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Licensed Unlicensed Requires Authentication Published by De Gruyter December 22, 2019

D(n)-quadruples in the ring of integers of ℚ(√2, √3)

  • Zrinka Franušić EMAIL logo and Borka Jadrijević
From the journal Mathematica Slovaca


Let 𝓞𝕂 be the ring of integers of the number field 𝕂 = Q(2,3). A D(n)-quadruple in the ring 𝓞𝕂 is a set of four distinct non-zero elements {z1, z2, z3, z4} ⊂ 𝓞𝕂 with the property that the product of each two distinct elements increased by n is a perfect square in 𝓞𝕂. We show that the set of all n ∈ 𝓞𝕂 such that a D(n)-quadruple in 𝓞𝕂 exists coincides with the set of all integers in 𝕂 that can be represented as a difference of two squares of integers in 𝕂.

The authors were supported by the Croatian Science Foundation under the project no. 6422. Z. F. acknowledges support from the QuantiXLie Center of Excellence, a project co-financed by the Croatian Government and European Union through the European Regional Development Fund - the Competitiveness and Cohesion Operational Programme (Grant KK.

  1. Communicated by Milan Paštéka


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Received: 2019-03-27
Accepted: 2019-05-30
Published Online: 2019-12-22
Published in Print: 2019-12-18

© 2019 Mathematical Institute Slovak Academy of Sciences

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