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

MathSciNet review:
777283

Full-text PDF Free Access

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]**Pierre Barrucand and Harvey Cohn,*Remarks on principal factors in a relative cubic field*, J. Number Theory**3**(1971), 226–239. MR**0276197****[2]**Taira Honda,*Pure cubic fields whose class numbers are multiples of three*, J. Number Theory**3**(1971), 7–12. MR**0292795****[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, Emma Lehmer, and Daniel Shanks,*Integer sequences having prescribed quadratic character*, Math. Comp.**24**(1970), 433–451. MR**0271006**, 10.1090/S0025-5718-1970-0271006-X**[5]**C. D. Patterson and H. C. Williams,*A report on the University of Manitoba Sieve Unit*, Congr. Numer.**37**(1983), 85–98. MR**703580****[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 √𝐷*, Math. Comp.**36**(1981), no. 154, 593–601. MR**606518**, 10.1090/S0025-5718-1981-0606518-7**[8]**Hugh C. Williams,*Continued fractions and number-theoretic computations*, Rocky Mountain J. Math.**15**(1985), no. 2, 621–655. Number theory (Winnipeg, Man., 1983). MR**823273**, 10.1216/RMJ-1985-15-2-621**[9]**H. C. Williams, G. W. Dueck, and B. K. Schmid,*A rapid method of evaluating the regulator and class number of a pure cubic field*, Math. Comp.**41**(1983), no. 163, 235–286. MR**701638**, 10.1090/S0025-5718-1983-0701638-2

Retrieve articles in *Mathematics of Computation*
with MSC:
11Y65,
11A55,
11J70

Retrieve articles in all journals with MSC: 11Y65, 11A55, 11J70

Additional Information

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

Article copyright:
© Copyright 1985
American Mathematical Society