Journal of Integer Sequences, Vol. 18 (2015), Article 15.9.5

Periodic Continued Fractions and Kronecker Symbols

Kurt Girstmair
Institut für Mathematik
Universität Innsbruck
Technikerstr. 13/7
A-6020 Innsbruck


We study the Kronecker symbol 􏰁(s|t)􏰂 for the sequence of the convergents s/t of a purely periodic continued fraction expansion. Whereas the corresponding sequence of Jacobi symbols is always periodic, it turns out that the sequence of Kronecker symbols may be aperiodic. Our main result describes the period length in the periodic case in terms of the period length of the sequence of Jacobi symbols and gives a necessary and sufficient condition for the occurrence of the aperiodic case.

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Received May 15 2015; revised versions received August 19 2015; August 20 2015. Published in Journal of Integer Sequences, August 20 2015.

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