Let p denote a prime number. P. Samuel recently solved the problem of determining all squares in the linear recurrence sequence {Tₙ}, where Tₙ and Uₙ satisfy Tₙ² - pUₙ² = 1. Samuel left open the problem of determining all squares in the sequence {Uₙ}. This problem was recently solved by the authors. In the present paper, we extend our previous joint work by completely solving the equation Uₙ = bx², where b is a fixed positive squarefree integer. This result also extends previous work of the second author.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm105-1-6,
author = {D. Poulakis and P. G. Walsh},
title = {On the Diophantine equation x2 - dy4 = 1 with prime discriminant II},
journal = {Colloquium Mathematicae},
volume = {106},
year = {2006},
pages = {51-55},
zbl = {1155.11316},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm105-1-6}
}
D. Poulakis; P. G. Walsh. On the Diophantine equation x² - dy⁴ = 1 with prime discriminant II. Colloquium Mathematicae, Tome 106 (2006) pp. 51-55. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm105-1-6/