Continued Fractions, Algebraic Numbers and Modular Invariants
- 1 September 1969
- journal article
- research article
- Published by Oxford University Press (OUP) in IMA Journal of Applied Mathematics
- Vol. 5 (3) , 318-328
- https://doi.org/10.1093/imamat/5.3.318
Abstract
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 fractions, quadratic forms and modular functions in the appropriate sections.Keywords
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