Document Type
Article
Publication Date
8-15-2025
Abstract
In this paper, we generalise to the family of Fermat quartics X4+Y4=2m,m∈Z, a result of Aigner [‘Über die Möglichkeit von x4+y4=z4 in quadratischen Körpern’, Jahresber. Deutsch. Math.-Ver. 43 (1934), 226–228], which proves that there is only one quadratic field, namely Q(−7−−−√), that contains solutions to the Fermat quartic X4+Y4=1. The m≡0(mod4) case is due to Aigner. The m≡2(mod4) case follows from a result of Emory [‘The Diophantine equation X4+Y4=D2Z4 in quadratic fields’, Integers 12 (2012), Article no. A65, 8 pages]. This paper focuses on the two cases m≡1,3(mod4), classifying for m≡1(mod4) the infinitely many quadratic number fields that contain solutions, and proving for m≡3(mod4) that Q(2–√) and Q(−2−−−√) are the only quadratic number fields that contain solutions.
Recommended Citation
Athipatla, Jayanth, "THE FERMAT QUARTIC X4 + Y4 = 2m IN QUADRATIC NUMBER FIELDS" (2025). Mathematics Faculty Publications. 85.
https://digitalcommons.unomaha.edu/mathfacpub/85
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.
Funded by the University of Nebraska at Omaha Open Access Fund
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