Quote:
Originally Posted by fivemack
You don't need the lower series  you just need to advance the high series until the difference is _some_ square, and checking whether a number is a square isn't too difficult. If a number is a possiblesquaremodp for lots of p, it's pretty likely to be square, and each p rules out about half the numbers ... you can keep track of the values of the high series modulo lots of p quite efficiently since you're only doing addition.

Thanks fivemack, but I realize that I can do addition/testing for a long time to find the right lower square. I'm looking for a way to calculate the coincidence rather than stepping to it, which would take a thousand years for some of the larger composites. I've been trying to see if I can find a way via modular means, but I'm too darn rusty with any background in math I may have had long ago. Maybe there isn't a way to do what I'm looking for and that's why I can't find it.
In a way, I'm picturing two graphs converging to the crossover, but the two series are really just two sections of the same graph, without the composite offset. If I can "visualize" a way to work this, maybe a function can be derived. I suspect, that function already is known by everyone, but I don't know enough to describe it properly or recognize it, yet.
Thank you very much for all your help.