On the Biquadratic Diophantine Equation x4 + 9x2y2 + 27y4 = z2

Chatchawan Panraksa

doi:10.69793/ijmcs/04.2025/cp

On the Biquadratic Diophantine Equation x⁴ + 9x²y² + 27y⁴ = z²

Chatchawan Panraksa

Science Division

Research output: Contribution to journal › Article › peer-review

Abstract

We demonstrate that the biquadratic Diophantine equation x⁴ + 9x²y² + 27y⁴ = z² 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.

Original language	English
Pages (from-to)	1053-1056
Number of pages	4
Journal	International Journal of Mathematics and Computer Science
Volume	20
Issue number	4
DOIs	https://doi.org/10.69793/ijmcs/04.2025/cp
Publication status	Published - Jan 2025

Keywords

3-adic valuations
Diophantine equations
Fermat descent
local obstructions
quartic Thue equations

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10.69793/ijmcs/04.2025/cp

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abstract = "We demonstrate that the biquadratic Diophantine equation x4 + 9x2y2 + 27y4 = z2 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.",

keywords = "3-adic valuations, Diophantine equations, Fermat descent, local obstructions, quartic Thue equations",

author = "Chatchawan Panraksa",

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doi = "10.69793/ijmcs/04.2025/cp",

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N2 - We demonstrate that the biquadratic Diophantine equation x4 + 9x2y2 + 27y4 = z2 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.

AB - We demonstrate that the biquadratic Diophantine equation x4 + 9x2y2 + 27y4 = z2 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.

KW - 3-adic valuations

KW - Diophantine equations

KW - Fermat descent

KW - local obstructions

KW - quartic Thue equations

UR - https://www.scopus.com/pages/publications/105023066537

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JO - International Journal of Mathematics and Computer Science

JF - International Journal of Mathematics and Computer Science

IS - 4

ER -

On the Biquadratic Diophantine Equation x⁴ + 9x²y² + 27y⁴ = z²

Abstract

Keywords

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