On a Sequence Arising in Series for π

  • Morris Newman
  • Daniel Shanks


In a recent investigation of dihedral quartic fields [6] a rational sequence {a n } was encountered. We show that these a n are positive integers and that they satisfy surprising congruences modulo a prime p. They generate unknown p-adic numbers and may therefore be compared with the cubic recurrences in [1], where the corresponding p-adic numbers arc known completely [2]. Other unsolved problems are presented. The growth of the a n is examined and a new algorithm for computing a n is given. An appendix by D. Zagier, which carries the investigation further, is added.


Modular Form Eisenstein Series Modular Function Congruence Subgroup General Congruence 
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Copyright information

© Springer Science+Business Media New York 1984

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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