engleski

Triples and quadruples which are D(n)-sets for several n's

For a nonzero integer n, a set of distinct nonzero integers {; ; a_1, a_2, ... , a_m}; ; such that a_ia_j + n is a perfect square for all 1 <= i < j <= m, is called a Diophantine m-tuple with the property D(n) or simply a D(n)-set. D(1)-sets are known as Diophantine m-tuples. It is natural to ask if there exists a Diophantine m-tuple (i.e. D(1)-set) which is also a D(n)-set for some n <> 1. For example, {; ; 8, 21, 55}; ; is a D(1) and D(4321)-triple, while {; ; 1, 8, 120}; ; is a D(1) and D(721)- triple. We will present infinite families of Diophantine triples {; ; a, b, c}; ; which are also D(n)-sets for two distinct n's with n <> 1, as well as some Diophantine triples which are also D(n)-sets for three distinct n's with n <> 1. We will consider similar problem with quadruples and we will show that there are infinitely many essentially different quadruples which are simultaneously D(n_1)-quadruples and D(n_2)-quadruples with n_1 <> n_2. This is joint work with Nikola Adžaga, Dijana Kreso, Vinko Petričević and Petra Tadić.

Diophantine m-tuples

nije evidentirano