Naslov
Polynomial D(-3)-quadruples

Autori
Filipin, Alan ; Jurasić, Ana

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

Izvornik
Booklet of Abstracts Number Theory Conference 2022 In honour of Professors Kálmán Győry, János Pintz and András Sárközy / - , 2022, 18-18

Skup
Number Theory Conference 2022 In honour of Professors Kálmán Győry, János Pintz and András Sárközy

Mjesto i datum
Debrecen, Mađarska, 04.07.2022. - 08.07.2022

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
Diophantine m-tuples ; polynomials ; Pell-like equations

Sažetak
In this talk we prove that there does not exist a set of four non-zero polynomials from Z[X], not all constant, such that the product of any two of its distinct elements decreased by 3 is a square of a polynomial from Z[X]. For non-zero integer n, a set of m positive integers is called D(n)-m-tuple if products of any two of its distinct elements increased by $n$ is a perfect square. There are many results concerning the upper bounds for such sets. It is easy to prove that if n=2(mod 4), then there does not exist a D(n)-quadruple. On the other hand, Dujella proved that if n is not 2(mod 4}; and n is not in S={;-4, -3, -1, 3, 5, 8, 12, 20};, then there exist at least one D(n)-quadruple. Moreover, he conjectured that there does not exist a D(n)-quadruple, if n is from S. That conjecture is still open, but was recently confirmed for n=-1 and n=-4. Here we consider the polynomial version of this problem, and proving that there does not exist a polynomial D(-3)-quadruple in Z[X], together with previous results of more authors, we finish the proof that there does not exist such polynomial D(n)-quadruple for n in S.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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