Let D denote a positive nonsquare integer such that D ≡ 1 (mod4). Let l(√d) (resp. l(1/2(1 + √d))) denote the length of the period of the continued fraction expansion of VS (resp. 1/2(1 + √d)). Recently Ishii, Kaplan, and Williams (On Eisenstein’s problem, Acta Arith. 54 (1990), 323-345) established inequalities between 1(VS) and l(1/2(l + √d)). In this note it is shown that these inequalities are best possible in a strong sense.

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Keywords Continued fractions
Persistent URL dx.doi.org/10.1090/S0002-9939-1994-1169053-7
Journal Proceedings of the American Mathematical Society
Citation
Williams, K.S, & Uck, N. (Nicholasb). (1994). Comparison of the lengths of the continued fractions of √d and 1/2(1 + √d). Proceedings of the American Mathematical Society, 120(4), 995–1002. doi:10.1090/S0002-9939-1994-1169053-7