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0. Introduction. In
this paper we introduce the concept of “Cycles in the Period” of the simple
continued fraction expansion of a real quadratic irrational. This is expressed in
the
Definition. Let M, D, d be positive rational integers, M square free,
M = D2 + d, d ≦ 2D. Let k, a, s be nonnegative rational integers, 0 ≦ a ≦ k − 1; let
f = f(k,a,s;d,D) be a polynomial with rational integral coefficients. For a fixed s,
the finite sequence of polynomials
will be called “Cycle in the Period” of the simple continued fraction expansion of
 if, for s0 ≧ 1, this expansion has the form
![√ --- ------------------------------------------
(0.1) M = [b0,b1,⋅⋅⋅ ,F(0),⋅⋅⋅ ,F (s0 − 1),f(k,a,s0;d,D ),⋅⋅⋅,
f(k,a+-b,s0;d,D-),⋅⋅⋅ ,f(k,a,s0;d,D-),F-′(s0 −-1),⋅⋅⋅ ,
------------------------------
F ′(0),f(k,a − 1,0;d,D ),⋅⋅⋅ ,b1,2b0 ]](a032x.png)
b ≧ 1;b ≦ k − 1;k is the length of the cycle; F′(s) means that the order of the f −s
must be reversed.
In the first part of this paper, the main result is the construction of infinitely
many classes of quadratic fields Q( ), each containing infinitely many M of a
simple structure. Among the various classes thus constructed, there are a few in
whose expansion of  cycles in the period surprisingly have the length ≦ 12.
Functions f(k,a,s;d,D), f(k,a + 1,s;d,D),⋯ are of course stated explicitly; hence
we are able to construct numbers  such that the primitive period of
their expansion has any given length m which is a function of the parameter
k.
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