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)

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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

In reply to Re^5: Closest factor less than sqrt(N) [proof] by hv
in thread OT: Finding Factor Closest To Square Root by QM

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