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A new characterization of the integer 5906

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Abstract

It is shown that 5906 is the. least integer expressible as the sum of two rational fourth powers but not as the sum of two integer fourth powers. The relevant Diophantine equation x4+y4=D represents a curve of genus 3, and extensive arithmetic calculations are involved: in particular, class-number, units and ideal-class stucture are. computed for four particular eighth degree extension fields of the rationals. The result provides several examples of curves of genus 3, everywhere locally solvable, but with no rational points.

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Authors and Affiliations

  1. Emmanuel College, CB2 3AP, Cambridge, England

    Andrew Bremner

  2. California Institute of Technology Mathematics, 91125, Pasadena, California, USA

    Patrick Morton

Authors
  1. Andrew Bremner
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  2. Patrick Morton
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Bremner, A., Morton, P. A new characterization of the integer 5906. Manuscripta Math 44, 187–229 (1983). https://doi.org/10.1007/BF01166081

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  • DOI: https://doi.org/10.1007/BF01166081

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