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On the diophantine equation x(2) + C=2y(n)

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Abu Muriefah, Fadwa S., Luca, Florian, Siksek, Samir and Tengely, Szabolcs (2009) On the diophantine equation x(2) + C=2y(n). International Journal of Number Theory, Vol.5 (No.6). pp. 1117-1128. doi:10.1142/S1793042109002572 ISSN 1793-0421.

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Official URL: http://dx.doi.org/10.1142/S1793042109002572

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Abstract

In this paper, we study the Diophantine equation x(2) + C = 2y(n) in positive integers x, y with gcd(x, y) = 1, where n >= 3 and C is a positive integer. If C = 1 (mod 4), we give a very sharp bound for prime values of the exponent n; our main tool here is the result on existence of primitive divisors in Lehmer sequences due to Bilu, Hanrot and Voutier. We illustrate our approach by solving completely the equations x(2) + 17(a1) = 2y(n), x(2) + 5(a1)13(a2) = 2y(n) and x(2) + 3(a1)11(a2) = 2y(n).

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Journal or Publication Title: International Journal of Number Theory
Publisher: World Scientific Publishing Co. Pte. Ltd.
ISSN: 1793-0421
Official Date: September 2009
Dates:
DateEvent
September 2009Published
Volume: Vol.5
Number: No.6
Number of Pages: 12
Page Range: pp. 1117-1128
DOI: 10.1142/S1793042109002572
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
Funder: CONACyT, Engineering and Physical Sciences Research Council (EPSRC), MarieCurie International Reintegration
Grant number: 46755, MIRG-CT-2006-044530

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