REAL QUADRATIC FIELDS, CONTINUED FRACTIONS, AND A CONSTRUCTION OF PRIMARY SYMMETRIC PARTS OF ELE TYPE

摘要

For a non-square positive integer d with 4 inverted iota d, put omega(d) := (1 + root d)/2 if d is congruent to 1 modulo 4 and omega(d):= root d otherwise. Let a(1), a(2),..., a(l-1) be the symmetric part of the simple continued fraction expansion of omega(d). We say that the sequence a(1), a(2),..., a([l/2]) is the primary symmetric part of the simple continued fraction expansion of omega(d). A notion of 'ELE type' for a finite sequence was introduced in Kawamoto et al (Comment. Math. Univ. St. Pauli 64(2) (2015), 131-155). The aims of this paper are to introduce a notion of 'pre-ELE type' for a finite sequence and to give a way of constructing primary symmetric parts of ELE type. As a byproduct, we show that there exist infinitely many real quadratic fields with period l of minimal type for each even l >= 6.