A new characterization of the integer 5906

Andrew Bremner; Patrick Morton

doi:10.1007/BF01166081

A new characterization of the integer 5906

Andrew Bremner, Patrick Morton

Research output: Contribution to journal › Article › peer-review

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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.

Original language	English (US)
Pages (from-to)	187-229
Number of pages	43
Journal	Manuscripta Mathematica
Volume	44
Issue number	1-3
DOIs	https://doi.org/10.1007/BF01166081
State	Published - Feb 1 1983
Externally published	Yes

ASJC Scopus subject areas

General Mathematics

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10.1007/BF01166081

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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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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.

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A new characterization of the integer 5906

Abstract

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