Advertisement

Acta Mathematica Hungarica

, Volume 159, Issue 1, pp 89–108 | Cite as

On the extensibility of \(D(-1)\)-pairs containing Fermat primes

  • M. Jukić Bokun
  • I. SoldoEmail author
Article
  • 15 Downloads

Abstract

We study the extendibility of a \(D(-1)\)-pair {1, p}, where p is a Fermat prime, to a \(D(-1)\)-quadruple in \(\mathbb{Z} [\sqrt{-t}], t > 0\).

Mathematics Subject Classification

11D09 11R11 11J86 

Key words and phrases

Diophantine quadruple quadratic field simultaneous Pellian equation linear form in logarithm 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The authors are grateful to Andrej Dujella and the referee for making valuable suggestions and comments leading to a better presentation of the paper.

References

  1. 1.
    F. S. Abu Muriefah and A. Al-Rashed, Some Diophantine quadruples in the ring \(\mathbb{Z} [\sqrt{-2}]\), Math. Commun., 9 (2004), 1–8Google Scholar
  2. 2.
    A. Baker and H. Davenport, The equations \(3x^2-2=y^2\) and \(8x^2-7=z^2\), Quart. J. Math. Oxford Ser. (2), 20 (1969), 129–137Google Scholar
  3. 3.
    Bayad, A., Filipin, A., Togbe, A.: Extension of a parametric family of Diophantine triples in Gaussian integers. Acta Math. Hungar. 148, 312–327 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bayad, A., Dossavi-Yovo, A., Filipin, A., Togbe, A.: On the extensibility of \(D(4)\)-triple \(\{k-2, k+2, 4k\}\) over Gaussian integers. Notes Number Theory Discrete Math. 23, 1–26 (2017)zbMATHGoogle Scholar
  5. 5.
    Dujella, A.: The problem of Diophantus and Davenport for Gaussian integers. Glas. Mat. Ser. III(32), 1–10 (1997)MathSciNetzbMATHGoogle Scholar
  6. 6.
    A. Dujella, Diophantine \(m\)-tuples page, available at http://web.math.pmf.unizg.hr/~duje/dtuples.html
  7. 7.
    A. Dujella, M. Jukić Bokun, and I. Soldo, A Pellian equation with primes and applications to \(D(-1)\)-quadruples, Bull. Malays. Math. Sci. Soc. (to appear)Google Scholar
  8. 8.
    Dujella, A., Pethő, A.: Generalization of a theorem of Baker and Davenport. Quart. J. Math. Oxford 49, 291–306 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    A. Dujella and I. Soldo, Diophantine quadruples in \(\mathbb{Z} [\sqrt{-2}]\), An. Ştiinţ. Univ. ``Ovidius'' Constanţa Ser. Mat., 18 (2010), 81–98Google Scholar
  10. 10.
    Franušić, Z.: On the extensibility of Diophantine triples \(\{k-1, k+1, 4k\}\) for Gaussian integers. Glas. Mat. Ser. III(43), 265–291 (2008)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Franušić, Z., Kreso, D.: Nonextensibility of the pair \(\mathbb{Z} [\sqrt{-2}]\), J. Comb. Number Theory 3, 1–15 (2011)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Filipin, A., Fujita, Y., Mignotte, M.: The non-extendibility of some parametric families of \(D(-1)\)-triples. Quart. J. Math. 63, 605–621 (2012)CrossRefzbMATHGoogle Scholar
  13. 13.
    Lebesgue, V.A.: Sur l'impossbilité en nombres entiers de l'équation \(x^m = y^2 + 1\). Nouv. Ann. Math. 9, 178–181 (1850)Google Scholar
  14. 14.
    Matthews, K.: The Diophantine equation \(x^2-Dy^2=N, D>0\). Expo. Math. 18, 323–332 (2000)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Matveev, E.M., An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II, Izv. Ross. Akad. Nauk Ser. Mat., 64, : 125–180 (in Russian). English translation in Izv. Math. 64(2000), 1217–1269 (2000)Google Scholar
  16. 16.
    T. Nagell, Introduction to Number Theory, Wiley (New York, 1951)Google Scholar
  17. 17.
    N. J. A. Sloane, The on-line encyclopedia of integer sequences, available at oeis.org/A019434Google Scholar
  18. 18.
    N. P. Smart, The Algorithmic Resolution of Diophantine Equations, Cambridge University Press (Cambridge, 1998)Google Scholar
  19. 19.
    Soldo, I.: On the existence of Diophantine quadruples in \(\mathbb{Z} [\sqrt{-2}]\), Miskolc Math. Notes 14, 261–273 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Soldo, I.: On the extensibility of \(D(-1)\)-triples \(\{1, b, c\}\) in the ring \(\mathbb{Z} [\sqrt{-2}]\), \(t>0\). Studia Sci. Math. Hungar. 50, 296–330 (2013)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Soldo, I.: \(D(-1)\)-triples of the form \(\{1, b, c\}\) in the ring \(\mathbb{Z} [\sqrt{-2}]\), \(t>0\). Bull. Malays. Math. Sci. Soc. 39, 1201–1224 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OsijekOsijekCroatia

Personalised recommendations