On the Biquadratic Diophantine Equation x^4 + 9x^2 y^2 + 27 y^4 = z^2.

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.