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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

Part of: Diophantine equations Algebraic number theory: global fields

Published online by Cambridge University Press:  06 November 2020

ANDREW LI Open the ORCID record for ANDREW LI [Opens in a new window]
ANDREW LI*
Affiliation:
Department of Mathematics, University of Nebraska Omaha, Omaha, NE68182, USA
*
e-mail: [email protected]
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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.

Keywords

quartic Diophantine equationsquadratic number fields

MSC classification

Primary: 11D25: Cubic and quartic equations
Secondary: 11R11: Quadratic extensions 11D45: Counting solutions of Diophantine equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 1 , August 2021 , pp. 21 - 28
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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References

Aigner, A., ‘Über die möglichkeit von ${x}^4+{y}^4={z}^4$ in quadratischen körpern’, Jahresber. Dtsch. Math.-Ver. 43 (1934), 226–229.Google Scholar
Faddeev, D. K., ‘Group of divisor classes on the curve defined by the equation ${x}^4+{y}^4=1$ ’, Soviet Math. Dokl. 1 (1960), 1149–1151.Google Scholar
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The LMFDB Collaboration, ‘The L-functions and modular forms database’, http://www.lmfdb.org, (online; accessed 28 July 2020). Google Scholar