D(z)-quadruples in the ring A[√(-2)], for some exceptional cases of z

By the work of Abu Muriefah, Al-Rashed, Dujella and Soldo, the problem of the existence of D ( z ) -quadruples in the ring Z [ − 2 ] has been solved, except for the cases z = 24 a + 2 + ( 12 b + 6 ) − 2 , z = 24 a + 5 + ( 12 b + 6 ) − 2 , z = 48 a + 44 + ( 24 b + 12 ) − 2 . sent 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. These new formulas then together with previous results on this subject allow us to almost completely characterize elements z of Z [ − 2 ] for which a Diophantine quadruple with the property D ( z ) exists.