Quote Originally Posted by Max Dettweiler View Post
I would think that even if the number has been tested once, a factor for that number can still save us from having to do the secondpass test.
That's right. Yet, it will happen only in a few years, and by that time, there is a good chance that a prime will be found or that sieving would have found that factor. And then, P-1 is not superfast. By the time it takes to find this factor, one could have done this one LLR test; but not two.

So, if one would want to look for factors of Doublecheck numbers, one would do that right before doublecheck hits them, and then again, one would not do it at all, because you are better off with LLR directly.

I hope I look not too wise-assed now... H.