In a recent investigation of dihedral quartic fields  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 , where the corresponding p-adic numbers are known completely . 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.
Modular Form Eisenstein Series Modular Function Congruence Subgroup General Congruence
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