Bugeaud, Y. ; Shorey, T. N.
(2001)
*On the number of solutions of the generalized Ramanujan-Nagell equation*
Journal fur die Reine und Angewandte Mathematik (Crelle's Journal), 2001
(539).
pp. 55-74.
ISSN 0075-4102

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Official URL: http://www.reference-global.com/doi/abs/10.1515/cr...

Related URL: http://dx.doi.org/10.1515/crll.2001.079, 20/09/2001

## Abstract

Let D_{1} and D_{2} be coprime positive integers and let k be an odd positive integer coprime with D_{1}D_{2}. We consider the Diophantine equation D_{1}x^{2} + D_{2} = k^{n} in the unknowns x≥1, n≥1. We give a necessary and sufficient condition on D_{1}, D_{2} and k under which this equation has at most 2^{ω(k)−1} solutions where ω(k) denoted the number of distinct prime divisors of k. Thus, under a necessary and sufficient conditon, the equation has at most one solution whenever k is a prime. We also consider some related equations and we prove that the Diophantine equation x^{2}+7=4y^{n} has no solution in integers x≥1, y > 2 and n >1.

Item Type: | Article |
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Source: | Copyright of this article belongs to Walter de Gruyter GmbH & Co. KG. |

ID Code: | 67730 |

Deposited On: | 31 Oct 2011 13:26 |

Last Modified: | 31 Oct 2011 13:26 |

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