### 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 x^{4}+y^{4}=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.

Original language | English (US) |
---|---|

Pages (from-to) | 187-229 |

Number of pages | 43 |

Journal | Manuscripta Mathematica |

Volume | 44 |

Issue number | 1-3 |

DOIs | |

State | Published - Feb 1983 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Manuscripta Mathematica*,

*44*(1-3), 187-229. https://doi.org/10.1007/BF01166081

**A new characterization of the integer 5906.** / Bremner, Andrew; Morton, Patrick.

Research output: Contribution to journal › Article

*Manuscripta Mathematica*, vol. 44, no. 1-3, pp. 187-229. https://doi.org/10.1007/BF01166081

}

TY - JOUR

T1 - A new characterization of the integer 5906

AU - Bremner, Andrew

AU - Morton, Patrick

PY - 1983/2

Y1 - 1983/2

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=34250153369&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34250153369&partnerID=8YFLogxK

U2 - 10.1007/BF01166081

DO - 10.1007/BF01166081

M3 - Article

AN - SCOPUS:34250153369

VL - 44

SP - 187

EP - 229

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

SN - 0025-2611

IS - 1-3

ER -