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I am reading this article:

https://www.ams.org/notices/199612/pomerance.pdf

The author mentions that trying to factor a number finding two squares such that its difference is the number is worse than trial division because most numbers have a small factor but only a fraction of them have a divisor near their root.

I am wondering how one can make this statement more precise.

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  • $\begingroup$ The theory of smooth numbers is useful for the number of numbers with a small factor. I am not sure how to estimate the number of numbers with a factor "near" the square root. $\endgroup$ – Peter Sep 19 '19 at 17:49

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