It seems like finding these points and finding factors would take longer than the book says. I wasn't quite sure when the book said that they looked at 10000!P. Wouldn't that take a long time to do? This seems like one of those things where you just have to try it yourself to see that it works. Did the book choose convenient cases, or is it pretty fast to factor smaller numbers?
I got a little lost when the book started talking about singular curves. I understand that the equation has multiple roots, but should we try to use a singular curve to factor n or should we avoid them?
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