purely periodic continued fractions
We know that periodic continued fractions represent quadratic irrationals; this article characterizes purely periodic continued fractions. We will use freely the results on convergents to a continued fraction.
Suppose first that is represented by a purely periodic continued fraction
Note that since it appears again in the continued fraction. Thus . The complete convergent is again , so that we have
Now suppose that and , and let the continued fraction for be . Let be the complete convergent of , and . Thus . Then
so that . Inductively, we have for all . Suppose now that the continued fraction for is not purely periodic, but rather has the form
for . Then and so
But , otherwise would have been the first element of the repeating period. Thus is a nonzero integer and thus is as well. But , which is a contradiction. Thus and the continued fraction is purely periodic. ∎
- 1 A.M. Rockett & P. Szüsz, Continued Fractions, World Scientific Publishing, 1992.
|Title||purely periodic continued fractions|
|Date of creation||2013-03-22 18:04:44|
|Last modified on||2013-03-22 18:04:44|
|Last modified by||rm50 (10146)|