engleski

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

Diophantine m-tuples

nije evidentirano