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THE DIOPHANTINE EQUATION $\boldsymbol{x}^{\boldsymbol{4}} \boldsymbol{+} \boldsymbol{2}^{\boldsymbol{n}}\boldsymbol{y}^{\boldsymbol{4}} \boldsymbol{=} \boldsymbol{1}$ IN QUADRATIC NUMBER FIELDS

Published online by Cambridge University Press:  06 November 2020

ANDREW LI*
Affiliation:
Department of Mathematics, University of Nebraska Omaha, Omaha, NE68182, USA

Abstract

Aigner showed in 1934 that nontrivial quadratic solutions to $x^4 + y^4 = 1$ exist only in $\mathbb Q(\sqrt {-7})$ . Following a method of Mordell, we show that nontrivial quadratic solutions to $x^4 + 2^ny^4 = 1$ arise from integer solutions to the equations $X^4 \pm 2^nY^4 = Z^2$ investigated in 1853 by V. A. Lebesgue.

Type
Research Article
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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References

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