Let be a positive integer which is not a perfect square.
Let denote the -th convergent and the length
of the shortest period in the simple continued fraction expansion
of .
It was proved by Mikusinski in 1954 that if
,
then all Newton's approximants

are convergents of , and moreover
for all nonegative integers .
If is a convergent of , then we say that is a
"good approximant". In 2001, we proved the converse of Mikusinski's
result, namely that if all approximants are good, then
.
It is easy to see that
. Therefore, good approximants
satisfy
for an integer .
If
, then
. For ,
we proved the upper bound
, and we presented a sequence of 's
(given in terms of Fibonacci numbers) which shows that this
upper bound for is sharp.
Let denote the number of good approximants among the numbers
,
. We will present some results and
conjectures (based on experimental data) about the magnitude of
compared with and .
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Please send comments and corrections to Thomas Klausner.