Some results concerning Voronoĭ's continued fraction over
Author:
H. C. Williams
Journal:
Math. Comp. 36 (1981), 631652
MSC:
Primary 12A45
MathSciNet review:
606521
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Abstract: Let D be a cubefree integer and let be the fundamental unit of the pure cubic field . It is well known that Voronoi's algorithm can be used to determine . In this work several results concerning Voronoi's algorithm in are derived and it is shown how these results can be used to increase the speed of calculating for many values of D. Among these D values are those such that is not a prime and the class number of is not divisible by 3. A frequency table of all class numbers not divisible by 3 for all with is also presented.
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G. F. Voronoi, On a Generalization of the Algorithm of Continued Fractions, Doctoral Dissertation, Warsaw, 1896. (Russian).
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H.
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evaluating 𝐿(1,𝜒) and the class number of a real quadratic
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(1976), no. 136, 887–893. MR 0414522
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Comp. 35 (1980), no. 152, 1423–1434. MR 583520
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 [1]
 Pierre Barrucand & Harvey Cohn, "A rational genus, class divisibility, and unit theory for pure cubic fields," J. Number Theory, v. 2, 1970, pp. 721. MR 0249398 (40:2643)
 [2]
 Pierre Barrucand & Harvey Cohn, "Remarks on principal factors in a relative cubic field," J. Number Theory, v. 3, 1971, pp. 226239. MR 0276197 (43:1945)
 [3]
 Pierre Barrucand, H. C. Williams & L. Baniuk, "A computational technique for determining the class number of a pure cubic field," Math. Comp., v. 30, 1976, pp. 312323. MR 0392913 (52:13726)
 [4]
 H. Brunotte, J. Klingen & M. Steurich, "Einige Bemerkungen zu Einheiten in reinen kubischen Körpern," Arch. Math., v. 29, 1977, pp. 154157. MR 0457399 (56:15604)
 [5]
 B. N. Delone & D. K. Faddeev, The Theory of Irrationalities of the Third Degree, Transl. Math. Monos., Vol. 10, Amer. Math. Soc., Providence, R. I., 1964. MR 0160744 (28:3955)
 [6]
 F. HalterKoch, "Eine Bemerkung über kubische Einheiten," Arch. Math., v. 27, 1976, pp. 593595. MR 0429827 (55:2837)
 [7]
 Taira Honda, "Pure cubic fields whose class numbers are multiples of three," J. Number Theory, v. 3, 1971, pp. 712. MR 0292795 (45:1877)
 [8]
 N. S. Jeans & M. D. Hendy, Some Inequalities Related to the Determination of the Fundamental Unit of a Pure Cubic Field, Occasional Publications in Mathematics No. 7, Massey University, Palmerston North, New Zealand, 1979.
 [9]
 Oskar Perron, Die Lehre von den Kettenbrüchen, 2nd ed., Munich, 1929; reprint, Chelsea, New York.
 [10]
 R. Steiner, "On the units in algebraic number fields," Proc. 6th Manitoba Conf. on Numerical Math., Utilitas Math., Winnipeg, 1976, pp. 413435. MR 532716 (81b:12008)
 [11]
 G. F. Voronoi, On a Generalization of the Algorithm of Continued Fractions, Doctoral Dissertation, Warsaw, 1896. (Russian).
 [12]
 H. C. Williams & J. Broere, "A computational technique for evaluating and the class number of a real quadratic field," Math. Comp., v. 30, 1976, pp. 887893. MR 0414522 (54:2623)
 [13]
 H. C. Williams, G. Cormack & E. Seah, "Computation of the regulator of a pure cubic field," Math. Comp., v. 34, 1980, pp. 567611. MR 559205 (81d:12003)
 [14]
 H. C. Williams, "Improving the speed of calculating the regulator of certain pure cubic fields," Math. Comp., v. 34, 1980, pp. 14231434. MR 583520 (82a:12003)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198106065217
PII:
S 00255718(1981)06065217
Article copyright:
© Copyright 1981
American Mathematical Society
