periodic continued fractions represent quadratic irrationals
A periodic simple continued fraction is a simple continued fraction
such that for some there is such that whenever , we have . Informally, a periodic continued fraction is one that eventually repeats. A purely periodic simple continued fraction is one for which ; that is, one whose repeating period starts with the initial element.
is a periodic continued fraction, we write it as
The forward direction is pretty straightforward. Given such a continued fraction, let be the complete convergent, i.e.
Note first that must be irrational since the continued fraction for any rational number terminates. Then the article on convergents to a continued fraction shows that
where the are the convergents to the continued fraction for . Thus
and thus is irrational and satisfies a quadratic equation so is a quadratic irrational. A simple computation then shows that is as well.
In the other direction, suppose that is a quadratic irrational satisfying
and with continued fraction representation
Then for any , we have
where the are the complete convergents of the continued fraction, so that from the quadratic equation we have
Note that for each since otherwise
and the quadratic equation would have a rational root, contradicting the fact that is irrational.
The remainder of the proof is an elaborate computation that shows we can bound each of independent of . Assuming that, it follows that there are only a finite number of possibilities for the triples , so we can choose such that
Then each of is a root of (say)
so that two of them must be equal. But then (say), and
and the continued fraction is periodic.
We proceed to find the bounds. We know that
for some (depending on ) with . Thus
and thus also
It remains to bound . But
Substituting the values of on the right, and using the fact that
we get after a computation
and we have thus bounded all of independent of . ∎
- 1 G.H. Hardy & E.M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Oxford Science Publications, 1979.
|Title||periodic continued fractions represent quadratic irrationals|
|Date of creation||2013-03-22 18:04:40|
|Last modified on||2013-03-22 18:04:40|
|Last modified by||rm50 (10146)|
|Defines||periodic continued fraction|