Glasnik Matematicki, Vol. 49, No. 1 (2014), 25-36.

THE EXTENDIBILITY OF DIOPHANTINE PAIRS I: THE GENERAL CASE

Alan Filipin, Yasutsugu Fujita and Alain Togbé

Faculty of Civil Engineering, University of Zagreb, Fra Andrije Kačića-Miošića 26, 10000 Zagreb, Croatia
e-mail: filipin@master.grad.hr

Department of Mathematics, College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba, Japan
e-mail: fujita.yasutsugu@nihon-u.ac.jp

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

Abstract.   Let a and b be positive integers with a<b , such that ab+1 is a perfect square. In this paper we give an upper bound for the minimal positive integer c such that {a,b,c,d} is the set of positive integers which has the property that the product of any two of its elements increased by 1 is a perfect square and d≠ a+b+c+2(abc±√((ab+1)(ac+1)(bc+1))).

2010 Mathematics Subject Classification.   11D09, 11J68.

Key words and phrases.   Diophantine tuples, simultaneous Diophantine equations.

Full text (PDF) (free access)

DOI: 10.3336/gm.49.1.03

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