Skip to content
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

Abstract

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.01.1.1.01.0004).


  1. Communicated by Milan Paštéka

References

[1] Abu Muriefah, F. S.—Al-Rashed, A.: Some Diophantine quadruples in the ring ℤ[2], Math. Commun. 9 (2004), 1–8.Search in Google Scholar

[2] Dujella, A.: Generalization of a problem of Diophantus, Acta Arith. 65 (1993), 15–27.10.4064/aa-65-1-15-27Search in Google Scholar

[3] Dujella, A.: Some polynomial formulas for Diophantine quadruples, Grazer Math. Ber. 328 (1996), 25–30.Search in Google Scholar

[4] Dujella, A.: The problem of Diophantus and Davenport for Gaussian integers, Glas. Mat. Ser. III 32 (1997), 1–10.Search in Google Scholar

[5] Dujella, A.: Diophantine quadruples and quintuples modulo 4, Notes Number Theory Discrete Math. 4 (1998), 160–164.Search in Google Scholar

[6] Dujella, A.—Filipin, A.—Fuchs, C.: Effective solution of the D(−1)-quadruple conjecture, Acta Arith. 128 (2007), 319–338.10.4064/aa128-4-2Search in Google Scholar

[7] Dujella, A.—Franušić, Z.: On differences of two squares in some quadratic fields, Rocky Mountain J. Math. 37 (2007), 429–453.10.1216/rmjm/1181068760Search in Google Scholar

[8] Dujella, A.—Soldo, I.: Diophantine quadruples in ℤ[2], An. Stiint. Univ. “Ovidius" Constanta Ser. Mat. 18 (2010), 81–98.Search in Google Scholar

[9] Franušić, Z.: Diophantine quadruples in the ring ℤ[2], Math. Commun. 9 (2004), 141–148.Search in Google Scholar

[10] Franušić, Z.: A Diophantine problem in ℤ[(1 + d)/2], Studia Sci. Math. Hungar. 46 (2009), 103–112.Search in Google Scholar

[11] Franušić, Z.: Diophantine quadruples inZ[4k+3],, Ramanujan J. 17 (2008), 77–88.10.1007/s11139-007-9015-ySearch in Google Scholar

[12] Franušić, Z.: Diophantine quadruples in the ring of integers of the pure qubic field ℚ(23), Miskolc Math. Notes 14 (2013), 893–903.10.18514/MMN.2013.753Search in Google Scholar

[13] Franušić, Z.—Soldo, I.: The problem of Diophantus for integers of ℚ(3), Rad HAZU, Matematičke znanosti. 18 (2014), 15–25.Search in Google Scholar

[14] Jukić Matić, Lj.: Non-existence of certain Diophantine quadruples in rings of integers of pure cubic fields, Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), 163–167.10.3792/pjaa.88.163Search in Google Scholar

[15] Mohanty, S. P.—Ramamsamy, M. S.: On Pr,k sequences, Fibonacci Quart. 23 (1985), 36–44.Search in Google Scholar

[16] Mootha, V. K.—Berzsenyi, G.: Characterization and extendibility of Pt-sets, Fibonacci Quart. 27 (1989), 287–288.Search in Google Scholar

[17] Soldo, I.: On the existence of Diophantine quadruples in ℤ[2], Miskolc Math. Notes 14 (2013), 265–277.10.18514/MMN.2013.565Search in Google Scholar

[18] Williams, K. S.: Integers of biquadratics fields, Canad. Math. Bull. 13 (1970), 519–526.10.4153/CMB-1970-094-8Search in Google Scholar

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

Downloaded on 30.5.2023 from https://www.degruyter.com/document/doi/10.1515/ms-2017-0307/html
Scroll to top button