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 convergentsMathworldPlanetmathPlanetmath to a continued fractionMathworldPlanetmath.

Theorem 1.

(Galois) A quadratic irrational t is represented by a purely periodic simple continued fraction if and only if t>1 and its conjugatePlanetmathPlanetmath s under the transformation d-d satisfies -1<s<0.


Suppose first that t is represented by a purely periodic continued fraction


Note that a01 since it appears again in the continued fraction. Thus t>1. The rth complete convergent is again t, so that we have


so that


Consider the polynomialPlanetmathPlanetmath f(x)=qr-1x2+(qr-2-pr-1)x-pr-2. f(t)=0, so the other root of f(x) is the conjugate s of t. But f(-1)=(pr-1-pr-2)+(qr-1-qr-2)>0 since the pi and the qi are both strictly increasing sequencesMathworldPlanetmath, while f(0)=-pr-2<0. Thus s lies between -1 and 0 and we are done.

Now suppose that t>1 and -1<s<0, and let the continued fraction for t be [a0,a1,]. Let tn be the nth complete convergent of t, and sn=tn¯. Thus s0=s. Then


so that


and thus


so that -1<s1<0. Inductively, we have -1<sn=tn¯<0 for all n0. Suppose now that the continued fraction for t is not purely periodic, but rather has the form


for k1. Then tk=tk+j and so


But ak-1ak+j-1, otherwise ak-1 would have been the first element of the repeating period. Thus tk-1-tk+j-1 is a nonzero integer and thus sk-1-sk+j-1 is as well. But -1<sk-1-sk+j-1<1, which is a contradictionMathworldPlanetmathPlanetmath. Thus k=0 and the continued fraction is purely periodic. ∎


  • 1 A.M. Rockett & P. Szüsz, Continued Fractions, World Scientific Publishing, 1992.
Title purely periodic continued fractions
Canonical name PurelyPeriodicContinuedFractions
Date of creation 2013-03-22 18:04:44
Last modified on 2013-03-22 18:04:44
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 6
Author rm50 (10146)
Entry type Theorem
Classification msc 11Y65
Classification msc 11A55