ON THE DIOPHANTINE EQUATION x2 + C = 2yn-Reference-Cited by-同舟云学术

ON THE DIOPHANTINE EQUATION x2 + C = 2yn

Published:2009-09 Issue:06 Volume:05 Page:1117-1128
ISSN:1793-0421
Container-title:International Journal of Number Theory
language:en
Short-container-title:Int. J. Number Theory

Author:

ABU MURIEFAH FADWA S.¹,LUCA FLORIAN²,SIKSEK SAMIR³,TENGELY SZABOLCS⁴

Affiliation:

1. Riyadh University for Women, Science Section (Mathematics), P. O. Box 60561, Riyadh 11555, Saudi Arabia

2. Instituto de Matemáticas, Universidad Nacional Autónoma de México, C. P. 58089, Morelia, Michoacán, Mexico

3. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

4. Mathematical Institute, University of Debrecen and the Number Theory Research Group of the Hungarian Academy of Sciences, P. O. Box 12, 4010 Debrecen, Hungary

Abstract

In this paper, we study the Diophantine equation x2 + C = 2yn 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 x2 + 17a1 = 2yn, x2 + 5a113a2 = 2yn and x2 + 3a111a2 = 2yn.

Publisher

World Scientific Pub Co Pte Lt

Subject

Algebra and Number Theory

Link

https://www.worldscientific.com/doi/pdf/10.1142/S1793042109002572

Reference27 articles.

1. On the Diophantine Equation x2+q2k+1=yn

2. Ternary Diophantine Equations via Galois Representations and Modular Forms

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