I’m trying to characterize positive integers $c$ and $b$, with $\gcd(c,b) \mid 2$, such that $c^4+4bc^3-4b^4$ is an integer square.
The solutions I’ve found [by brute force calculation] are $$(c,b) = (1,1), (2,2), (17,6), (34,12), (113,66), (226,132), …$$
In lieu of a full characterization, I’m also content [for the purposes of the puzzle I’m working on] to show that $c < 3b$ always holds, which appears to be true from the limited data set I have so far.
I’ve tried writing it out as $$(c^2+2bc)^2 = k^2+(2b^2)^2+(2bc)^2$$ for some integer $k$, and then applying the parameterization of the 2.1.3 Diophantine equation, but nothing useful popped out. Any advice or hints (or complete answers!) would be appreciated.
EDIT: From an earlier, similar question — which I only now recalled, because MSE linked it automatically in the ‘Related’ section! — I see that Fermat’s tangent method should apply. I’ll work on that, but I’d appreciate it if anyone can provide some hints/shortcuts on how to prove (or, I suppose, disprove?!) that $c<3b$, either for all rational solutions or just for the integer solutions.