On some Diophantine equations over complex quadratic number fields

MalineeÿChaiya^a, SomjateÿChaiya^a, SupawadeeÿPrugsapitak^b,*

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 x²−axy+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	DepartmentÿofÿMathematics, FacultyÿofÿScience, SilpakornÿUniversity, NakhonÿPathomÿ73000ÿThailand
^b	AlgebraÿandÿApplicationsÿResearchÿUnit, DepartmentÿofÿMathematicsÿandÿStatistics, FacultyÿofÿScience, PrinceÿofÿSongklaÿUniversity, Songkhlaÿ90110ÿThailand

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

Received 15 Jan 2017, Accepted 31 Oct 2017

On some Diophantine equations over complex quadratic number fields

MalineeÿChaiyaa, SomjateÿChaiyaa, SupawadeeÿPrugsapitakb,*

MalineeÿChaiya^a, SomjateÿChaiya^a, SupawadeeÿPrugsapitak^b,*