Vol. 63, No. 1, 1976

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Fundamental units and cycles in the period of real quadratic number fields. I

Leon Bernstein

Vol. 63 (1976), No. 1, 37–61

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

(0.1)   F (s) = f(k,a,s;d,D ),f(k,a+ 1,s;d,D),⋅⋅⋅ ,f (k,a + k− 1,s;d,D )

will be called “Cycle in the Period” of the simple continued fraction expansion of   ---
√ M 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 ]

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(√ ---
M), each containing infinitely many M of a simple structure. Among the various classes thus constructed, there are a few in whose expansion of √---
M 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 √ ---
M such that the primitive period of their expansion has any given length m which is a function of the parameter k.

Mathematical Subject Classification
Primary: 12A45, 12A45
Secondary: 10A30
Published: 1 March 1976
Leon Bernstein