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

Alan Filipin

International Journal of Mathematics and Mathematical Sciences, 2007, vol. 2007, 1-12

Abstract:

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 { k − 4 , k + 4 , 4 k , d }, for k ≥ 5, is a D ( 16 ) -quadruple, then d = k 3 − 4 k . Furthermore, if { k − 4 , 4 k , 9 k − 12 } , for k > 5 , is a D ( 16 ) -quadruple, then d = 9 k 3 − 48 k 2 + 76 k − 32 . 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 } .

Date: 2007
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:063739

DOI: 10.1155/2007/63739

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