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On the difference of two fourth powers

Part of: Curves Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  10 November 2023

Nguyen Xuan Tho Open the ORCID record for Nguyen Xuan Tho [Opens in a new window]
Nguyen Xuan Tho*
Affiliation:
Hanoi University of Science and Technology Hanoi, Vietnam (tho.nguyenxuan1@hust.edu.vn)
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Abstract

We investigate the equation $D=x^4-y^4$ in field extensions. As an application, for a prime number p, we find solutions to $p=x^4-y^4$ if $p\equiv 11$ (mod 16) and $p^3=x^4-y^4$ if $p\equiv 3$ (mod 16) in all cubic extensions of $\mathbb{Q}(i)$.

Keywords

Diophantine equationsquartic curvessolutions in field extensions

MSC classification

Primary: 14G25: Global ground fields
Secondary: 14H50: Plane and space curves

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 67 , Issue 1 , February 2024 , pp. 142 - 150
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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