the equation has no integer solution.

Index Terms—Exponential Diophantine equation, integer solutions, integer ring, algebraic number theory.

Ran Yinxia is with Longnan Teachers College, Department of Mathematics, Gansu Longnan, China (email: ranyinxia@163.com).

Cite: Ran Yinxia, "On the Generalized LEBESGUE-Ramanujan-Nagell Equation," International Journal of Applied Physics and Mathematics vol. 7, no. 2, pp. 112-117, 2017.

General Information

ISSN: 2010-362X
Frequency: Bimonthly (2011-2014); Quarterly (Since 2015)
DOI: 10.17706/IJAPM
Editor-in-Chief: Prof. Haydar Akca
Abstracting/ Indexing: Index Copernicus, EI (INSPEC, IET), Chemical Abstracts Services (CAS), Electronic Journals Library, Nanowerk Database, Google Scholar, EBSCO, and ProQuest
E-mail: ijapm@iap.org
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