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

Autori
Dujella, Andrej

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Izvornik
International Conference on Class Groups of Number Fields and Related Topics-2021 / Chakraborty, Kalyan - Kozhikode : Kerala School of Mathematics, 2021, 10-10

Skup
International Conference on Class Groups of Number Fields and Related Topics (ICCGNFRT 2021)

Mjesto i datum
Kozhikode, Indija, 21.10.2021. - 24.10.2021

Vrsta sudjelovanja
Pozvano predavanje

Vrsta recenzije
Podatak o recenziji nije dostupan

Ključne riječi
Diophantine m-tuples

Sažetak
For an integer n, a set of distinct nonzero integers {; ; ; a_1, a_2, ... , a_m}; ; ; such that a_i a_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 also called Diophantine m-tuples. The first Diophantine quadruple, the set {; ; ; 1, 3, 8, 120}; ; ; was found by Fermat. He, Togbe and Ziegler proved recently that there does not exist a Diophantine quintuple. On the other hand, it is known that there exist infinitely many rational Diophantine sextuples. 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 Adžaga, Kreso and Tadić, 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 Petričević we proved 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. Moreover, the elements in some of these quadruples are squares, so they are also D(0)-quadruples. E.g. {; ; ; 54^2, 100^2, 168^2, 364^2}; ; ; is a D(8190^2), D(40320^2) and D(0)-quadruple. In a recent joint work in with Kazalicki and Petričević, we considered D(n)-quintuples with square elements (so they are also D(0)- quintuples). We proved that there are infinitely many such quintuples. One example is a D(480480^2)-quintuple {; ; ; 225^2, 286^2, 819^2, 1408^2, 2548^2}; ; ; . In this talk, we will describe methods used in constructions of mentioned triples, quadruples and quintuples.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

POVEZANOST RADA