<s>[INST] If $4xy+3=c^2+3d^2$, is $xy$ necessarily a square? I have a polynomial which, simplified, ends up in the form
$$4xy+3 = c^2+3d^2.$$
Evidently $4xy+3$ is of the form $a^2+3b^2$, in light of the equality. But does
$$
    c^2 + 3d^2 = 4xy + 3 = xy(2)^2 + 3(1)^2
$$
necessarily force $xy$ to be a square? I just can't see how to prove it.
Thanks.
 [/INST] For the equation: $4xy+3=c^2+3d^2$
Can be written like such a simple solution:
$x=2a^2+6t^2-6at-2a+3t-1$
$y=2a^2+6t^2-6at-a+3t-1$
$c=2a^2+6t^2-6at-2$
$d=2a^2+6t^2-6at-2a+4t-1$
$a,t$ - what some integers any sign.
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