## Abstract

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.

## 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 1984