On a Sequence Arising in Series for π

  • Morris Newman
  • Daniel Shanks
Chapter

Abstract

In a recent investigation of dihedral quartic fields [6] a rational sequcnce (an) was encountered. We show that these a„ arc positive integers and that thcy satisfy surprising congruences modulo a prime p. Thcy generate unknown p-adic numbers and may therefore be comparcd with the cubic recurrences in [1], where the corresponding p-adic numbers are known completely [2]. Other unsolved problems are presented. Thc growth of thc a„ is examined and a new algorithm for computing a„ is givcn. An appcndix by D. Zagier, which carries thc investigation furthcr, is added.

Keywords

Modular Form Eisenstein Series Modular Function Congruence Subgroup General Congruence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Morris Newman
    • 1
    • 2
  • Daniel Shanks
    • 1
    • 2
  1. 1.Department of MathematicsUniversity of CaliforniaSanta BarbaraUSA
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA

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