Download the whole paper in PDF format

Chatchawan Panraksa,
On the Biquadratic Diophantine Equation x^4 + 9x^2 y^2 + 27 y^4 = z^2.
Int. J. Math. Comput. Sci., 20, no. 4, (2025), 1053-1056.

DOI:

https://doi.org/10.69793/ijmcs/04.2025/cp

Keywords and phrases:

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

Abstract:

We demonstrate that the biquadratic Diophantine equation x^4 + 9x^2 y^2 + 27 y^4 = z^2 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.