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THE FERMAT QUARTIC X4 + Y4 = 2m IN QUADRATIC NUMBER FIELDS
Published online by Cambridge University Press: 15 August 2025
Abstract
In this paper, we generalise to the family of Fermat quartics 
$X^4 + Y^4 = 2^m, m \in \mathbb {Z}$, a result of Aigner [‘Über die Möglichkeit von 
$x^4 + y^4 = z^4$ in quadratischen Körpern’, Jahresber. Deutsch. Math.-Ver. 43 (1934), 226–228], which proves that there is only one quadratic field, namely 
$\mathbb {Q}(\sqrt {-7})$, that contains solutions to the Fermat quartic 
$X^4 + Y^4 = 1$. The 
$m \equiv 0 \pmod 4$ case is due to Aigner. The 
$m \equiv 2 \pmod 4$ case follows from a result of Emory [‘The Diophantine equation 
$X^4 + Y^4 = D^2Z^4$ in quadratic fields’, Integers 12 (2012), Article no. A65, 8 pages]. This paper focuses on the two cases 
$m \equiv 1, 3 \pmod 4$, classifying for 
$m \equiv 1 \pmod 4$ the infinitely many quadratic number fields that contain solutions, and proving for 
$m \equiv 3 \pmod 4$ that 
$\mathbb {Q}(\sqrt {2})$ and 
$\mathbb {Q}(\sqrt {-2})$ are the only quadratic number fields that contain solutions.
MSC classification
Primary:
11D25: Cubic and quartic equations
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- Research Article
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- Copyright
- © The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc
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