## Abstract

In a recent investigation of dihedral quartic fields [6] a rational sequcnce (a_{n}) 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
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## References

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## Copyright information

© Springer Science+Business Media New York 2004