Hi! This is my first post on Math Overflow. I have two equations: $a(3a-1) + b(3b-1) = c(3c-1)$ and $a(3a-1) - b(3b-1) = d(3d-1)$. I'm trying to find properties of $a$ and $b$ that lead to solutions, where $a, b, c, d \in \mathbb{N}$. I'm having trouble applying any of the techniques in my abstract algebra book, as they mostly only apply to linear Diophantine equations.

So far, I only really have managed to deduce the following things:

$2b(3b-1) + d(3d-1) = a(3a-1) +b(3b-1)$

$2b(3b-1) = c(3c-1) - d(3d-1)$

$2b(3b-1) = (c-d)(3(c+d)-1)$

Any ideas on where to go from here would be greatly appreciated. Thanks!