engleski

D(n)-sets with square elements

For an integer n, a set of distinct nonzero integers {; ; a1, a2, ...am}; ; such that aiaj+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. When considering D(n)-sets, usually an integer n is fixed in advance. However, we may ask if a set can have the property D(n) for several different n’s. For example, {; ; 8, 21, 55}; ; is a D(1)-triple and D(4321)-triple. In a joint work with Adzaga, Kreso and Tadic, we presented several families of Diophantine triples which are D(n)- sets for two distinct n’s with n≠1. In a joint work with Petricevic we proved that there are infinitely many (essentially different) quadruples which are simultaneously D(n1)-quadruples and D(n2)-quadruples with n1≠n2. Morever, the elements in some of these quadruples are squares, so they are also D(0)-quadruples. E.g. {; ; 542, 1002, 1682, 3642}; ; is a D(81902), D(403202) and D(0)-quadruple. In this talk, we will describe methods used in constructions of mentioned triples and quadruples. We will also mention a work in progress with Kazalicki and Petricevic on D(n)- quintuples with square elements (so they are also D(0)-quintuples). There are infinitely many such quintuples. One example is a D(480480^2)-quintuple {; ; 225^2, 286^2, 819^2, 1408^2, 2548^2}; ; .

diophantine triples ; diophantine quadruples ; elliptic curves

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