On the equation $a^{3} + b^{3n} = c^{2}$

Volume 163 / 2014

Michael A. Bennett, Imin Chen, Sander R. Dahmen, Soroosh Yazdani Acta Arithmetica 163 (2014), 327-343 MSC: Primary 11D41; Secondary 11D61, 11G05, 14G05. DOI: 10.4064/aa163-4-3

Abstract

We study coprime integer solutions to the equation $a^3 + b^{3n} = c^2$ using Galois representations and modular forms. This case represents perhaps the last natural family of generalized Fermat equations descended from spherical cases which is amenable to resolution using the so-called modular method. Our techniques involve an elaborate combination of ingredients, ranging from $\mathbb Q$-curves and a delicate multi-Frey approach, to appeal to intricate image of inertia arguments.

Authors

  • Michael A. BennettDepartment of Mathematics
    University of British Columbia
    Vancouver, British Columbia, V6T 1Z2, Canada
    e-mail
  • Imin ChenDepartment of Mathematics
    Simon Fraser University
    Burnaby, British Columbia, Canada
    e-mail
  • Sander R. DahmenDepartment of Mathematics
    VU University Amsterdam
    De Boelelaan 1081a
    1081 HV Amsterdam, The Netherlands
    e-mail
  • Soroosh YazdaniDepartment of Mathematics and Computer Science
    University of Lethbridge
    Lethbridge, Alberta, T1K 3M4, Canada
    e-mail

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