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

Abstract

For a non-square positive integer d with 4 d , put ω(d) := (1+ √ d)/2 if d is congruent to 1 modulo 4 and ω(d) := √ d otherwise. Let a1, a2, . . . , a`−1 be the symmetric part of the simple continued fraction expansion of ω(d). We say that the sequence a1, a2, . . . , a[`/2] is the primary symmetric part of the simple continued fraction expansion of ω(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 ` of minimal type for each even `≥ 6.