Publications of Hugh C. Williams



 

Books, or Chapters or Articles in Books

 

1.               R.S.D. Thomas and H.C. Williams, eds., Proceedings of the Manitoba Conference on Numerical Mathematics, Winnipeg, Utilitas Mathematica Publishing Inc., 1971, 676 pages.

 

2.               R.S.D. Thomas and H.C. Williams, eds., Proceedings of the Second Manitoba Conference on Numerical Mathematics, Winnipeg, Utilitas Mathematica Publishing Inc., 1972, 468 pages.

 

3.               R.S.D. Thomas and H.C. Williams, eds., Proceedings of the Third Manitoba Conference on Numerical Mathematics, Winnipeg, Utilitas Mathematica Publishing Inc., 1973, 455 pages.

 

4.                H.C. Williams, Tables of indices and power residues, in CRC Standard Mathematical Tables, 21st edition, edited by S.M. Selby, Chemical Rubber Co., 1973. pp. 146-150.

 

5.                H.C. Williams, Diophantine equations, in CRC Standard Mathematical Tables, 21st edition, S.M. Selby (ed.), Chemical Rubber Co., 1973, pp.157-164.

 

6.                H.C. Williams, Continued fractions, in CRC Standard Mathematical Tables, 21st edition, S.M. Selby (ed.), Chemical Rubber co., 1973, pp. 165-171.

 

7.                B.L. Hartnell and H.C. Williams, eds., Proceedings of the Fourth Manitoba Conference on Numerical Mathematics, Winnipeg, Utilitas Mathematica Publishing Inc., 1974, 410 pages.

 

8.                B.L. Hartnell and H.C. Williams, eds., Proceedings of the Fifth Manitoba Conference on Numerical Mathematics, Winnipeg, Utilitas Mathematica Publishing Inc., 1976, 658 pages.

 

9.                B.L. Hartnell and H.C. Williams, eds., Proceedings of the Sixth Manitoba Conference on Numerical Mathematics, Winnipeg, Utilitas Mathematica Publishing Inc., 1976, 504 pages.

 

10.             D. McCarthy and H.C. Williams, eds., Proceedings of the Seventh Manitoba Conference on Numerical Mathematics, Winnipeg, Utilitas Mathematica Publishing Inc., 1977, 607 pages.

 

11.             D. McCarthy and H.C. Williams, eds., Proceedings of the Eighth Manitoba Conference on Numerical Mathematics and Computing, Congressus Numerantium XXII, Winnipeg, 1978, 487 pages.

 

12.            G.H.J. van Rees and H.C. Williams, Proceedings of the Ninth Manitoba Conference on Numerical Mathematics and Computing, Congressus Numerantium XXVII, Winnipeg, 1980, 434 pages.

 

13.             D.S. Meek and H.C. Williams, Proceedings of the Tenth Manitoba Conference on Numerical Mathematics and Computing, Congressus Numerantium Vols. 30 and 31, Winnipeg, 1981, pages 317 and 313.

 

14.             H.C. Williams, Computationally hard problems as a source for cryptosystems, Secure Communications and Asymmetric Cryptosystems, G.J. Simons (ed.) AAAS Publishing, 1982.

 

15.             H.C. Williams, Advances in Cryptology CRYPTO '85 Proceedings, Lecture Notes in Computer Science 218, Springer-Verlag, New York, 1985.

 

16.             Mathematics of Computation Special Issue dedicated to Daniel Shanks, 48, Nos. 177-178, 447 pages (1987).

 

17.             A. Petto, M.E. Pohst, H.C. Williams, H.G. Zimmer, eds., Computational Number Theory, Walter de Gruyter, Berlin, 1991.

 

18.              Mathematics of Computation Special Memorial Issue dedicated to the memory of D.H. Lehmer, 61, No. 203, 491 pages (1993).

 

19.             H.C. Williams, Number theory and finite fields, Part IV, ch. 2, CRC Handbook of Combinatorial Designs, C.J. Colbourn and J.H. Dinitz, eds., pp. 615-644 (1996).

 

20.             H.C. Williams, Édouard Lucas and Primality Testing, Canadian Mathematical Society Series of Monographs and Advanced Texts, 22, Wiley-Interscience, New York, 525 pages (1998).

 

21.             K. Alster, J. Urbanowicz, H.C. Williams, eds., Public-key cryptography and computational number theory, Proceedings of the International Conference organized by the International Mathematical Stefan Banach Center, Warsaw, Poland, September 11-15, 2000, Walter de Gruyter, Berlin, 331 pages.

 

22.             M. J. Jacobson, Jr. and H.C. Williams Solving the Pell Equation , Springer, New York, 2008, 520pp.
ERRATA to the First Printing of "Solving the Pell Equation"


 

 

Papers in Refereed Journals

 

1.                   H.C. Williams, R.A. German and C.R. Zarnke, Solution of the cattle problem of Archimedes, Math. Comp., 19, 671-674 (1965).

 

2.                   R.G. Stanton and H.C. Williams, A packing problem, Can. Math. Bull., 10, 287-297 (1967).

 

3.                   H.C. Williams and C.R. Zarnke, A report on prime numbers of the forms M=(6a+1)s2m-1-1 and M'=(6a-1)22m-1, Math. Comp., 420-422 (1968).

 

4.                   H.C. Williams, Note on a Diophantine equation, Elemente der Math., 25, 123-125 (1970).

 

5.                   H.C. Williams, On a generalization of the Lucas functions, Acta Arithmetica, 29, 33-52 (1972).

 

6.                   H.C. Williams, The primality of N = 2A3n-1, Can. Math. Bull., 15, 585-589 (1972).

 

7.                   H.C. Williams, Some properties of the general Lucas polynomials, Matrix and Tensor Quarterly, 21, 91-93 (1971).

 

8.                   H.C. Williams, Fibonacci numbers obtained from Pascal's triangle and generalizations, Fibonacci Quarterly, 10, 405-412 (1972).

 

9.                   H.C. Williams and C.R. Zarnke, Some prime numbers of the form 2A3n-1, Math. Comp., 26, 995-998 (1972).

 

10.               H.C. Williams, The quadratic character of a certain quadratic surd, Utilitas Mathematica, 5, 49-55 (1974).

 

11.               H.C. Williams, The rank of apparition of a generalized Fibonacci sequence, Fibonacci Quarterly, 13, 240-242 (1975).

 

12.               H.C. Williams, On Fibonacci numbers of the form x2+1, Fibonacci Quarterly, 13, 213-214 (1975).

 

13.               H.C. Williams, Some algorithms for solving a cubic congruence modulo p, Utilitas Mathematica, 6, 285-306 (1974).

 

14.               H.C. Williams and J.S. Judd, Determination of the primality of N by using factors of N2±1, Math. Comp., 30, 157-172 (1976).

 

15.               Barrucand, H.C. Williams and L. Baniuk, A computational technique for determining the class number of a pure cubic field, Math. Comp., 30, 312-323  
  (1976).

 

16.               R.G. Stanton and H.C. Williams, An application of combinatorics in number theory, Ars Combinatoria, 1, 321-330 (1976).

 

17.               H.C. Williams, A generalization of Lehmer's functions, Acta Arith., 29, 315-341 (1976).

 

18.               H.C. Williams, Some results on fundamental units in cubic fields, journal für die Reine und Angewandte Mathematik, 286/287, 75-85 (1976).

 

19.               H.C. Williams, and J.S. Judd, Some algorithms for prime testing using generalized Lehmer functions, Math. Comp., 30, 867-886 (1976).

 

20.               H.C. Williams, and J. Broere, A computational technique for evaluating L(1,ﭏ) and the class number of a real quadratic field, Math. Comp., 30, 887-893
  (1976).

 

21.               R.G. Stanton, C. Sudler and H.C. Williams, An upper bound for the period of the simple continued fraction for √D, Pacific Journal of Math., 67, 525-536
  (1976).

 

22.               H.C. Williams, Certain pure cubic fields with class number one, Math. Comp., 31, 578-580 (1977).

 

23.               H.C. Williams and G. Matthew, Some new primes of the form k·2n + 1, Math. Comp., 31, 797-798 (1977).

 

24.               H.C. Williams, Properties of some functions similar to Lucas sequences, Fibonacci Quarterly, 15, 97-112 (1977).

 

25.               H.C. Williams and R. Holte, Computation of the solution of x3 + Dy3 =1, Math. Comp., 31, 778-785 (1977).

 

26.               H.C. Williams, On numbers analogous to the Carmichael numbers, Can. Math. Bull., 20, 133-143 (1977).

 

27.               D.A. Buell, H.C. Williams and K.S. Williams, On the imaginary bicyclic biquadratic fields with class number 2, Math. Comp., 31, 1034-1042 (1977).

 

28.               H.C. Williams, Some properties of a special set of recurring sequences, Pac. Journal Math., 77, 1306-1310 (1978).


29.               H.C. Williams, Some results concerning the nearest integer continued fraction expansion of √D, J. Reine Agnew. Math., 315, 1-15 (1980).

 

30.               H.C. Williams and R. Holte, Some observations on primality testing, Math. Comp., 32, 905-917 (1978).

 

31.               M. Newman, D. Shanks and H.C. Williams, Simple groups of square order and an interesting sequence of primes Sp, Acta Arith., 38, 129-140 (1980).

 

32.               H.C. Williams and P.A. Buhr, Calculation of the regulator of Q(√D) by use of the nearest integer continued fraction algorithm, Math. Comp., 33, 369-381
  (1979).

 

33.               H.C. Williams, Some primes with interesting digit patterns, Math. Comp., 32, 1306-1310 (1978).

 

34.               H.C. Williams, Primality testing on a computer, Ars Combinatoria, 5, 127-185 (1978).

 

35.               F. Diaz y Diaz, Daniel Shanks and H.C. Williams, Quadratic fields with 3-rank equal to 4, Math. Comp., 33, 836-840 (1979).

 

36.               H.C. Williams and B. Schmid, Some remarks concerning the MIT public-key cryptosystem, BIT, 19, 525-538 (1979).

 

37.               D. Shanks and H.C. Williams, A note on the class-number one in pure cubic fields, Math. Comp., 33, 1317-1320 (1979).

 

38.               R. Purdy, A. Terras, R. Terras and H.C. Williams, Graphing L-functions of Kronecker symbols in the real part of the critical strip, Math. Student, 47,
  101-131 (1979).             

 

39.               H.C. Williams and E. Seah, Some primes of the form (an-1)/(a-1), Math. Comp., 33, 1337-1342 (1979).

 

40.               H.C. Williams, Improving the speed of calculating the regulator of certain pure cubic fields, Math. Comp., 34, 1423-1434 (1980).

 

41.               H.C. Williams and G.V. Cormack, Some very large primes of the form k·2m+1, Math. Comp., 34, 1419-1422 (1980).

 

42.               H.C. Williams, G. Cormack and E. Seah, Calculation of the regulator of a pure cubic field, Math. Comp., 34, 567-611 (1980).

 

43.               H.C. Williams, A modification of the RSA public-key encryption procedure, IEEE Trans. Information Theory, IT-26, 726-729 (1980).

 

44.               Daniel Shanks and H.C. Williams, Gunderson's function in Fermat's last theorem, Math. Comp., 36, 291-295 (1981).

 

45.               H.C. Williams, Some results concerning Voronoi's continued fraction over Q(D1/3), Math. Comp., 36, 632-652 (1981).

 

46.               H.C. Williams, A numerical investigation into the length of the period of the continued fraction of √D, Math. Comp., 36, 593-601 (1981).

 

47.               R. Baillie, H.C. Williams and G. Cormack, The problem of Sierpinski concerning k·2n+1, Math. Comp., 37, 229-231 (1981).

 

48.               H.C. Williams, The primality of certain integers of the form 2Arn –1, Acta Arith., 39, 7-17 (1981).

 

49.               H.C. Williams, A note on the Fibonacci quotient Fp-ε(p),Can. Math. Bull., 25, 366-370 (1982).

 

50.              H.C. Williams, A class of primality tests for trinomials which include the Lucas-Lehmer test, Pacific Journal of Mathematics, 98, 477-494 (1982).

 

51.               H.C. Williams, Determination of principal factors in Q(3√D), Math. Comp., 38, 261-274 (1982).

 

52.               H.C. Williams, The influence of computers on the development of number theory, Computers and Mathematics with Applications, 8, 75-93 (1982).

 

53.               H.C. Williams, A p+1 method of factoring, Math. Comp., 39, 225-234 (1982).

 

54.               E. Seah, L. Washington and H.C. Williams, The calculation of a large class number with an application to real cyclotomic fields, Math. Comp., 41, 303-305
  (1983).

 

55.               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, 235-286
  (1983).

 

56.               H.C. Williams, Factoring on a computer, The Mathematical Intelligencer, 6, 29-36 (1984).

 

57.               H.C. Williams and G.W. Dueck, An analogue of the nearest integer continued fraction for certain cubic irrationalities, Math. Comp., 42, 683-705 (1984).

 

58.               H.C Williams, Continued fractions and number-theoretic computations, Rocky Mountain Journal of Math., 15, 621-655 (1985).

 

59.               H.C. Williams, Some public-key crypto-functions as intractable as factorization, Cryptologia, 9, 223-237 (1985).

 

60.               C.D. Patterson and H.C. Williams, Some periodic continued fractions with long periods, Math. Comp., 44, 523-532 (1985).

 

61.               H.C. Williams, A note on the period length of the continued fraction expansion of certain √D, Utilitas Math., 28, 201-209 (1985).

 

62.               G. Dueck and H.C. Williams, Computation of the class number and class group of a complex cubic field, Math. Comp., 45, 223-231 (1985). Corrigendum,    

              ibid. 50, 655-657 (1988).
 

63.               H.C. Williams, On mid period criteria of the nearest integer continued fraction expansion of D1/2, Utilitas Math., 27, 169-185 (1985).

 

64.               M. Tennenhouse and H.C. Williams, A note on class-number one in certain real quadratic and pure cubic fields, Math. Comp., 46, 333-336 (1986).

 

65.               G.C. Kurtz, D. Shanks and H.C. Williams, Fast primality tests for numbers greater than 50.109, Math. Comp., 46, 691-701 (1986).

 

66.               H.C. Williams and H. Dubner, The primality of R1031, Math. Comp., 47, 703-711 (1986).

 

67.               H.C. Williams, The spacing of the minima in certain cubic lattices, Pacific Journal of Math., 124, 483-496 (1986).

 

68.               J. Buchmann and H.C. Williams, On principal ideal testing in totally complex quartic fields and the determination of certain cyclotomic constants, Math.
  Comp., 48,
55-56 (1987).  

 

69.               Pierre Barrucand, John Loxton and H.C. Williams, Some explicit upperbounds on the class number and regulator of cubic field with negative discriminant,      Pacific Journal of Math., 128, 209-222 (1987).

 

70.               H.C. Williams, Effective primality tests for some integers of the forms A5n-1 and A7n-1, Math. Comp., 48, 385-403 (1987).

 

71.               H.C. Williams and M.C. Wunderlich, On the parallel generation of the residues for the continued fraction factoring algorithm, Math. Comp., 48, 405-423 
  (1987).

 

72.               M. Wunderlich and H.C. Williams, A parallel version of the continued fraction integer factoring algorithm, Journal of Super-Computing, 1, 217-230 (1987).

 

73.               J. Buchmann and H.C. Williams, On principal ideal testing in algebraic number fields, J. Symbolic Computation, 4, 11-19 (1987).

 

74.               J. Buchmann and H.C. Williams, On the infrastructure of the principal ideal class of an algebraic number field of unit rank one, Math. Comp., 50, 569-579
  (1988).

 

75.               A.J. Stephens and H.C. Williams, Some computational results on a problem concerning powerful numbers, Math. Comp., 50, 619-632 (1988).

 

76.               H.C. Williams, A note on the primality of 62n + 1 and 102n + 1, Fibonacci Quarterly, 26, 296-305 (1988).

 

77.               R.A. Mollin and H.C. Williams, A conjecture of S. Chowla via the generalized Riemann Hypothesis, Proceedings of the Amer. Math. Soc., 102, 794-796
  (1988).

 

78.               A.J. Stephens and H.C. Williams, Computation of real quadratic fields of class number one, Math. Comp., 51, 809-824 (1988).

 

79.               R.A. Mollin and H.C. Williams, On prime values polynomials and class numbers of real quadratic fields, Nagoya, 112, 143-151 (1988).

 

80.               J. Buchmann and H.C. Williams, A key exchange system based on imaginary quadratic fields, Journal of Cryptology, 1, 107-118 (1988).

 

81.               R.A. Mollin and H.C. Williams, Real quadratic fields of class number one and continued fraction period less than six, C.R. Math. Rep. Acad. Sci. Canada,       Vol. XI, 51-56 (1989).

 

82.               R.A. Mollin and H.C. Williams, Quadratic non-residues and prime-producing polynomials, Can. Math. Bull., 32, 474-478 (1989).

 

83.               J. Buchmann and H.C. Williams, On the computation of the class number of an algebraic number field, Math. Comp., 53, 679-688 (1989).

 

84.               R.A. Mollin and H.C. Williams, Period four and real quadratic fields of class number one, Proc. Japan. Acad., 65, 89-93 (1989).

 

85.               G.W. Fung and H.C. Williams, Quadratic polynomials which have a high density of prime values, Math. Comp., 55, 345-353 (1990).

 

86.               J. Buchmann, J.W. Sands and H.C. Williams, p-adic computation of real quadratic class numbers, Math. Comp., 54, 885-868 (1990).

 

87.               G.W. Fung and H.C. Williams, On the computation of a table of complex cubic fields with discriminants D>106, Math. Comp., 55, 313-325 (1990).

 

88.               Stephens and H.C. Williams, An open architecture number sieve, Lecture notes of the London Math. Soc., 154, 38-75 (1990).

 

89.               R. Mollin and H.C. Williams, Class number problems for real quadratic fields, Lecture notes of the London Math. Soc., 154, 177-195 (1990).

 

90.               R. Mollin and H C. Williams, Solution of a problem of Yokoi, Proc. Japan Acad., 66, 141-145 (1990).

 

91.               H.C. Williams, The period length of Voronoi's algorithm for certain cubic orders, Publicationes Mathematicae Debrecen, 37, 245-265 (1990).


92.         H. C. Williams, Eisenstein's problem and continued fractions. Util.Math., 37 (1990), 145-157.

 

93.               R.A. Mollin and H.C. Williams, Continued fractions of period five and real quadratic fields of class number one, Acta. Arith., 56, 55-63 (1990).

 

94.               G.W. Fung, H. Stroher, H.C. Williams and H.G. Zimmer, Torsion groups of elliptic curves with integer j-invariant over pure cubic fields, J. of Number 
  Theory,
36, 12-45, (1990).

 

95.               R.A. Mollin and H.C. Williams, On a determination of real quadratic fields of class number one and related continued fraction period less than 25, Proc.
  Japan Acad.,
67A, 20-25, (1991).

 

96.               R.A. Mollin and H.C. Williams, Affirmative solution of a conjecture related to a sequence of Shanks, Proc. Japan Acad., 67A, 70-72 (1991).

 

97.               H.C. Williams, Some formulas concerning the fundamental unit of a real quadratic field, Discrete Math., 92, 431-440 (1991).

 

98.               J. Buchmann and H.C. Williams, Some remarks concerning the complexity of computing class groups of quadratic fields, Journal of Complexity, 7,   
  311-315, (1991).

 

99.               R.A. Mollin and H.C. Williams, On the divisor function and class numbers of real quadratic fields III, Proceed. Japan Acad., 67, 338-342 (1991).

 

100.              R.A. Mollin and H.C. Williams, Computation of the class number of a real quadratic field, Utilitas Math., 41, 259-308 (1992).

 

101.           S. Louboutin, R.A. Mollin and H.C. Williams, Class numbers of real quadratic fields, continued fractions, reduced ideals, prime producing quadratic  
  polynomials and quadratic covers, Can. J. Math,  44, 824-842, (1992).

 

102.            R.A. Mollin and  H.C. Williams, Consecutive powers in continued fractions, Acta. Arith., 61, 233-264 (1992).

 

103.            R. Scheidler and H.C. Williams, A method of tabulating a function of Erdös, Math. Comp., 59, 251-257 (1992).

 

104.            R.A.. Mollin and H.C. Williams, A complete generalization of Yokoi's p-invariants, Colloquium Math., 63, 285-294 (1992).

 

105.            R.A.. Mollin and H.C. Williams, On real quadratic fields of class number two, Math. Comp., 59, 625-632 (1992).

 

106.            R.A.. Mollin and H.C. Williams, On the period length of some special continued fractions, Sém. Théorie des Nombres, Bordeaux , 4, 19-42 (1992).

 

107.            Granville, G. Fung and H.C. Williams, Computation of the first factor of the class number of cyclotomic fields, J. of Number Theory, 42, 297-312 (1992).

 

108.            H.C. Williams, How was F6 factored?, Math. Comp., 61, 463-474 (1993).

 

109.            R.A. Mollin and H.C. Williams, Classification and enumeration of real quadratic fields having exactly one non-inert prime less than a Minkowski bound, Can. Math. Bull., 36, 108-115 (1993).

 

110.            R. Scheidler, J. Buchmann, H.C. Williams, A key exchange protocol using real quadratic fields, J. of Cryptology, 7, 171-199 (1994).

 

111.            R. A. Mollin and H.C. Williams, Quadratic residue covers in certain real quadratic fields, Math. Comp., 62, 885-897 (1994).

 

112.            H.C. Williams, Some generalizations of the Sn sequence of Shanks, Acta Arith., 69, 199-215 (1995).

 

113.            R.F. Lukes, C.D. Patterson and H.C. Williams, Numerical sieving devices: their history and some applications, Nieuw Arch. voor Wiskunde (4), 13, 113-139 (1995).

 

114.            R.A. Mollin, A..J. van der Poorten and H.C. Williams, Halfway to a solution of a Diophantine equation, Séminaire de Théorie des Nombres de Bordeaux, 6, 421-459 (1994).

 

115.            R. Scheidler and H.C. Williams, A public-key cryptosystem utilizing cyclotomic fields, Designs, Codes and Cryptography, 6, 117-131 (1995).

 

116.            J.O. Shallit, H.C. Williams and F. Morain, Discovery of a lost factoring machine, Mathematical Intelligencer, 17, #3, 41-47 (1995).

 

117.            M.J. Jacobson, Jr., R.F. Lukes and H.C. Williams, Some numerical experiments concerning quadratic fields, J. Experimental Math., 4, 211-225 (1995).

 

118.            R.A. Mollin and H.C. Williams, Proof, disproof and advances concerning certain conjectures on real quadratic fields, Can. J. Math, 47, 1023-1036 (1995).

 

119.            E. Bach, R.F. Lukes, J.O. Shallit, H.C. Williams, Some results and estimates on pseudopowers, Math. Comp., 65, 1737-1747 (1996).

 

120.            R.F. Lukes, C.D. Patterson and H.C. Williams, Some results on pseudosquares, Math. Comp., 65, 361-372 (1996).

 

121.            R. Scheidler, A. Stein and H.C. Williams, Key exchange in real quadratic function fields, Designs, Codes and Cryptography, 7, 153-174 (1996).

 

122.            R.F. Lukes, R. Scheidler and H.C. Williams, Further tabulation of the Erdös-Selfridge function, Math. Comp., 66, 1709-1717 (1997).

 

123.            H.C. Williams, Daniel Shanks (1917-1996), Math. Comp., 66, 929-934 (1997) and Notices of the AMS, 44, 813-816 (1997).

 

124.            E. Teske and H.C. Williams, A problem concerning a character sum, Experimental Math., 8, 63-72 (1999).

 

125.            Stein and H.C. Williams, Some methods for evaluating the regulator of a real quadratic function field, Experimental Math., 89, 119-133 (1999).

 

126.            A.J. van der Poorten and H.C. Williams, On certain continued fraction expansions of fixed period length, Acta Arith., 89, 23-35 (1999).

 

127.            H.C. Williams, A number theoretic function arising from continued fractions, Fibonacci Quarterly, 38, 201-211 (2000).

 

128.            Granville, R.A. Mollin and H.C. Williams, An upper bound on the least inert prime in a real quadratic field, Canadian Journal of Math., 52, 369-380 (2000).

 

129.            Stein and H.C. Williams, Explicit primality criteria for (p-1)pn-1, Math. Comp., 69, 1721-1734 (2000).

 

130.            M.J. Jacobson, Jr. and H.C. Williams, The size of fundamental solutions of consecutive Pell equations, Experimental Math., 9, 631-640 (2000).

 

131.            A.J. van der Poorten, H.J.J. te Riele, H.C. Williams, Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100,000,000,000, Math. Comp., 7, 1311-1328 (2001).

 

132.            J.A. Sellers and H.C. Williams, On the infinitude of composite NSW numbers, Fibonacci Quarterly, 40, 253-254 (2002).

 

133.            M.J. Jacobson, Jr. and H.C. Williams, New quadratic polynomials with high densities of prime values, Math. Comp., 72, 499-519 (2003).

 

134.            M.J. Jacobson, Jr. and H.C. Williams, Modular arithmetic on elements of small norm in quadratic fields, Designs, Codes and Cryptography, 27, 93-110 (2002).

 

135.            H. J. J. te Riele and H.C. Williams, New computations concerning the Cohen-Lenstra heuristics , Experimental Mathematics, 12, 99-113 (2003).

 

136.            K. Cheung and H.C. Williams, Some Results Concerning Certain Periodic Continued Fractions, Acta Arithmetica, 117 (2005), 247-264.

 

137.            M. J. Jacobson, Jr., Reginald E. Sawilla, and H.C. Williams, Efficient Ideal Reduction in Quadratic Fields, International Journal of Mathematics and Computer Science, 1 (2006), 83-116.

 

138.            M. J. Jacobson, Jr., R. Scheidler and H.C. Williams, An Improved Real Quadratic Field Based Key Exchange Procedure, Journal of Cryptography, 19 (2006), 211-239.

 

139.            R. de Haan, M. J. Jacobson, Jr. and H.C. Williams, A Fast, Rigorous Technique for Computing the Regulator of a Real Quadratic Field, Math. Comp., 76 (2007), 2139-2160.

 

140.            R. Patterson, A. J. van der Poorten and H.C. Williams, Characterization of a Generalized Shanks Sequence, Pacific Journal of Mathematics, 230 (2007),  185-216.

 

141.            P. Berrizbeitia, S. Müller and H.C. Williams, Pseudopowers and Primality Proving, Contributions to Discrete Mathematics, 2 (2007), 219-236.


142.       R. D. Patterson, A. J. van der Poorten and H. C. Williams, Sequences of Jacobian varieities with torsion divisors of quadratic order, Functiones et Approximatio 39 (2008), 345-360.

 


 

 

Papers in Refereed Conference Proceedings

 

1.                   R.G. Stanton and H.C. Williams, Computation of some number theoretic coverings, Combinatorial Mathematics, Lecture notes #840, Springer-Verlag, 8-13.

 

2.                   C. Patterson and H.C. Williams, A report on the University of Manitoba Sieve Unit, Proc. 11th Manitoba Conference on Numerical Mathematics and Computing, Congressus Numerantium 37, 1983, pp. 85-98.

 

3.                   H.C. Williams, An overview of factoring, Advances in Cryptology, edited by D. Chaum, Plenum Publishing, 1984, pp. 71-80.

 

4.                   H.C. Williams, An M3 public-key encryption scheme, Advances in Cryptology, Springer-Verlag, 1986, pp. 358-368.

 

5.                   R. Mollin and H.C. Williams, Prime-producing polynomials and real quadratic fields of class number one, International Number Theory Conference Proceedings, Quebec, 1987, pp. 654-663.

 

6.                   A.J. Stephens and H.C. Williams, Some computational results on a problem of Eisenstein, Proceedings of the International Number Theory Conference, International Number Theory Conference, Laval University. Sponsored by the Canadian Mathematical Society, 1987, pp. 869-886.

 

7.                   R. Mollin and H.C. Williams, Solution to the class number one problem for real quadratic fields of Richaud-Degert Type (with one possible exception), Number Theory, Proceedings of the First Conference on the Canadian Number Theory Association, Walter de Gruyter, New York, 1990, pp. 417-425.

 

8.                   R. Mollin and H.C. Williams, Class number one for real quadratic fields, continued fractions and reduced ideals, Proceedings of the NATO ASI on Number Theory and Applications, 265, 1989, pp. 481-496.

 

9.                   J. Buchmann and H.C. Williams, On the existence of a short proof for the value of the class number and regulator of a real quadratic field, NATO ASI Series C, Number Theory and Applications, Dortrecht, 265, 1989, pp. 327-345.

 

10.               J. Buchmann and H.C. Williams, A key exchange system based on real quadratic fields, Proc. CRYPTO '89, Springer-Verlag, New York, 1990, pp. 335-343.

 

11.               R.A. Mollin and H.C. Williams, Solution of the class number one problem for real quadratic fields of extended Richaud-Degert Type (with one possible exception), Number Theory, Proceedings of the First Conference of the Canadian Number Theory Association, R.A. Mollin ed., Walter de Gruyter, New York, 1990, pp. 417-425.

 

12.               J. Buchmann and H.C. Williams, Quadratic fields and cryptography, Number Theory and Cryptography, London Math Soc. Lecture Note Series 154, 1990, pp. 9-25.

 

13.               J. Buchmann, S. Düllmann and H.C. Williams, On the complexity of a new key exchange system, Proc. EUROCRYPT 89, 1990, pp. 597-616.

 

14.               J. Buchmann, R. Scheidler and H.C. Williams, Implementation of a key exchange protocol using real quadratic fields, Proc. EUROCRYPT 90, 1990, pp. 98-109.

 

15.               R.A. Mollin and H.C. Williams, On a solution of a class number two problem for a family of real quadratic fields, Computational Number Theory, Walter De Gruyter & Co., 1991, pp. 95-101.

 

16.               R.A. Mollin and H. C. Williams, Powers of 2, continued fractions and the class number one problem for real quadratic fields Q(D1/2)) with d≡1 (mod 8), The Mathematical Heritage of C.F. Gauss, G.M. Rassias (ed), World Scientific Publishing Co., Singapore, 1991, pp. 505-516.

 

17.               S. Louboutin, R.A. Mollin and H. C. Williams, Class groups of exponent two in real quadratic fields, Advances in Number Theory, Proc. Third Canadian Number Theory Association Conference, Oxford, 1993, pp. 499-513.

 

18.               H.C. Williams and J.O. Shallit, Factoring integers before computers, Math. Comp. 1943-1993, Proc. Symp. Appl. Math, 48, 1994, pp. 481-531.

 

19.               J. Buchmann, C. Thiel and H.C. Williams, Short representation of quadratic integers, Computational Algebra and Number Theory, Mathematics and its Applications 325, Kluwer Academic Publishers, 1995, pp. 159-185.

 

20.               A. Stein and H.C. Williams, An improved method of evaluating the regulator of a real quadratic function field, Proc. ANTS III, Lecture Notes in Computer Science, 1423, Springer-Verlag, 1998, pp. 607-620.

 

21.               E. Teske and H.C. Williams, A note on Shanks's chains of primes, Proc.ANTS IV, Algorithmic Number Theory, Lecture Notes in Computer Science, 1838, Springer, Berlin, 2000, pp. 563-580.

 

22.               H.C. Williams, Solving the Pell equation, Proc. Millennial Conference on Number Theory, A.K. Peters, Natick MA, 2002, pp. 397-435.

 

23.               M. Jacobson, Jr., R. Scheidler and H.C. Williams, The efficiency and security of real quadratic field based key exchange protocol, Public Key Cryptography and Computational Number Theory, Walter de Gruyter, Warsaw, 2001, pp. 89-112.

 

24.               M. Jacobson, H.C. Williams and K. Wooding, Imaginary cyclic quartic fields with large minus class number, Proc. ANTS VI, Algorithmic Number Theory Symposium, Lecture Notres in Computer Science 3076, Springer, Berling, 2004, pp. 280-292.

 

25.               P. Berrizbeitia, S. Müller and H.C. Williams, Pseudocubes and Primality Testing, Proc. ANTS VI, Algorithmic Number Theory Symposium, Lecture Notes in Computer Science 3076, Springer, Berlin, 2004, pp. 102-116.

 

26.               J-C Bajard, L. Imbert, G.A. Jullien and H.C. Williams, A CRT-based Montgomery multiplication for finite fields of small characteristic, Proc. 17th IMACS World Congress, Scientific Computation, Applied Mathematics and Simulation, 2005, pp.101-107.

 

27.               M. J. Jacobson, Jr., S. Ramachandran, and H.C. Williams, Numerical Results on Class Groups of Imaginary Quadratic Fields , Proc. ANTS-VII, Algorithmic Number Theory Symposium, Lecture Notes in Computer Science 4076, Springer Berlin, 2006, pp. 87-101.

 

28.               K.Wooding and H.C. Williams, Doubly-focused Enumeration of Pseudosquares and Pseudocubes, Proc. ANTS-VII, Algorithmic Number Theory Symposium, Lecture Notes in Computer Science 4076 Springer, Berlin, 2006, pp. 208-211.


29.         R. Sawilla, A. Silvester and H. C. Williams, A New Look at an Old Equation, Algorithmic Number Theory, Proceedings of Ants-VIII, Algorithmic Number Theory Symposium, Lecture Notes in Computer Science 5011, Springer, Berlin, 2008, pp. 39-59.

 


 

 

Other Contributions

 

1.            H.C. Williams, A generalization of the Fibonacci numbers, Proceedings Louisiana Conference on Combinatorics, Graph Theory, and Computing, Baton Rouge, 1970, pp. 340-356.

 

2.         H.C. Williams, An algorithm for determining certain large primes, Proceedings Second Louisiana Conference on Combinatorics, Graph Theory, and Computing, Baton Rouge, 1971, pp. 533-536.

 

3.         H.C. Williams and C.R. Zarnke, Computer determination of some large primes, Proceedings Second Louisiana Conference on Combinatorics, Graph Theory, and Computing, Baton Rouge, 1971, pp. 563-570.

 

4.         B.D. Beach and H.C. Williams, Some computer results on periodic continued fractions, Proceedings Second Louisiana Conference on Combinatorics, Graph Theory and Computing, Baton Rouge, 1971, pp.133-146.

 

5.         D.G. Gryte, R.A. Kingsley and H.C. Williams, On Fibonacci numbers of certain forms, Proceedings Second Louisiana Conference on Combinatorics, Graph Theory and Computing, Baton Rouge, 1971, pp. 339-344.

 

6.         B.D. Beach, H.C. Williams and C.R. Zarnke, Some computer results on units in quadratic and cubic fields, Proceedings 25th summer meeting of the Canadian Mathematical Congress, 1971, pp. 609-648.

 

7.         B.D. Beach and H.C Williams, A computer algorithm for determining the solution of the Diophantine equation x4 - dy4 = 1, Proceedings Manitoba Conference on Numerical Mathematics, Winnipeg, 1971, pp. 663-670.

 

8.         H.C. Williams and C.R. Zarnke, Computation of the solutions of the Diophantine equation x3 + dy3 = 1, Proceedings Manitoba Conference on Numerical Mathematics, Winnipeg, 1971, pp. 671-676.

 

9.         H.C. Williams, and B.D. Beach, A numerical investigation of the Diophantine equation x2 -Dy2 = 1, Proceedings Third Southeastern Conference on Combinatorics, Graph Theory and Computing, Boca Raton, 1972, pp. 37-52.

 

10.        H.C. Williams and C.R. Zarnke, Computation of the solutions of the Diophantine equation x2 - Dy4 = 1, Proceedings Third Southeastern Conference on Combinatorics, Graph Theory and Computing, Boca Raton, 1972, pp. 463-484.

 

11.        H.C. Williams, Some algorithms for solving xq = N (mod p), Proceedings Third Southeastern Conference on Combinatorics, Graph Theory and Computing, Boca Raton, 1972, pp. 451-462.

 

12.        H.C. Williams and C.R. Zarnke, Computer solution of the Diophantine equation x2 - dy4 = -1, Proceedings of the Second Manitoba Conference on Numerical Mathematics, Cong. Numerantium, 7, 1972, pp. 405-416.

 

13.        H.C. Williams and C.R. Zarnke, Computer calculation of units in cubic fields, Proceedings of the Second Manitoba Conference on Numerical Mathematics, Cong. Numerantium, 7, 1972, pp. 433-468.

 

14.        F. Morain, J. Shallit and H.C. Williams, La machine à congruences, Musée des Arts et Métiers La Revue, 14, 1996, pp. 14-19.