I became interested in the nearest square (NSCF), nearest integer
(NICF) and optimal continued fraction (OCF) through Jim White of the
ANU, around October 2007. Coincidentally John Robertson
had also been studying the NSCF and NICF and joined the discussion.
NICF refers here to a half-regular continued fraction where the partial
numerators are allowed to be ±1, in contrast to the nearest integer
continued fraction of Adolf Hurwitz and B. Minnegerode, where the
partial numerators are all -1. I should also mention Selenius' ideal
relative approximation continued fraction (IACF or SK) which is
the same as the OCF, but somewhat harder to describe. Selenius first
obtains the regular continued fraction (RCF) and replaces all 1's by a
process of *singularisation*.
For quadratic surds, the period-lengths of OCF and SK are sometimes
double that of the period-length of the corresponding NSCF. Contrary to
a disparaging review by D.H. Lehmer of A.A.K. Ayyangar's paper below,
NSCF has some nice properties. For example, the period-length of the
NSCF expansion for √D is not greater than and can be much shorter than
the RCF period-length.
Also there are three mid-point criteria for solving Pell's equation -
see paper 13 below.