engleski

D(z)-quadruples in the ring Z[√ -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 = 24a + 2 + (12b + 6)√−2, z = 24a + 5 + (12b + 6)√−2, z = 48a + 44 + (24b + 12)√−2. 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. 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.

Diophantine quadruples; quadratic field; congruence

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