A note on the Diophantine equation $$\varvec{x^2=4p^n-4p^m+\ell ^2}$$ x 2 = 4 p n - 4 p m + ℓ 2

Fadwa S. Abu Muriefah, Maohua Le () and Gökhan Soydan ()
Additional contact information
Fadwa S. Abu Muriefah: Princess Nourah Bint Abdulrahman University
Maohua Le: Institute of Mathematics, Lingnan Normal College
Gökhan Soydan: Bursa Uludağ University

Indian Journal of Pure and Applied Mathematics, 2022, vol. 53, issue 4, 915-922

Abstract: Abstract Let $$\ell $$ ℓ be a fixed odd positive integer. In this paper, using some classical results on the generalized Ramanujan-Nagell equation, we completely derive all solutions (p, x, m, n) of the equation $$x^2=4p^n-4p^m+\ell ^2$$ x 2 = 4 p n - 4 p m + ℓ 2 with $$\ell ^2 1$$ ℓ > 1 , where p is a prime, x, m, n are positive integers satisfying $$\gcd (x,\ell )=1$$ gcd ( x , ℓ ) = 1 and $$m 4p^m$$ l 2 > 4 p m . As an example of using this method, we find all solutions (p, x, m, n) of the equation for $$\ell \in \{5,7\}$$ ℓ ∈ { 5 , 7 } .

Keywords: Polynomial-exponential Diophantine equation; Generalized Ramanujan-Nagell equation; Baker’s method; 11D61 (search for similar items in EconPapers)
Date: 2022
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s13226-021-00197-3 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:indpam:v:53:y:2022:i:4:d:10.1007_s13226-021-00197-3

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/13226

DOI: 10.1007/s13226-021-00197-3

Access Statistics for this article

Indian Journal of Pure and Applied Mathematics is currently edited by Nidhi Chandhoke

More articles in Indian Journal of Pure and Applied Mathematics from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().