Glasnik Matematicki, Vol. 48, No. 2 (2013), 291-299.

ON THE DIOPHANTINE INEQUALITY |X2-cXY2+Y4| ≤ c+2

Bo He, István Pink, Ákos Pintér and Alain Togbé

Department of Mathematics, ABa Teacher's College, Wenchuan, Sichuan 623000, P. R. China
e-mail: bhe@live.cn

Institute of Mathematics, P. O. Box 12, H-4010 Debrecen, Hungary
e-mail: pinki@science.unideb.hu

Institute of Mathematics, MTA-DE Research Group, "Equations, Functions and Curves", Hungarian Academy of Sciences and University of Debrecen, P. O. Box 12, H-4010 Debrecen, Hungary
e-mail: apinter@science.unideb.hu

Department of Mathematics, Purdue University North Central, 1401 S. U.S. 421, Westville, IN 46391, USA
e-mail: atogbe@pnc.edu

Abstract.   Generalizing some earlier results, we find all the coprime integer solutions of the Diophantine inequality

|X2-cXY2+Y4| ≤ c+2,   (X,Y)=1,

except when c ≡ 2 (mod 4), in which case we bound the number of integer solutions. Our work is based on the results on the Diophantine equation AX4-BY2=C, where A, B are positive integers and C ±{1, 2, 4}.

2010 Mathematics Subject Classification.   11D25, 11J86.

Key words and phrases.   Diophantine equations, quartic equations.

Full text (PDF) (free access)

DOI: 10.3336/gm.48.2.05

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