I have the following equation:
- $A^2-3B^2+7C^2-21D^2=0$
and I have to prove that it does not have any non-trivial solution in $\mathbb{Z}^4$ . Using Congruence I deduced that $X,Z \equiv 0$ (mod $3$) and then plugging it into the forms $X^2 = 9D^2$ and $Z=9E^2$ means that the equation "reproduces" itself with a factor before it. After this I don't know how to proceed.
Does $gcd(1,-3,7,21) = 1 $ have any part to play in this?
Thank you in advance.