Geodesic
On the extension of the Diophantine pair \(\{1,3\}\) in \(\mathbb Z[\sqrt d]\)
Journal of integer sequences, Tome 13 (2010) no. 9
In this paper, we consider Diophantine triples of the form ${1,3,c}$ in the ring $Z[\sqrt d]$. We prove that the Diophantine pair ${1,3}$ cannot be extended to the Diophantine quintuple in $Z[\sqrt d]$ with $d 0$ and $d \ne -2$.
Classification : 11D09, 11R11
Keywords: Diophantine sets, quadratic fields, pellian equations 
@article{JIS_2010__13_9_a3,
     author = {Franu\v{s}i\'c,  Zrinka},
     title = {On the extension of the {Diophantine} pair \(\{1,3\}\) in \(\mathbb {Z[\sqrt} d]\)},
     journal = {Journal of integer sequences},
     year = {2010},
     volume = {13},
     number = {9},
     zbl = {1217.11030},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2010__13_9_a3/}
}
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Franušić,  Zrinka. On the extension of the Diophantine pair \(\{1,3\}\) in \(\mathbb Z[\sqrt d]\). Journal of integer sequences, Tome 13 (2010) no. 9. http://geodesic.mathdoc.fr/item/JIS_2010__13_9_a3/