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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