I was looking at this webpage which lists the first 200 Lucas Numbers color-coded with their prime factors and I noticed that all the Lucas numbers with power of two or prime indexes were relatively prime to all previous Lucas numbers.

Clearly the author(s) of the page know this too because they did the color-coding, which shows the pattern clearly. I was wondering if this pattern holds for all Lucas numbers, and where I might find a proof or counterexample? My Google-Fu hasn't turned up anything on this particular question. Is this an open problem?