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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
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.
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- Research Article
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- © 2020 Australian Mathematical Publishing Association Inc.
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