From: utzoo!decvax!ucbvax!G:shallit
Newsgroups: net.math
Title: Re: puzzle
Article-I.D.: populi.190
Posted: Sat May 29 21:51:57 1982
Received: Sun May 30 03:00:33 1982

To answer your questions, it is not hard to prove that the continued
fraction for x terminates if and only if x is rational.  One way to see
this is to associate CF's with the Euclidean algorithm for GCD; the two
algorithms are sort of "mirrors" of each other, since in one you are
interested in remainders; in the other, the quotients.

Second, if the CF for x is eventually periodic, then x is the root of
a quadratic equation.  The converse is also true.

Third, it is true that e = [ 2, 1, 2, 1, 1, 4, 1, 1, 6, ... ].  This
representation is due to Euler and Hurwitz.  There are similar
expansions of interest for exp (1/m) where m is a positive integer
and related transcendental quantities.

Two books that have interesting things about CF's are "Continued Fractions
by C. D. Olds, and Knuth, Art of Computer Programming, V. II.
/Jeff Shallit, Department of Mathematics, University of California, Berkeley.