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The numbers I wish to compute range around 188 multiplied by 3 to the power of 6548 plus 1, as an example. so lets say in excess of 3000 digits.

My problem with this is, using my quad-core machine, I'm finding my petty division through subtraction routine fairly slow, and (x+a/x)/2 as a solution is failing badly.

Can someone please find time to discuss with me a way of doing this?

Can a prime have a GCD with a smaller number(like the sqrt), I somehow just don't see it...

So if there is an easy solution, please let me know it =)