I have the following
Conjecture: If $w$ and $z$ are non-negative integers satisfying the equation $$ w(w+6) = z(16z^2+36z+27), \tag{$\star$} $$ then $w=z=0$.
I believe it to be true for the following two reasons:
It is a reformulation of a known result. (n.b. I don't want to share the original result here, to avoid influencing answers.)
I've done computer verification for all $w,z < 10000$
What are some methods of attack on this type of question? Ideally, I'm looking for a descent-type method, by which I can show that $w=0$ or $z=0$ [or both] or $w=z$ [which would evidently force $w=z=0$].
EDIT: Here's the kind of thing I'm playing with…
Let $w=ab$ and $w+6=cd$ for positive integers $a,b,c,d$. Then ($\star$) implies $z=ac$ and $16z^2+36z+27=bd$. Now obviously $cd-ab=6$, which has a general solution; I could plug in that solution and hack away to try to show that $ab=0$. Also, it's trivial that $16a^2c^2 + 36ac + 27 = bd$. But so far, the effort this approach requires seems too inelegant for my taste.
You are trying to find integer points on an elliptic curve - this is a much studied problem, see, for example: https://mathoverflow.net/questions/7907/how-to-find-all-integer-points-on-an-elliptic-curve