I am having difficulty solving the following problem:
Prove rigorously that there is no integer solution for the Diophantine Equation $x^2 + 3y^2 = 11z^2$ except when $x = y=z = 0$.
Proving directly would seem far to difficult. I would assume proof by contradiction would be the most efficient way to solve this problem. To keep this simple, I shall assume that $(x,y,z)$ is a primitive integer solution. For any prime number $p$, there is a non-zero solution for the equation $x^2 + 3y^2 = 11z^2\pmod p$. Then simply find a good modulus $p$ to deduce a contradiction.
This is where I have a difficult time, trying to deduce the contradiction. Any suggestions?
Thank you for your time, and thanks in advance for your feedback.