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An introduction to continued fractions. (English) Zbl 0596.10008

Diophantine analysis, Proc. Number Theory Sect. Aust. Math. Soc. Conv., Univ. New South Wales 1985, Lond. Math. Soc. Lect. Note Ser. 109, 99-138 (1986).
[For the entire collection see Zbl 0583.00005.]
Let \([c_ 0;c_ 1,c_ 2,...]\), \(c_ 0\in {\mathbb{Z}}\), \(c_ 1,c_ 2,...\in {\mathbb{N}}\) be a regular continued fraction with sequence of convergents \((p_ n/q_ n)_{n\geq -1}\). More and more one begins to realize that the elementary arithmetic of continued fractions becomes much more transparent if one observes that \[ \left( \begin{matrix} c_ 0\\ 1\end{matrix} \begin{matrix} 1\\ 0\end{matrix} \right)\left( \begin{matrix} c_ 1\\ 1\end{matrix} \begin{matrix} 1\\ 0\end{matrix} \right)\left( \begin{matrix} c_ 2\\ 1\end{matrix} \begin{matrix} 1\\ 0\end{matrix} \right)...\left( \begin{matrix} c_ n\\ 1\end{matrix} \begin{matrix} 1\\ 0\end{matrix} \right)=\left( \begin{matrix} p_ n\\ q_ n\end{matrix} \begin{matrix} p_{n-1}\\ q_{n-1}\end{matrix} \right),\quad n\geq 0. \] This expository paper gives an introduction to the theory of the regular continued fraction based on the above formula. Subjects treated are e.g. periodic continued fraction and the relation between the regular continued fraction for the number \(2\sum^{\infty}_{n=0}2^{-2^ n}\) and the paper folding sequence. The paper also contains the delightful but little known proof of H. J. S. Smith (1855) of Fermat’s result that every prime \(p\equiv 1(mod 4)\) is the sum of two squares.
The reviewer recommends this article to everyone who has to talk on continued fractions in an elementary course.
Reviewer: H.Jager

MSC:

11A55 Continued fractions
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.

Citations:

Zbl 0583.00005