On the Biquadratic Diophantine Equation x4 + 9x2y2 + 27y4 = z2
Issued Date
2025-01-01
Resource Type
ISSN
18140424
eISSN
18140432
Scopus ID
2-s2.0-105023066537
Journal Title
International Journal of Mathematics and Computer Science
Volume
20
Issue
4
Start Page
1053
End Page
1056
Rights Holder(s)
SCOPUS
Bibliographic Citation
International Journal of Mathematics and Computer Science Vol.20 No.4 (2025) , 1053-1056
Suggested Citation
Panraksa C. On the Biquadratic Diophantine Equation x4 + 9x2y2 + 27y4 = z2. International Journal of Mathematics and Computer Science Vol.20 No.4 (2025) , 1053-1056. 1056. doi:10.69793/ijmcs/04.2025/cp Retrieved from: https://repository.li.mahidol.ac.th/handle/123456789/113381
Title
On the Biquadratic Diophantine Equation x4 + 9x2y2 + 27y4 = z2
Author(s)
Author's Affiliation
Corresponding Author(s)
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Abstract
We demonstrate that the biquadratic Diophantine equation x<sup>4</sup> + 9x<sup>2</sup>y<sup>2</sup> + 27y<sup>4</sup> = z<sup>2</sup> admits no non-trivial positive integer solutions. Employing a Fermat-style infinite descent, our proof combines congruences modulo 8 and 9, 3-adic valuations, and three distinct difference-of-squares factorizations to reveal local obstructions, culminating in the descent argument. This approach not only solves the equation but also exemplifies how tailored algebraic identities can unlock solutions to challenging quartic Thue equations.