Extensions of Some Parametric Families of D(16)-Triples
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+4,4k,d}, for k≥5, is a D(16)-quadruple, then d=...
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Wiley
2007-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2007/63739 |
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| author | Alan Filipin |
| author_facet | Alan Filipin |
| author_sort | Alan Filipin |
| collection | DOAJ |
| description | 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+4,4k,d}, for k≥5, is a D(16)-quadruple, then d=k3−4k. Furthermore, if {k−4,4k,9k−12}, for k>5, is a D(16)-quadruple, then d=9k3−48k2+76k−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}. |
| format | Article |
| id | doaj-art-bea1d6b1d81442dfba41875550585091 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2007-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-bea1d6b1d81442dfba418755505850912025-02-03T05:46:00ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252007-01-01200710.1155/2007/6373963739Extensions of Some Parametric Families of D(16)-TriplesAlan Filipin0Faculty of Civil Engineering, University of Zagreb, Fra Andrije Kačića-Miošića 26, Zagreb 10000, CroatiaLet 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+4,4k,d}, for k≥5, is a D(16)-quadruple, then d=k3−4k. Furthermore, if {k−4,4k,9k−12}, for k>5, is a D(16)-quadruple, then d=9k3−48k2+76k−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}.http://dx.doi.org/10.1155/2007/63739 |
| spellingShingle | Alan Filipin Extensions of Some Parametric Families of D(16)-Triples International Journal of Mathematics and Mathematical Sciences |
| title | Extensions of Some Parametric Families of D(16)-Triples |
| title_full | Extensions of Some Parametric Families of D(16)-Triples |
| title_fullStr | Extensions of Some Parametric Families of D(16)-Triples |
| title_full_unstemmed | Extensions of Some Parametric Families of D(16)-Triples |
| title_short | Extensions of Some Parametric Families of D(16)-Triples |
| title_sort | extensions of some parametric families of d 16 triples |
| url | http://dx.doi.org/10.1155/2007/63739 |
| work_keys_str_mv | AT alanfilipin extensionsofsomeparametricfamiliesofd16triples |