The length of the period of the simple continued fraction of d^1∕2

John H. E. Cohn

Vol. 71 (1977), No. 1, 21–32

DOI: 10.2140/pjm.1977.71.21

Abstract

Let p(d) denote the length of the period of the simple continued fraction for d^1∕2 and 𝜖 the fundamental unit in the ring Z [d^1∕2]. We prove that as d →∞,

Theorem 1. p(d) ≦ 7∕2π⁻²d^1∕2 log d + O(d^1∕2).

Theorem 2. log 𝜖 ≦ 3π⁻²d^1∕2 log d + O(d^1∕2).

Theorem 3. p(d)≠o(d^1∕2∕log log d).

Theorem 4. If log 𝜖≠o(d^1∕2 log d) then also p(d)≠o(d^1∕2 log d).

Mathematical Subject Classification

Primary: 10A30, 10A30

Secondary: 12A45

Milestones

Received: 23 April 1974

Revised: 21 December 1976

Published: 1 July 1977