International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 63739, 12 pages
Research Article

Extensions of Some Parametric Families of D(16)-Triples

Alan Filipin

Faculty of Civil Engineering, University of Zagreb, Fra Andrije Kačića-Miošića 26, Zagreb 10000, Croatia

Received 7 September 2006; Revised 24 November 2006; Accepted 29 November 2006

Academic Editor: Dihua Jiang

Copyright © 2007 Alan Filipin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let n be an integer. A set of m positive integers is called a D(n)-m-tuple if the product of any two of them increased by n is a perfect square. In this paper, we consider extensions of some parametric families of D(16)-triples. We prove that if {k4,k+4,4k,d}, for k5, is a D(16)-quadruple, then d=k34k. Furthermore, if {k4,4k,9k12}, for k>5, is a D(16)-quadruple, then d=9k348k2+76k32. But for k=5, this statement is not valid. Namely, the D(16)-triple {1,20,33} has exactly two extensions to a D(16)-quadruple: {1,20,33,105} and {1,20,33,273}.