Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A note on a theorem of Perron


Authors: K. C. Prasad and M. Lari
Journal: Proc. Amer. Math. Soc. 97 (1986), 19-20
MSC: Primary 11J70
MathSciNet review: 831378
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Abstract: Given an infinite simple continued fraction $ \left[ {{a_0},{a_1}, \ldots ,{a_n}, \ldots } \right]$, let $ {M_n}$ denote $ \left[ {0,{a_n},{a_{n - 1}}, \ldots ,{a_1}} \right] + \left[ {{a_{n + 1}},{a_{n + 2}}, \ldots } \right]$. A well-known result due to Perron [1, III, 212] states: If $ {a_{n + 2}} = m$, then there is a $ k$ in $ \left\{ {n,n + 1,n + 2} \right\}$ for which $ {M_k} > \sqrt {{m^2} + 4} $. In this note we give a new proof for this result and add that there is a $ j$ in $ \left\{ {n,n + 1,n + 2} \right\}$ for which $ {M_j} < \sqrt {{m^2} + 4} $.


References [Enhancements On Off] (What's this?)

  • [1] J. F. Koksma, Diophantische Approximationen, Chelsea, New York, 1936.
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  • [3] F. Bagemihl and J. R. McLaughlin, Generalization of some classical theorems concerning triples of consecutive convergents to simple continued fractions, J. Reine Angew. Math. 221 (1966), 146–149. MR 0183999

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DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0831378-5
Article copyright: © Copyright 1986 American Mathematical Society