Characterization of a generalized Shanks sequence

Roger D. Patterson, Alfred J. van der Poorten, Hugh C. Williams

    Research output: Contribution to journalArticlepeer-review

    5 Citations (Scopus)


    We consider generalizations of Shanks' sequence of quadratic fields ℚ(√Sn) where Sn = (2n + 1)2 + 2n+2. Quadratic fields of this type are of interest because it is possible to explicitly determine the fundamental unit. If a sequence of quadratic fields given by Dn = A2x2n+ Bxn+C2 satisfies certain conditions (notably that the regulator is of order {circled dash}(n2)), then we determine the exact form such a sequence must take.

    Original languageEnglish
    Pages (from-to)185-215
    Number of pages31
    JournalPacific Journal of Mathematics
    Issue number1
    Publication statusPublished - 2007


    • Periodic continued fraction
    • Quadratic order
    • Units


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