Some periodic continued fractions with long periods

Authors:
C. D. Patterson and H. C. Williams

Journal:
Math. Comp. **44** (1985), 523-532

MSC:
Primary 11Y65; Secondary 11A55, 11J70

DOI:
https://doi.org/10.1090/S0025-5718-1985-0777283-1

MathSciNet review:
777283

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be the period length of the continued fraction for . Under the extended Riemann Hypothesis for one would expect that . In order to test this it is necessary to find values of *D* for which is large. This, in turn, requires that we be able to find solutions to large sets of simultaneous linear congruences. The University of Manitoba Sieve Unit (UMSU), a machine similar to D. H. Lehmer's DLS-127, was used to find such values of *D*. For example, if , then 25679652. Some results are also obtained for the Voronoi continued fraction for .

**[1]**P. Barrucand & H. Cohn, "Remarks on principal factors in a relative cubic field,"*J. Number Theory*, v. 3, 1971, pp. 226-239. MR**0276197 (43:1945)****[2]**T. Honda, "Pure cubic fields whose class numbers are multiples of three,"*J. Number Theory*, v. 3, 1971, pp. 7-12. MR**0292795 (45:1877)****[3]**D. H. Lehmer, "A history of the sieve process," in*A History of Computing in the Twentieth Century*, Academic Press, New York, 1980, pp. 445-456.**[4]**D. H. Lehmer, E. Lehmer & D. Shanks, "Integer sequences having prescribed quadratic character,"*Math. Comp.*, v. 24, 1970, pp. 433-451. MR**0271006 (42:5889)****[5]**C. D. Patterson & H. C. Williams, "A report on the University of Manitoba Sieve Unit,"*Congr. Numer.*, v. 37, 1983, pp. 85-98. MR**703580 (84g:10003)****[6]**G. F. Voronoi,*On a Generalization of the Algorithm of Continued Fractions*, Doctoral Dissertation, Warsaw, 1896. (Russian)**[7]**H. C. Williams, "A numerical investigation into the length of the period of the continued fraction expansion of ,"*Math. Comp.*, v. 36, 1981, pp. 593-601. MR**606518 (82f:10011)****[8]**H. C. Williams,*Continued Fractions and Number Theoretic Computations*, Proc. Number Theory Conf., Edmonton, 1983;*Rocky Mountain J. Math.*(To appear.) MR**823273 (87h:11129)****[9]**H. C. Williams, G. W. Dueck & B. K. Schmid, "A rapid method of evaluating the regulator and class number of a pure cubic field,"*Math. Comp.*, v. 41, 1983, pp. 235-286. MR**701638 (84m:12010)**

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DOI:
https://doi.org/10.1090/S0025-5718-1985-0777283-1

Article copyright:
© Copyright 1985
American Mathematical Society