Original scientific paper

https://doi.org/10.3336/gm.46.2.03

The D(-k2)-triple {1,k2+1,k2+4} with k prime

Alan Filipin ; Faculty of Civil Engineering, University of Zagreb, Fra Andrije Kačića-Miošića 26, 10 000 Zagreb, Croatia
Yasutsugu Fujita ; Department of Mathematics, College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba, Japan

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.

Keywords

Diophantine tuples; simultaneous Diophantine equations

Hrčak ID:

74261

URI

https://hrcak.srce.hr/74261

Publication date:

23.11.2011.

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