http://qs1969.pair.com?node_id=432985


in reply to Re^4: Closest factor less than sqrt(N) [proof]
in thread OT: Finding Factor Closest To Square Root

Here's an easier way: take n = m^2, and a factor greater than the square root m + d, then the other factor is:

m^2/(m + d) = m - dm/(m + d) = m - d + d^2/(m + d)
Now d^2/(m + d) is greater than zero, so the cofactor is greater than m - d and hence closer to the square root.

Hugo