Naslov
Recent results on Diophantine sets

Autori
Filipin, Alan

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

Izvornik
Diophantine Analysis and Related Fields 2017 / - , 2017, 5-5

Skup
Diophantine Analysis and Related Fields 2017

Mjesto i datum
Tokyo, Japan, 07.01.2017. - 09.01.2017

Vrsta sudjelovanja
Pozvano predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
Diophantine m-tuples

Sažetak
Let n be a nonzero integer. A set {;a_1, ..., a_m}; is called a D(n)-m-tuple, if the product of any of its two distinct elements increased by n is a perfect square. One of the obvious and most interesting questions is how large those sets can be. The most well-known and studied case is when n = 1. There is a folklore conjecture that there does not exist a D(1)-quintuple. Actually, there is a stronger version of the conjecture, that every triple can be extended to a quadruple with a larger element in a unique way. Very recently it was reported that the weaker version of the conjecture, that there does not exist a quintuple, has been solved. In this talk we will present the recent results we have proved towards proving both versions of the conjecture. Except the case n = 1, the cases n = 4 and n = -1 have been studied thoroughly in recent years. The cases n = 4 and n = 1 are closely connected. In this talk we will also present the proof of the newest upper bound for the number of D(4)-quintuples. At the end we will also illustrate some elementary ideas that can help us solve the extendibility of some D(n) sets, when the standard methods do not work. The results presented in this talk are joint work with M. Cipu, Y. Fujita, M. Bliznac Trebješanin and N. Adžaga.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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