Amateur Mathematician wrote: These patterns are easy to derive, yes, ...
Out of curiosity, what is the size of the numbers you are dealing with?
OK, thanks. Are you aware of them having been described before? I've been through a lot of recent peer reviewed work discussing patterns in prime numbers and have not found this particular angle described. But perhaps my search has not been exhaustive.
I'm actually not especially interested in working on finding big prime numbers. I just like working with patterns in seemingly disordered systems.