Characterization of a generalized Shanks sequence

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

doi:10.2140/pjm.2007.230-185

Characterization of a generalized Shanks sequence

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

Research output: Contribution to journal › Article › peer-review

5 Citations (Scopus)

Abstract

We consider generalizations of Shanks' sequence of quadratic fields ℚ(√S_n) where S_n = (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 D_n = 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 language	English
Pages (from-to)	185-215
Number of pages	31
Journal	Pacific Journal of Mathematics
Volume	230
Issue number	1
DOIs	https://doi.org/10.2140/pjm.2007.230-185
Publication status	Published - 2007

Keywords

Periodic continued fraction
Quadratic order
Units

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10.2140/pjm.2007.230-185

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AB - 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.

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KW - Quadratic order

KW - Units

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Characterization of a generalized Shanks sequence

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Keywords

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