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The period of convergents modulo m of reduced quadratic irrationals. (English) Zbl 0734.11006

Let \(\alpha\) be a quadratic irrational number such that its simple continued fraction expansion is purely periodic and denote by \(p_ n/q_ n\) the n-th convergent of \(\alpha\). The authors show that for any positive integer m the sequence of convergents \(p_ n/q_ n\) is purely periodic modulo m. Denoting by k(m) the minimal period length of this sequence modulo m, many interesting results are proved for k(m). Among others it is shown that k(m) has similar properties as the minimal period length of the Fibonacci sequence modulo m. Some open problems are also presented.
Reviewer: P.Kiss (Eger)

MSC:

11A55 Continued fractions
11B50 Sequences (mod \(m\))
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