## 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 are 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. Zagicr, which carries the investigation further, is added.

## Keywords

Modular Form Eisenstein Series Modular Function Modular Group Algebraic Integer
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.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.William Adams & Daniel Shanks, “Strong primality tests that arc not sufficient,”
*Math. Comp*.,*v*. 39, 1982, pp. 255–300.MathSciNetMATHCrossRefGoogle Scholar - 2.William Adams & Daniel Shanks, “Strong primality tests. II—Algebraic identification of the p-adic limits and their implications.” (To appear.)Google Scholar
- 3.H. Behnke & F. Sommer,
*Theorie der analytischen Funktionen einer complexen Veränderlichen*,Springer, Berlin, 1965, viii + 603 pp.Google Scholar - 4.Marvin I. Knopp,
*Modular Functions in Analytic Number Theory*,Markham, Chicago, III., 1970, x + 150 pp.Google Scholar - 5.Derrick H. Lehmer & Emma Lehmer, “Cyclotomy with short periods,”
*Math. Comp*.,*v*. 41, 1983, pp. 743–758.MathSciNetMATHCrossRefGoogle Scholar - 6.Daniel Shanks, “Dihedral quartic approximations and series for
*7r*,*’ J. Number Theory*,*v*. 14, 1982, pp. 397–423.MathSciNetMATHCrossRefGoogle Scholar - 7.Daniel Shanks, “Review of A. O. L. Atkin’s table,”
*Math. Comp*.,*v*. 32, 1978, p. 315.MathSciNetCrossRefGoogle Scholar - 8.Thomas R. Parkin & Daniel Shanks, “On the distribution of parity in the partition function,”
*Math. Comp*.,*v*. 21, 1967, pp. 446–480.MathSciNetMATHCrossRefGoogle Scholar - 9.Daniel Shanks & Larry P. Schmid, “Variations on a theorem of Landau,”
*Math. Comp*.,*v*. 20, 1966, pp. 551–569.Google Scholar - 10.

## Copyright information

© Springer Science+Business Media New York 2000