Glasnik Matematicki, Vol. 46, No. 2 (2011), 311-323.

THE D(-k2)-TRIPLE {1,k2+1,k2+4} WITH k PRIME

Alan Filipin and Yasutsugu Fujita

Faculty of Civil Engineering, University of Zagreb, Fra Andrije Kačića-Miošića 26, 10 000 Zagreb, Croatia
e-mail: filipin@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


Abstract.   Let n be a nonzero integer. A set of m distinct positive integers is called a D(n)-m-tuple if the product of any two of them increased by n is a perfect square. Let k be a prime number. In this paper we prove that the D(-k2)-triple {1,k2+1,k2+4} cannot be extended to a D(-k2)-quadruple if k≠3. And for k=3 we prove that if the set {1,10,13,d} is a D(-9)-quadruple, then d=45.

2000 Mathematics Subject Classification.   11D09, 11J68.

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


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DOI: 10.3336/gm.46.2.03


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