## On the period length of some special continued fractions.(English)Zbl 0766.11003

For positive integers $$r$$, $$s$$ with $$s>r$$ let $$M(r,s)$$ be the length of the simple continued fraction expansion of $$s/r$$. Let $$gcd(a,q)=1$$ where $$a$$ and $$q$$ are positive integers with $$a=1$$.
Define natural numbers $$s_ i$$ by $$s_ i\equiv a^ i\pmod q$$ and $$0<s_ i<q$$. Further let $$W(a,q)=2\sum_{i=1}^ \omega \lfloor(M(s_ i,q)+1)/2\rfloor$$, where $$\omega$$ is the order of $$a$$ modulo $$q$$, and $$\lfloor\;\rfloor$$ denotes the greatest integer function. If $$q=a^ m\pm 1$$, then, it is easy to evaluate $$W(a,q)$$, e.g. $$W*a,a^ m+1)=2m-2$$. In the paper the authors show how to develop formulas for determining the value of $$W(a,(a^ m\pm1)/q)$$ in terms of $$W(a,q)$$ when $$a^ m\equiv\pm 1\pmod q$$.
Reviewer: Péter Kiss (Eger)

### MSC:

 11A55 Continued fractions 11J70 Continued fractions and generalizations

### Keywords:

period length; continued fraction expansion
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### References:

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