Naslov
Diophantine triples in the ring of integers of the quadratic field Q[√ -t], t>0

Autori
Soldo, Ivan

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

Izvornik
Computational Aspects of Diophantine Equations, University of Salzburg / - , 2016

Skup
Computational Aspects of Diophantine Equations

Mjesto i datum
Salzburg, Austrija, 15.02.2016. - 19.02.2016

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Nije recenziran

Ključne riječi
diophantine $m$-tuple; quadratic field

Sažetak
Let $R$ be a commutative ring and $z \in R$. A set $\{; ; a_1, a_2, \dots, a_m\}; ; $ in $R$ such that $a_i \ne 0$, $i=1, \dots, m, a_i \ne a_j$ and $a_i a_j+z$ is a square in $R$ for all $1\leq i < j\leq m$ is called a Diophantine $m$-tuple with the property $D(z)$, or simply a $D(z)$-$m$-tuple in the ring $R$. We study $D(-1)$-triples of the form $\{; ; 1, b, c\}; ; $ in the ring $\mathbb{; ; Z}; ; [\sqrt{; ; -t}; ; ], t>0$, for positive integer $b$ such that $b$ is a prime, twice prime and twice prime squared. We prove that in those cases $c$ has to be an integer. By using that result we obtained some results about the existence of $D(-1)$-quadruples in certain ring of integers $\mathbb{; ; Z}; ; [\sqrt{; ; -t}; ; ]$ of the quadratic field $\mathbb{; ; Q}; ; (\sqrt{; ; -t}; ; )$, $t>0$.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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