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Continued fractions of transcendental numbers

Published online by Cambridge University Press:  26 February 2010

A. Baker
A. Baker
Affiliation:
Trinity College, Cambridge
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Extract

It is well known that if, in a continued fraction

there is a subsequence of the an which increases very rapidly, then ξ is a transcendental number. A result of this kind follows from Liouville's Theorem on rational approximations to algebraic numbers, but the most precise result so far established is that which was deduced from Roth's Theorem by Davenport and Roth [1]. They proved (Theorem 3) that if ξ is algebraic, then

where qn is the denominator of the nth convergent to (1). Thus if

ξ is transcendental.

Type
Research Article
Information
Mathematika , Volume 9 , Issue 1 , June 1962 , pp. 1 - 8
Copyright
Copyright © University College London 1962

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References

1.Davenport, H. and Roth, K. P., “Rational approximations to algebraic numbers”, Mathematika, 2 (1955), 160–167.CrossRefGoogle Scholar
2.LeVeque, W. J., Topics in number theory (Reading, Mass., 1956), Vol. 2, Ch. 4.Google Scholar
3.Maillet, E., Introduction àla théorie des nombres transcendants (Paris, 1906), Chapitre VII.Google Scholar
4.Perron, O., Die Lehre von den Kettenbrüchen (Leipzig und Berlin, 1929), Kap. 4.Google Scholar