# Andrej Dujella: Newton's approximants and continued fractions

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 .