Naslov
The non-existence of $D(-1)$-quadruples extending certain pairs in imaginary quadratic rings

Autori
Yasutsugu Fujita, Ivan Soldo

Vrsta, podvrsta
Radovi u časopisima, znanstveni

Izvornik
Acta mathematica Hungarica (2023)

Status rada
Prihvaćen

Ključne riječi
system of Pellian equations, Diophantine $m$-tuple, Pad\'e approximation method

Sažetak
A $D(n)$-$m$-tuple, where $n$ is a non-zero integer, is a set of $m$ distinct elements in a commutative ring $R$ 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 $a<b< 16a^2-a-2+2\sqrt{;k(8a^2+3a+1)};$ and integers $c$ and $d$ satisfying $d<0<c$. By combining that result with \cite[Theorem~1.1]{;FS};, we were able to obtain a general result on the non-existence of a $D(-1)$-quadruple $\{;a, b, c, d\};$ in $\bZ[\sqrt{;-k};]$ with integers $a, b, c, d$ satisfying $a<b\le 8a-3$. Furthermore, for a non- negative integer $i$ and a positive integer $j$, we apply the obtained results in proving of the non-existence of $D(-1)$-quadruples containing powers of primes $p^i$, $q^j$ with an arbitrary different primes $p$ and $q$.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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