# On a theorem of Serret on continued fractions

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## Abstract

A classical theorem in continued fractions due to Serret shows that for any two irrational numbers x and y related by a transformation $\gamma$ in PGL(2,Z) there exist s and t for which the complete quotients x_s and y_t coincide. In this paper we give an upper bound in terms of $\gamma$ for the smallest indices s and t.
Original language English 379–384 6 Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 110 2 14 Jul 2015 https://doi.org/10.1007/s13398-015-0238-2 Published - 1 Sep 2016

### Bibliographical note

© Springer-Verlag Italia, 2015. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details

## Keywords

• math.NT
• continued fractions
• PGL (2,z)-equivalent numbers