Startseite Mathematik On the congruent number problem over integers of cyclic number fields
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

On the congruent number problem over integers of cyclic number fields

  • Albertas Zinevičius EMAIL logo
Veröffentlicht/Copyright: 17. August 2016
Veröffentlichen auch Sie bei De Gruyter Brill
Informationen für Autor*innen Erkunden Sie dieses Fachgebiet
Mathematica Slovaca
Aus der Zeitschrift Mathematica Slovaca Band 66 Heft 3

Abstract

Given a cyclic field extension K/ℚ of degree d and a nonzero rational integer m, we show that the equation mp2 = x4 - y2 has no nontrivial solutions in K/Q when p belongs to a subset of rational prime numbers of relative density at least φ(d)/(2d).

MSC 2010: Primary 11D45; 11H06
keywords: congruent numbers; cyclic extensions; rings of integers; prime numbers

(Communicated by Stanislav Jakubec)

acknowledgement

The author is thankful to dr. A. Bartel for an answer to a question about independence of Chebotarev densities and to prof. A. Dubickas for his remarks on the manuscript.

References

[1] Chandrasekar, V.: The congruent number problem, Resonance 3 (1998), 33–45.10.1007/BF02837344Suche in Google Scholar

[2] Narkiewicz, W.: Elementary and Analytic Theory of Algebraic Numbers (3rd ed.), Springer-Verlag, Berlin-Heidelberg, 2004.10.1007/978-3-662-07001-7Suche in Google Scholar

[3] Girondo, E.—Gonzalez-Diez, G.—Gonzalez-Jimenez, E.—Steuding, R.—Steuding, J.: Right triangles with algebraic sides and elliptic curves over number fields, Math. Slovaca 59 (2009), 299–306.10.2478/s12175-009-0126-3Suche in Google Scholar

[4] Green, B.—Tao, T.: The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2) 167 (2008), 481–547.10.4007/annals.2008.167.481Suche in Google Scholar

[5] Jarden, M.—Narkiewicz, W.: On sums of units, Monatsh. Math. 150 (2006), 327–332.10.1007/s00605-006-0402-zSuche in Google Scholar

[6] Jedrzejak, T.: Congruent numbers over real number fields, Colloq. Math. 128 (2012), 179–186.10.4064/cm128-2-3Suche in Google Scholar

[7] Tada, M.: Congruent numbers over real quadratic fields, Hiroshima Math. J. 31 (2001), 331–343.10.32917/hmj/1151105704Suche in Google Scholar

[8] Tschebotareff, N.: Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehören, Math. Ann. 95 (1925), 191–228.10.1007/BF01206606Suche in Google Scholar

[9] Tunnell, J. B.: A Classical Diophantine problem and Modular Forms of Weight 3/2, Invent. Math. 72 (1983), 323–334.10.1007/BF01389327Suche in Google Scholar

Received: 2013-1-16
Accepted: 2014-1-8
Published Online: 2016-8-17
Published in Print: 2016-6-1

© 2016 Mathematical Institute Slovak Academy of Sciences

Artikel in diesem Heft

  1. Research Article
  2. On the vertex-to-edge duality between the Cayley graph and the coset geometry of von Dyck groups
  3. Research Article
  4. Characterization of hereditarily reversible posets
  5. Research Article
  6. Families of sets which can be represented as sublattices of the lattice of convex subsets of a linearly ordered set
  7. Research Article
  8. Diophantine equation X4+Y4 = 2(U4 + V4)
  9. Research Article
  10. On the congruent number problem over integers of cyclic number fields
  11. Research Article
  12. On the Diophantine equation x2 + C= yn for C = 2a3b17c and C = 2a13b17c
  13. Research Article
  14. Representations and evaluations of the error term in a certain divisor problem
  15. Research Article
  16. A construction of a Peano curve
  17. Research Article
  18. Compositions of ϱ-upper continuous functions
  19. Research Article
  20. Products of Świa̧tkowski functions
  21. Research Article
  22. Subclasses of meromorphically p-valent functions involving a certain linear operator
  23. Research Article
  24. Oscillation results for second-order mixed neutral differential equations with distributed deviating arguments
  25. Research Article
  26. Oscillation criteria for higher order nonlinear delay dynamic equations on time scales
  27. Research Article
  28. On the solutions of a fourth order parabolic equation modeling epitaxial thin film growth
  29. Research Article
  30. Meromorphic solutions of q-shift difference equations
  31. Research Article
  32. Oscillation criteria for higher-order nonlinear delay dynamic equations on time scales
  33. Research Article
  34. Multiplier spaces and the summing operator for series
  35. Research Article
  36. Weighted αβ-statistical convergence of Kantorovich-Mittag-Leffler operators
  37. Research Article
  38. The local spectral radius of a nonnegative orbit of compact linear operators
  39. Research Article
  40. A note on Lie product preserving maps on Mn(ℝ)
  41. Research Article
  42. On vector optimality conditions for constrained problems with ℓ-stable data
  43. Research Article
  44. Reparameterization of weakly nonlinear regression models with constraints
  45. Research Article
  46. Convolutions of polynomial kernels
  47. Research Article
  48. Closed form evaluation of sums containing squares of Fibonomial coefficients
https://doi.org/10.1515/ms-2015-0158
Schlagwörter für diesen Artikel
congruent numbers; cyclic extensions; rings of integers; prime numbers

Artikel in diesem Heft

  1. Research Article
  2. On the vertex-to-edge duality between the Cayley graph and the coset geometry of von Dyck groups
  3. Research Article
  4. Characterization of hereditarily reversible posets
  5. Research Article
  6. Families of sets which can be represented as sublattices of the lattice of convex subsets of a linearly ordered set
  7. Research Article
  8. Diophantine equation X4+Y4 = 2(U4 + V4)
  9. Research Article
  10. On the congruent number problem over integers of cyclic number fields
  11. Research Article
  12. On the Diophantine equation x2 + C= yn for C = 2a3b17c and C = 2a13b17c
  13. Research Article
  14. Representations and evaluations of the error term in a certain divisor problem
  15. Research Article
  16. A construction of a Peano curve
  17. Research Article
  18. Compositions of ϱ-upper continuous functions
  19. Research Article
  20. Products of Świa̧tkowski functions
  21. Research Article
  22. Subclasses of meromorphically p-valent functions involving a certain linear operator
  23. Research Article
  24. Oscillation results for second-order mixed neutral differential equations with distributed deviating arguments
  25. Research Article
  26. Oscillation criteria for higher order nonlinear delay dynamic equations on time scales
  27. Research Article
  28. On the solutions of a fourth order parabolic equation modeling epitaxial thin film growth
  29. Research Article
  30. Meromorphic solutions of q-shift difference equations
  31. Research Article
  32. Oscillation criteria for higher-order nonlinear delay dynamic equations on time scales
  33. Research Article
  34. Multiplier spaces and the summing operator for series
  35. Research Article
  36. Weighted αβ-statistical convergence of Kantorovich-Mittag-Leffler operators
  37. Research Article
  38. The local spectral radius of a nonnegative orbit of compact linear operators
  39. Research Article
  40. A note on Lie product preserving maps on Mn(ℝ)
  41. Research Article
  42. On vector optimality conditions for constrained problems with ℓ-stable data
  43. Research Article
  44. Reparameterization of weakly nonlinear regression models with constraints
  45. Research Article
  46. Convolutions of polynomial kernels
  47. Research Article
  48. Closed form evaluation of sums containing squares of Fibonomial coefficients
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 15.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/ms-2015-0158/pdf
Button zum nach oben scrollen