Continued Fractions, Algebraic Numbers and Modular Invariants

@article{Churchhouse1969ContinuedFA,
  title={Continued Fractions, Algebraic Numbers and Modular Invariants},
  author={R. F. Churchhouse and S. T. E. Muir},
  journal={Ima Journal of Applied Mathematics},
  year={1969},
  volume={5},
  pages={318-328}
}
Brillhart discovered in 1965 that the continued fraction of the real root of the cubic equation x3—8x—10 = 0 has a number of very large partial quotients. In this paper we explain why this phenomenon is surprising and then show how its consideration naturally leads one into some very deep branches of the theory of numbers before the reason for the phenomenon becomes clear. In order to make the paper intelligible to non-specialists we give a brief account of the classical theories of continued… 
View via Publisher
pdmi.ras.ru
10 Citations
Background Citations
1
Methods Citations
1

Tables from this paper

10 Citations

Citation Type

Citation Type

Continued fractions for some algebraic numbers.
where <*! = 2 cos — =is a root of x -\x — 2x — l, #2 is a root of x — χ — l, OCB = j/2~+ j/3"is a root of X — 9z — 4# + 27z — 36z — 23. The first numbers were chosen s cubic irrationalities, oc1 s…
In the search of convergents to 23
The Minima of Forms and Units by Approximation
The least positive integers such that there exist rational integers representing the absolute values of a reduced binary quadratic form and a reduced binary cubic form can be calculated by the simple…
Computational mathematics and computer science
Some computational aspects of, and the use of computers in, algebraic number theory
TLDR
An explanation is given of how the application of a computer has contributed to the progress of research in this area and several number-theoretic questions are expounded for which the use of acomputer seems to be appropriate but has not extensively been made.
On the method of Thue-Siegel
The methods of Thue and Siegel, based on explicit Pade approximations to algebraic functions, are used to examine diophantine approximations to roots of
L’arithmétique sur une surface percée
RésuméL’article rappelle les principes de l’approximation diophantienne des nombres réels par des nombres rationnels. La mesure de cette approximation et le contexte général de la théorie de Markoff…
Reliable real arithmetic
C167. The probability of odd partial quotients in continued fractions
Transcendental numbers and diophantine approximations

References

Lehrbuch der algebra
Volume 1, Die Grundlagen: Rationale Functionen Determinanten Die Wurzeln algebraischer Gleichungen Symmetrische Functionen Lineare Transformation. Invariation Tschirnhausen-Transformation Die…