Newton's Formula and the Continued Fraction Expansion of √d

Householder's approximants and continued fraction expansion of quadratic irrationals

“…These results imply that if ℓ( [2] proved the converse of this result. Namely, if all approximants R…”

“…we know that some of approximants R (2) n are convergents and some of them are not. Using a result of Komatsu [8] from 1999, Dujella also showed that being a good approximant is a periodic and a palindromic property, so he defined the number b( √ d) as the number of good approximants in the period.…”

“…Using a result of Komatsu [8] from 1999, Dujella also showed that being a good approximant is a periodic and a palindromic property, so he defined the number b( √ d) as the number of good approximants in the period. Formulas (1.4) and (1.5) suggest that R (2) n should be convergent whose index is twice as large when it is a good approximant. However, this is not always true, and Dujella defined the number j( √ d) as half of the distance from two times larger index.…”