Some computational results on a problem concerning powerful numbers
HTML articles powered by AMS MathViewer
- by A. J. Stephens and H. C. Williams PDF
- Math. Comp. 50 (1988), 619-632 Request permission
Abstract:
Let D be a positive square-free integer and let $X + Y\sqrt D$ be the fundamental unit in the order with Z-basis $\{ 1,\sqrt D \}$. An algorithm, which is of time complexity $O({D^{1/4 + \varepsilon }})$ for any positive $\varepsilon$, is developed for determining whether or not $D|Y$. Results are presented for a computer run of this algorithm on all $D < {10^8}$. The conjecture of Ankeny, Artin and Chowla is verified for all primes $\equiv 1 \pmod 4$ less than ${10^9}$.References
- N. C. Ankeny, E. Artin, and S. Chowla, The class-number of real quadratic number fields, Ann. of Math. (2) 56 (1952), 479–493. MR 49948, DOI 10.2307/1969656
- B. D. Beach, H. C. Williams, and C. R. Zarnke, Some computer results on units in quadratic and cubic fields, Proceedings of the Twenty-Fifth Summer Meeting of the Canadian Mathematical Congress (Lakehead Univ., Thunder Bay, Ont., 1971) Lakehead Univ., Thunder Bay, Ont., 1971, pp. 609–648. MR 0337887 G. Chrystal, Textbook of Algebra, part 2, 2nd ed., Dover reprint, New York, 1969, pp. 423-490. P. Erdős, "Consecutive numbers," Eureka 38, 1975/76, pp. 3-8.
- Andrew Granville, Powerful numbers and Fermat’s last theorem, C. R. Math. Rep. Acad. Sci. Canada 8 (1986), no. 3, 215–218. MR 841645
- H. W. Lenstra Jr., On the calculation of regulators and class numbers of quadratic fields, Number theory days, 1980 (Exeter, 1980) London Math. Soc. Lecture Note Ser., vol. 56, Cambridge Univ. Press, Cambridge, 1982, pp. 123–150. MR 697260
- R. A. Mollin and P. G. Walsh, A note on powerful numbers, quadratic fields and the Pellian, C. R. Math. Rep. Acad. Sci. Canada 8 (1986), no. 2, 109–114. MR 831787
- L. J. Mordell, On a pellian equation conjecture, Acta Arith. 6 (1960), 137–144. MR 118699, DOI 10.4064/aa-6-2-137-144
- Oskar Perron, Die Lehre von den Kettenbrüchen, Chelsea Publishing Co., New York, N. Y., 1950 (German). 2d ed. MR 0037384
- H. Zantema, Class numbers and units, Computational methods in number theory, Part II, Math. Centre Tracts, vol. 155, Math. Centrum, Amsterdam, 1982, pp. 213–234. MR 702518
- Daniel Shanks, Class number, a theory of factorization, and genera, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969) Amer. Math. Soc., Providence, R.I., 1971, pp. 415–440. MR 0316385
- Daniel Shanks, The infrastructure of a real quadratic field and its applications, Proceedings of the Number Theory Conference (Univ. Colorado, Boulder, Colo., 1972) Univ. Colorado, Boulder, Colo., 1972, pp. 217–224. MR 0389842 R. Soleng, "A computer investigation of units in quadratic number fields," unpublished manuscript.
- H. C. Williams, A numerical investigation into the length of the period of the continued fraction expansion of $\sqrt {D}$, Math. Comp. 36 (1981), no. 154, 593–601. MR 606518, DOI 10.1090/S0025-5718-1981-0606518-7
- H. C. Williams and M. C. Wunderlich, On the parallel generation of the residues for the continued fraction factoring algorithm, Math. Comp. 48 (1987), no. 177, 405–423. MR 866124, DOI 10.1090/S0025-5718-1987-0866124-1
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Math. Comp. 50 (1988), 619-632
- MSC: Primary 11R11; Secondary 11A51, 11R27, 11Y16, 11Y40
- DOI: https://doi.org/10.1090/S0025-5718-1988-0929558-3
- MathSciNet review: 929558