Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia


IMPACT FACTOR 2018: 0.490

CiteScore 2018: 0.47

SCImago Journal Rank (SJR) 2018: 0.279
Source Normalized Impact per Paper (SNIP) 2018: 0.627

Mathematical Citation Quotient (MCQ) 2018: 0.29

Online
ISSN
1337-2211
See all formats and pricing
More options …
Volume 66, Issue 3

Issues

On the congruent number problem over integers of cyclic number fields

Albertas Zinevičius
  • Corresponding author
  • Department of Mathematics and Informatics Vilnius University Naugarduko 24 LT-03225 Vilnius LITHUANIA
  • Institute of Mathematics and Informatics Akademijos 4 LT-08663 Vilnius LITHUANIA
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2016-08-17 | DOI: https://doi.org/10.1515/ms-2015-0158

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

References

  • [1]

    Chandrasekar, V.: The congruent number problem, Resonance 3 (1998), 33–45.Google Scholar

  • [2]

    Narkiewicz, W.: Elementary and Analytic Theory of Algebraic Numbers (3rd ed.), Springer-Verlag, Berlin-Heidelberg, 2004.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.Google Scholar

  • [4]

    Green, B.—Tao, T.: The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2) 167 (2008), 481–547.Google Scholar

  • [5]

    Jarden, M.—Narkiewicz, W.: On sums of units, Monatsh. Math. 150 (2006), 327–332.Google Scholar

  • [6]

    Jedrzejak, T.: Congruent numbers over real number fields, Colloq. Math. 128 (2012), 179–186.Google Scholar

  • [7]

    Tada, M.: Congruent numbers over real quadratic fields, Hiroshima Math. J. 31 (2001), 331–343.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.Google Scholar

  • [9]

    Tunnell, J. B.: A Classical Diophantine problem and Modular Forms of Weight 3/2, Invent. Math. 72 (1983), 323–334.Google Scholar

About the article


Received: 2013-01-16

Accepted: 2014-01-08

Published Online: 2016-08-17

Published in Print: 2016-06-01


Citation Information: Mathematica Slovaca, Volume 66, Issue 3, Pages 561–564, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2015-0158.

Export Citation

© 2016 Mathematical Institute Slovak Academy of Sciences.Get Permission

Comments (0)

Please log in or register to comment.
Log in