MMN-565
On the existence of Diophantine quadruples in $Z[\sqrt(-2)]$
Abstract
By the work of Abu Muriefah, Al-Rashed, Dujella and the author, the problem of the existence of $D(z)$-quadruples in the ring $mathbb{Z}[sqrt{-2}]$ has been solved, except for the cases egin{align*} z&=24a+2+(12b+6)sqrt{-2},, z=24a+5+(12b+6)sqrt{-2},\ z&=48a+44+(24b+12)sqrt{-2}. end{align*}
In this paper, we present some new formulas for $D(z)$-quadruples in these remaining cases, involving some congruence conditions modulo 11 on integers $a$ and $b$. We show the existence of $D(z)$-quadruple for significant proportion
of the remaining three cases.
Vol. 14 (2013), No. 1, pp. 265-277
DOI: 10.18514/MMN.2013.565