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Research articles

ScienceAsia 43(2017): 383-386 |doi: 10.2306/scienceasia1513-1874.2017.43.383

On some Diophantine equations over complex quadratic number fields

Malinee Chaiyaa, Somjate Chaiyaa, Supawadee Prugsapitakb,*

ABSTRACT:     In this paper, we establish a necessary and sufficient condition for two algebraic integers in complex quadratic number fields to be consecutive terms of generalized Fibonacci numbers. We use this result to obtain all solutions of the Diophantine equation x2axy+by2=c over Gaussian integers, where b and c are units in ℤ[i] and a∈ℤ[i] with |a|2≥10.

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a Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom 73000 Thailand
b Algebra and Applications Research Unit, Department of Mathematics and Statistics, Faculty of Science, Prince of Songkla University, Songkhla 90110 Thailand

* Corresponding author, E-mail: supawadee.p@psu.ac.th

Received 15 Jan 2017, Accepted 31 Oct 2017