Some computational results on a problem concerning powerful numbers
Authors:
A. J. Stephens and H. C. Williams
Journal:
Math. Comp. 50 (1988), 619632
MSC:
Primary 11R11; Secondary 11A51, 11R27, 11Y16, 11Y40
MathSciNet review:
929558
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let D be a positive squarefree integer and let be the fundamental unit in the order with Zbasis . An algorithm, which is of time complexity for any positive , is developed for determining whether or not . Results are presented for a computer run of this algorithm on all . The conjecture of Ankeny, Artin and Chowla is verified for all primes less than .
 [1]
N.
C. Ankeny, E.
Artin, and S.
Chowla, The classnumber of real quadratic number fields, Ann.
of Math. (2) 56 (1952), 479–493. MR 0049948
(14,251h)
 [2]
B.
D. Beach, H.
C. Williams, and C.
R. Zarnke, Some computer results on units in quadratic and cubic
fields, Mathematical Congress (Lakehead Univ., Thunder Bay, Ont.,
1971) Lakehead Univ., Thunder Bay, Ont., 1971, pp. 609–648. MR 0337887
(49 #2656)
 [3]
G. Chrystal, Textbook of Algebra, part 2, 2nd ed., Dover reprint, New York, 1969, pp. 423490.
 [4]
P. Erdős, "Consecutive numbers," Eureka 38, 1975/76, pp. 38.
 [5]
Andrew
Granville, Powerful numbers and Fermat’s last theorem,
C. R. Math. Rep. Acad. Sci. Canada 8 (1986), no. 3,
215–218. MR
841645 (87h:11010)
 [6]
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
(86g:11080)
 [7]
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
(87g:11020)
 [8]
L.
J. Mordell, On a pellian equation conjecture, Acta Arith.
6 (1960), 137–144. MR 0118699
(22 #9470)
 [9]
Oskar
Perron, Die Lehre von den Kettenbrüchen, Chelsea
Publishing Co., New York, N. Y., 1950 (German). 2d ed. MR 0037384
(12,254b)
 [10]
R.
J. Schoof, Quadratic fields and factorization, Computational
methods in number theory, Part II, Math. Centre Tracts, vol. 155,
Math. Centrum, Amsterdam, 1982, pp. 235–286. MR 702519
(85g:11118b)
 [11]
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
(47 #4932)
 [12]
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
(52 #10672)
 [13]
R. Soleng, "A computer investigation of units in quadratic number fields," unpublished manuscript.
 [14]
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
(82f:10011), http://dx.doi.org/10.1090/S00255718198106065187
 [15]
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
(88i:11099), http://dx.doi.org/10.1090/S00255718198708661241
 [1]
 N. C. Ankeny, E. Artin & S. Chowla, "The class number of real quadratic number fields," Ann. of Math., v. 56, 1952, pp. 479493. MR 0049948 (14:251h)
 [2]
 B. D. Beach, H. C. Williams & C. R. Zarnke, Some Computer Results on Units in Quadratic and Cubic Fields, Proc. 25th Summer Meeting Canad. Math. Congr., Lakehead Univ., 1971, pp. 609648. MR 0337887 (49:2656)
 [3]
 G. Chrystal, Textbook of Algebra, part 2, 2nd ed., Dover reprint, New York, 1969, pp. 423490.
 [4]
 P. Erdős, "Consecutive numbers," Eureka 38, 1975/76, pp. 38.
 [5]
 A. Granville, "Powerful numbers and Fermat's Last Theorem," C. R. Math. Rep. Acad. Sci. Canada, v. 8, 1986, pp. 215218. MR 841645 (87h:11010)
 [6]
 H. W. Lenstra, Jr., On the Calculation of Regulators and Class Numbers of Quadratic Fields, London Math. Soc. Lecture Note Series, vol. 56, 1982, pp. 123150. MR 697260 (86g:11080)
 [7]
 R. A. Mollin & P. G. Walsh, "A note on powerful numbers, quadratic fields, and the Pellian," C. R. Math. Rep. Acad. Sci. Canada, v. 8, 1986, pp. 109111. MR 831787 (87g:11020)
 [8]
 L. J. Mordell, "On a Pellian equation conjecture," Acta Arith., v. 6, 1960, pp. 137144. MR 0118699 (22:9470)
 [9]
 Oskar Perron, Die Lehre von den Kettenbrüchen, 2nd ed., Chelsea, New York, 1950. MR 0037384 (12:254b)
 [10]
 R. J. Schoof, "Quadratic fields and factorization," Computational Methods in Number Theory (H. W. Lenstra, Jr. and R. Tijdemann, eds.), Math. Centrum Tracts, Number 155, Part II, Amsterdam, 1983, pp. 235286. MR 702519 (85g:11118b)
 [11]
 D. Shanks, Class Number, A Theory of Factorization and Genera, Proc. Sympos. Pure Math., vol. 20 (1969 Institute on Number Theory), Amer. Math. Soc., Providence, R. I., 1971, pp. 415440. MR 0316385 (47:4932)
 [12]
 D. Shanks, The Infrastructure of a Real Quadratic Field and Its Applications, Proc. 1972 Number Theory Conference, Boulder, 1972, pp. 217224. MR 0389842 (52:10672)
 [13]
 R. Soleng, "A computer investigation of units in quadratic number fields," unpublished manuscript.
 [14]
 H. C. Williams, "A numerical investigation into the length of the period of the continued fraction expansion of ," Math. Comp., v. 34, 1981, pp. 593601. MR 606518 (82f:10011)
 [15]
 H. C. Williams & M. C. Wunderlich, "On the parallel generation of the residues for the continued fraction factoring algorithm," Math. Comp., v. 48, 1987, pp. 405423. MR 866124 (88i:11099)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
11R11,
11A51,
11R27,
11Y16,
11Y40
Retrieve articles in all journals
with MSC:
11R11,
11A51,
11R27,
11Y16,
11Y40
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198809295583
PII:
S 00255718(1988)09295583
Article copyright:
© Copyright 1988 American Mathematical Society
