Vol. 71, No. 1, 1977

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The length of the period of the simple continued fraction of d12

John H. E. Cohn

Vol. 71 (1977), No. 1, 21–32
Abstract

Let p(d) denote the length of the period of the simple continued fraction for d12 and 𝜖 the fundamental unit in the ring Z [d12]. We prove that as d →∞,

Theorem 1. p(d) 72π2d12 log d + O(d12).

Theorem 2. log 𝜖 3π2d12 log d + O(d12).

Theorem 3. p(d)o(d12log log d).

Theorem 4. If log 𝜖o(d12 log d) then also p(d)o(d12 log d).

Mathematical Subject Classification
Primary: 10A30, 10A30
Secondary: 12A45
Milestones
Received: 23 April 1974
Revised: 21 December 1976
Published: 1 July 1977
Authors
John H. E. Cohn