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

  • Zrinka Franušić EMAIL logo and Borka Jadrijević
Published/Copyright: December 22, 2019
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 69 Issue 6

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.

Keywords: Diophantine quadruples; bicyclic biquadratic fields; difference of squares
MSC 2010: Primary 11E25; 11R16; 11Y99

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

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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

Articles in the same Issue

  1. Regular papers
  2. RNDr. Kvetoslava Dvořáková passed away
  3. On the Riesz structures of a lattice ordered abelian group
  4. On Diophantine equation x4 + y4 = n(u4 + v4)
  5. On a Waring-Goldbach problem involving squares and cubes
  6. D(n)-quadruples in the ring of integers of ℚ(√2, √3)
  7. Geometry of ℙ2 blown up at seven points
  8. Preservation of Rees exact sequences
  9. Pointwise multipliers between weighted copson and cesàro function spaces
  10. Some properties associated to a certain class of starlike functions
  11. Asymptotic properties of noncanonical third order differential equations
  12. Existence and regularity results for unilateral problems with degenerate coercivity
  13. Direct and inverse approximation theorems of functions in the Orlicz type spaces 𝓢M
  14. Some approximation properties of a kind of (p, q)-Phillips operators
  15. Best proximity points for a new type of set-valued mappings
  16. Weakly demicompact linear operators and axiomatic measures of weak noncompactness
  17. Fixed point results for F𝓡-generalized contractive mappings in partial metric spaces
  18. Einstein-Weyl structures on trans-Sasakian manifolds
  19. Characterizations of linear Weingarten space-like hypersurface in a locally symmetric Lorentz space
  20. ∗-Ricci solitons and gradient almost ∗-Ricci solitons on Kenmotsu manifolds
  21. Uncorrelatedness sets of discrete random variables via Vandermonde-type determinants
  22. A note on the consistency of wavelet estimators in nonparametric regression model under widely orthant dependent random errors
  23. On adaptivity of wavelet thresholding estimators with negatively super-additive dependent noise
  24. Generalized Meir-Keeler type contractions and discontinuity at fixed point II
https://doi.org/10.1515/ms-2017-0307
Keywords for this article
Diophantine quadruples; bicyclic biquadratic fields; difference of squares

Articles in the same Issue

  1. Regular papers
  2. RNDr. Kvetoslava Dvořáková passed away
  3. On the Riesz structures of a lattice ordered abelian group
  4. On Diophantine equation x4 + y4 = n(u4 + v4)
  5. On a Waring-Goldbach problem involving squares and cubes
  6. D(n)-quadruples in the ring of integers of ℚ(√2, √3)
  7. Geometry of ℙ2 blown up at seven points
  8. Preservation of Rees exact sequences
  9. Pointwise multipliers between weighted copson and cesàro function spaces
  10. Some properties associated to a certain class of starlike functions
  11. Asymptotic properties of noncanonical third order differential equations
  12. Existence and regularity results for unilateral problems with degenerate coercivity
  13. Direct and inverse approximation theorems of functions in the Orlicz type spaces 𝓢M
  14. Some approximation properties of a kind of (p, q)-Phillips operators
  15. Best proximity points for a new type of set-valued mappings
  16. Weakly demicompact linear operators and axiomatic measures of weak noncompactness
  17. Fixed point results for F𝓡-generalized contractive mappings in partial metric spaces
  18. Einstein-Weyl structures on trans-Sasakian manifolds
  19. Characterizations of linear Weingarten space-like hypersurface in a locally symmetric Lorentz space
  20. ∗-Ricci solitons and gradient almost ∗-Ricci solitons on Kenmotsu manifolds
  21. Uncorrelatedness sets of discrete random variables via Vandermonde-type determinants
  22. A note on the consistency of wavelet estimators in nonparametric regression model under widely orthant dependent random errors
  23. On adaptivity of wavelet thresholding estimators with negatively super-additive dependent noise
  24. Generalized Meir-Keeler type contractions and discontinuity at fixed point II
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