Pregled bibliografske jedinice broj: 1110058
$D(-1)$-tuples in the ring $\bZ[\sqrt{;;;-k};;;]$ with $k>0$
$D(-1)$-tuples in the ring $\bZ[\sqrt{;;;-k};;;]$ with $k>0$ // Publicationes mathematicae, 100 (2022), 1-2; 49-67 doi:10.5486/PMD.2022.8975 (međunarodna recenzija, članak, znanstveni)
CROSBI ID: 1110058 Za ispravke kontaktirajte CROSBI podršku putem web obrasca
Naslov
$D(-1)$-tuples in the ring $\bZ[\sqrt{;;;-k};;;]$ with
$k>0$
Autori
Fujita, Yasutsugu ; Soldo, Ivan
Izvornik
Publicationes mathematicae (0033-3883) 100
(2022), 1-2;
49-67
Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni
Ključne riječi
System of Pellian equations, Diophantine $m$-tuple, quadratic extensions
Sažetak
Let $n$ be a non-zero integer and $R$ a commutative ring. A $D(n)$-$m$-tuple in $R$ is a set of $m$ non-zero elements in $R$ such that the product of any two distinct elements plus $n$ is a perfect square in $R$. In this paper, we prove that there does not exist a $D(-1)$-quadruple $\ {; ; ; a, b, c, d\}; ; ; $ in the ring $\bZ[\sqrt{; ; ; -k}; ; ; ]$, $k\ge 2$ with positive integers $a<b\le8a-3$ and negative integers $c$ and $d$. By using that result we were able to prove that such a $D(-1)$- pair $\{; ; ; a, b\}; ; ; $ cannot be extended to a $D(-1)$- quintuple $\{; ; ; a, b, c, d, e\}; ; ; $ in $\bZ[\sqrt{; ; ; - k}; ; ; ]$ with integers $c, d$ and $e$. Moreover, we apply the obtained result to the $D(-1)$-pair $\ {; ; ; p^i, q^j\}; ; ; $ with an arbitrary different primes $p$, $q$ and positive integers $i$, $j$.
Izvorni jezik
Engleski
Znanstvena područja
Matematika
POVEZANOST RADA
Projekti:
HRZZ-IP-2018-01-1313 - Diofantska geometrija i primjene (DIOPHANT) (Kazalicki, Matija, HRZZ - 2018-01) ( CroRIS)
Ustanove:
Sveučilište u Osijeku, Odjel za matematiku
Profili:
Ivan Soldo
(autor)
Citiraj ovu publikaciju:
Časopis indeksira:
- Web of Science Core Collection (WoSCC)
- Science Citation Index Expanded (SCI-EXP)
- SCI-EXP, SSCI i/ili A&HCI
- Scopus
