# Hugh Williams

#### Professor Emeritus of Mathematics & Statistics

Department of Mathematics and Statistics

#### PhD

University of Waterloo, 1969

#### MMath

University of Waterloo, 1967

#### BSc (Hons)

University of Waterloo, 1966

## Contact information

#### Email

hwilliam@ucalgary.ca

#### Phone

Office: 403.220.6322

Office : MS360

## Research and teaching

### Research areas

• Computational Number Theory
• Design and development of special-purpose hardware devices
• Cryptography
• History of mathematics and computation

## Publications

### Book

• Williams, Hugh. Solving the {P}ell equation  A K Peters, 2002. 397-435. Print.
• Teske, Edlyn and Williams, Hugh. A note on {S}hanks's chains of primes 1838. Springer, 2000. 563-580. Print.
• Stein, Andreas and Williams, Hugh. An improved method of computing the regulator of a real quadratic function field 1423. Springer, 1998. 607-620. Print.
• Teske, E. and Williams, Hugh. A problem concerning a character sum (extended abstract) 1423. Springer, 1998. 351-357. Print.
• Williams, Hugh. \'{E}douard {L}ucas and primality testing , John Wiley \& Sons Inc., 1998. x+525. Print.
• Buchmann, Johannes, Thiel, Christoph and Williams, Hugh. Short representation of quadratic integers 325. Kluwer Acad. Publ., 1995. 159-185. Print.
• Williams, Hugh and Shallit, J. O.. Factoring integers before computers 48. Amer. Math. Soc., 1994. 481-531. Print.
• Louboutin, S., Mollin, Richard and Williams, Hugh. Class groups of exponent two in real quadratic fields , Oxford Univ. Press, 1993. 499-513. Print.
• Mollin, Richard and Williams, Hugh. On a solution of a class number two problem for a family of real quadratic fields de Gruyter, 1991. 95-101. Print.
• Mollin, Richard and Williams, Hugh. Powers of {$2,$} continued fractions, and the class number one problem for real quadratic fields {${\bf Q}(\sqrt d),$} with {$d\equiv 1\pmod 8$} World Sci. Publishing, 1991. 505-516. Print.
• Buchmann, Johannes A. and Williams, Hugh. A key exchange system based on real quadratic fields (extended abstract) 435. Springer, 1990. 335-343. Print.
• Mollin, Richard and Williams, Hugh. Solution of the class number one problem for real quadratic fields of extended {R}ichaud-{D}egert type (with one possible exception) de Gruyter, 1990. 417-425. Print.
• Buchmann, Johannes A., D{\"u}llmann, Stephan and Williams, Hugh. On the complexity and efficiency of a new key exchange system 434. Springer, 1990. 597-616. Print.
• Mollin, Richard and Williams, Hugh. Class number problems for real quadratic fields 154. Cambridge Univ. Press, 1990. 177-195. Print.
• Stephens, A. J. and Williams, Hugh. An open architecture number sieve 154. Cambridge Univ. Press, 1990. 38-75. Print.
• Buchmann, Johannes and Williams, Hugh. Quadratic fields and cryptography 154. Cambridge Univ. Press, 1990. 9-25. Print.
• Mollin, Richard and Williams, Hugh. Class number one for real quadratic fields, continued fractions and reduced ideals 265. Kluwer Acad. Publ., 1989. 481-496. Print.
• Buchmann, Johannes and Williams, Hugh. On the existence of a short proof for the value of the class number and regulator of a real quadratic field 265. Kluwer Acad. Publ., 1989. 327-345. Print.
• Stephens, A. J. and Williams, Hugh. Some computational results on a problem of {E}isenstein de Gruyter, 1989. 869-886. Print.
• Mollin, Richard and Williams, Hugh. Prime producing quadratic polynomials and real quadratic fields of class number one de Gruyter, 1989. 654-663. Print.
• Williams, Hugh. An {$M\sp 3$} public-key encryption scheme 218. Springer, 1986. 358-368. Print.
• Williams, Hugh. Some public-key crypto-functions as intractable as factorization (extended abstract) 196. Springer, 1985. 66-70. Print.
• Williams, Hugh. An overview of factoring Plenum, 1984. 71-80. Print.
• Williams, Hugh. Computationally hard'' problems as a source for cryptosystems 69. Westview, 1982. 11-39. Print.
• Stanton, R. G. and Williams, Hugh. Computation of some number-theoretic coverings 884. Springer, 1981. 8-13. Print.
• Williams, Hugh. A generalization of the {F}ibonacci numbers Louisiana State Univ., 1970. 340-356. Print.

### Book chapter

• Jacobson, Michael, Scheidler, Renate and Williams, Hugh. "The efficiency and security of a real quadratic field-based key exchange protocol" Public-Key Cryptography and Computational Number Theory Berlin: de Gruyter & Co. Publishers, 2001. 89-112. Print.

### Conference proceedings

• Wooding, Kjell and Williams, Hugh. Doubly-focused enumeration of Pseudosquares and Pseudocubes 4076 Springer-Verlag, 2006. 208-221.
• Scheidler, Renate, Buchmann, Johannes and Williams, Hugh. Implementation of a key exchange protocol using real quadratic fields (extended abstract) 1991.

### Journal article

• Jacobson, M.J., Scheidler, Renate and Williams, Hugh. "An improved real quadratic field based key exchange procedure". Journal of Cryptology. The Journal of the International Association for Cryptologic Research 19.2 (2006): 211-239. Print.
• Poorten, A. J., Riele, H. J. J. and Williams, Hugh. "Corrigenda and addition to: {C}omputer verification of the {A}nkeny-{A}rtin-{C}howla conjecture for all primes less than {$100\,000\,000\,000$}'' [{M}ath.\ {C}omp.\ {\bf 70} (2001), no. 235, 1311-1328; \refmr {MR}1709160 (2001j:11125)\endrefmr]". Mathematics of Computation 72.241 (2003): 521-523 (electronic). Print.
• Jacobson, Jr. and Williams, Hugh. "New quadratic polynomials with high densities of prime values". Mathematics of Computation 72.241 (2003): 499-519 (electronic). Print.
• Riele, Herman and Williams, Hugh. "New computations concerning the {C}ohen-{L}enstra heuristics". Experimental Mathematics 12.1 (2003): 99-113. Print.
• Jacobson, Jr. and Williams, Hugh. "Modular arithmetic on elements of small norm in quadratic fields". Designs, Codes and Cryptography. An International Journal 27.1-2 (2002): 93-110. Print.
• Sellers, James A. and Williams, Hugh. "On the infinitude of composite {NSW} numbers". The Fibonacci Quarterly. The Official Journal of the Fibonacci Association 40.3 (2002): 253-254. Print.
• Poorten, A. J., Riele, H. J. J. and Williams, Hugh. "Computer verification of the {A}nkeny-{A}rtin-{C}howla conjecture for all primes less than {$100\,000\,000\,000$}". Mathematics of Computation 70.235 (2001): 1311-1328. Print.
• Stein, Andreas and Williams, Hugh. "Explicit primality criteria for {$(p-1)p\sp n-1$}". Mathematics of Computation 69.232 (2000): 1721-1734. Print.
• Granville, Andrew, Mollin, Richard and Williams, Hugh. "An upper bound on the least inert prime in a real quadratic field". Canadian Journal of Mathematics. Journal Canadien de Math\'ematiques 52.2 (2000): 369-380. Print.
• Jacobson, Jr. and Williams, Hugh. "The size of the fundamental solutions of consecutive {P}ell equations". Experimental Mathematics 9.4 (2000): 631-640. Print.
• Williams, Hugh. "A number theoretic function arising from continued fractions". The Fibonacci Quarterly. The Official Journal of the Fibonacci Association 38.3 (2000): 201-211. Print.
• Poorten, A. J. and Williams, Hugh. "On certain continued fraction expansions of fixed period length". Acta Arithmetica 89.1 (1999): 23-35. Print.
• Teske, Edlyn and Williams, Hugh. "A problem concerning a character sum". Experimental Mathematics 8.1 (1999): 63-72. Print.
• Stein, Andreas and Williams, Hugh. "Some methods for evaluating the regulator of a real quadratic function field". Experimental Mathematics 8.2 (1999): 119-133. Print.
• Williams, Hugh. "Daniel {S}hanks (1917-1996)". Mathematics of Computation 66.219 (1997): 929-934. Print.
• Lukes, Richard F., Scheidler, Renate and Williams, Hugh. "Further tabulation of the Erdos-Selfridge function". Mathematics of Computation 66.220 (1997): 1709-1717. Print.
• Lukes, R. F., Patterson, C. D. and Williams, Hugh. "Some results on pseudosquares". Mathematics of Computation 65.213 John Wiley & Sons Inc., (1996): 361-372, S25-S27. Print.
• Bach, Eric, Lukes, Richard, Shallit, Jeffrey and Williams, Hugh. "Results and estimates on pseudopowers". Mathematics of Computation 65.216 John Wiley & Sons Inc., (1996): 1737-1747. Print.
• Scheidler, Renate, Stein, Andreas and Williams, Hugh. "Key-exchange in real quadratic congruence function fields". Designs, Codes and Cryptography 7.1-2 (1996): 153-174. Print.
• Mollin, Richard and Williams, Hugh. "Corrigenda for: {A} conjecture of {S}. {C}howla via the generalized {R}iemann hypothesis'' [{P}roc.\ {A}mer.\ {M}ath.\ {S}oc. {\bf 102} (1988), no.\ 4, 794-796; \refmr {MR}0934844 (89d:11090)\endrefmr]". Proc. Amer. Math. Soc. 123.3 (1995): 975. Print.
• Mollin, Richard and Williams, Hugh. "Proof, disproof and advances concerning certain conjectures on real quadratic fields {$\bold Q(\sqrt{N\sp 2+4})$}". Canadian Journal of Mathematics. Journal Canadien de Math\'ematiques 47.5 (1995): 1023-1036. Print.
• Williams, Hugh. "Some generalizations of the {$S\sb n$} sequence of {S}hanks". Acta Arithmetica 69.3 John Wiley & Sons Inc., (1995): 199-215. Print.
• Lukes, R. F., Patterson, C. D. and Williams, Hugh. "Numerical sieving devices: their history and some applications". Nieuw Archief voor Wiskunde. Vierde Serie 13.1 John Wiley & Sons Inc., (1995): 113-139. Print.
• Jacobson, Jr., Lukes, Richard F. and Williams, Hugh. "An investigation of bounds for the regulator of quadratic fields". Experimental Mathematics 4.3 John Wiley & Sons Inc., (1995): 211-225. Print.
• Shallit, Jeffrey, Williams, Hugh and Morain, Fran. "Discovery of a lost factoring machine". The Mathematical Intelligencer 17.3 John Wiley & Sons Inc., (1995): 41-47. Print.
• Scheidler, Renate and Williams, Hugh. "A public-key cryptosystem utilizing cyclotomic fields". Designs, Codes and Cryptography 6.2 (1995): 117-131. Print.
• Mollin, Richard, Poorten, A. J. and Williams, Hugh. "Halfway to a solution of {$X\sp 2-DY\sp 2=-3$}". Journal de Th\'eorie des Nombres de Bordeaux 6.2 (1994): 421-457. Print.
• Fung, G. W. and Williams, Hugh. "Errata: {O}n the computation of a table of complex cubic fields with discriminant {$D>-10\sp 6$}'' [{M}ath.\ {C}omp.\ {\bf 55} (1990), no.\ 191, 313-325; \refmr {MR}1023760 (90m:11155)\endrefmr]". Mathematics of Computation 63.207 John Wiley & Sons Inc., (1994): 433. Print.
• Mollin, Richard and Williams, Hugh. "Quadratic residue covers for certain real quadratic fields". Mathematics of Computation 62.206 (1994): 885-897. Print.
• Scheidler, Renate, Buchmann, Johannes and Williams, Hugh. "A key exchange protocol using real quadratic fields". Journal of Cryptology 7.3 (1994): 171-199. Print.
• Mollin, Richard and Williams, Hugh. "Classification and enumeration of real quadratic fields having exactly one noninert prime less than a {M}inkowski bound". Canadian Mathematical Bulletin. Bulletin Canadien de Math\'ematiques 36.1 (1993): 108-115. Print.
• Williams, Hugh. "How was {$F\sb 6$} factored?". Mathematics of Computation 61.203 John Wiley & Sons Inc., (1993): 463-474. Print.
• Mollin, Richard and Williams, Hugh. "Computation of the class number of a real quadratic field". Utilitas Mathematica. A Canadian Journal of Applied Mathematics, Computer Science, and Statistics 41. (1992): 259-308. Print.
• Mollin, Richard and Williams, Hugh. "A complete generalization of {Y}okoi's {$p$}-invariants". Colloquium Mathematicum 63.2 (1992): 285-294. Print.
• Mollin, Richard and Williams, Hugh. "On the period length of some special continued fractions". S\'eminaire de Th\'eorie des Nombres de Bordeaux. S\'erie 2 4.1 (1992): 19-42. Print.
• Mollin, Richard and Williams, Hugh. "On real quadratic fields of class number two". Mathematics of Computation 59.200 (1992): 625-632. Print.
• Louboutin, S., Mollin, Richard and Williams, Hugh. "Class numbers of real quadratic fields, continued fractions, reduced ideals, prime-producing quadratic polynomials and quadratic residue covers". Canadian Journal of Mathematics. Journal Canadien de Math\'ematiques 44.4 (1992): 824-842. Print.
• Fung, Gilbert, Granville, Andrew and Williams, Hugh. "Computation of the first factor of the class number of cyclotomic fields". Journal of Number Theory 42.3 John Wiley & Sons Inc., (1992): 297-312. Print.
• Mollin, Richard and Williams, Hugh. "Consecutive powers in continued fractions". Acta Arithmetica 61.3 (1992): 233-264. Print.
• Scheidler, Renate and Williams, Hugh. "A method of tabulating the number-theoretic function g(k)". Mathematics of Computation 59.199 (1992): 251-257. Print.
• Williams, Hugh. "Some formulas concerning the fundamental unit of a real quadratic field". Discrete Math. 92.1-3 (1991): 431-440. Print.
• Mollin, Richard and Williams, Hugh. "Corrigenda for: {S}olution of a problem of {Y}okoi'' [{P}roc.\ {J}apan {A}cad.\ {S}er.\ {A} {M}ath.\ {S}ci.\ {\bf 66} (1990), no.\ 6, 141-145; \refmr {MR}1065792 (91k:11094)\endrefmr]". Japan Academy. Proceedings. Series A. Mathematical Sciences 67.7 (1991): 253. Print.
• Mollin, Richard and Williams, Hugh. "On the divisor function and class numbers of real quadratic fields. {III}". Japan Academy. Proceedings. Series A. Mathematical Sciences 67.10 (1991): 338-342. Print.
• Mollin, Richard and Williams, Hugh. "Affirmative solution of a conjecture related to a sequence of {S}hanks". Japan Academy. Proceedings. Series A. Mathematical Sciences 67.3 (1991): 70-72. Print.
• Mollin, Richard and Williams, Hugh. "On a determination of real quadratic fields of class number one and related continued fraction period length less than {$25$}". Japan Academy. Proceedings. Series A. Mathematical Sciences 67.1 (1991): 20-25. Print.
• Buchmann, Johannes A. and Williams, Hugh. "Some remarks concerning the complexity of computing class groups of quadratic fields". Journal of Complexity 7.3 (1991): 311-315. Print.
• Fung, Gilbert W. and Williams, Hugh. "On the computation of a table of complex cubic fields with discriminant {$D>-10\sp 6$}". Mathematics of Computation 55.191 (1990): 313-325. Print.
• Fung, G. W. and Williams, Hugh. "Quadratic polynomials which have a high density of prime values". Mathematics of Computation 55.191 (1990): 345-353. Print.
• Buchmann, J., Sands, J. W. and Williams, Hugh. "{$p$}-adic computation of real quadratic class numbers". Mathematics of Computation 54.190 (1990): 855-868. Print.
• Fung, G. W., Str{\"o}her, H., Williams, Hugh and Zimmer, H. G.. "Torsion groups of elliptic curves with integral {$j$}-invariant over pure cubic fields". Journal of Number Theory 36.1 (1990): 12-45. Print.
• Mollin, Richard and Williams, Hugh. "Solution of a problem of {Y}okoi". Japan Academy. Proceedings. Series A. Mathematical Sciences 66.6 (1990): 141-145. Print.
• Williams, Hugh. "The period length of {V}orono\u\i's algorithm for certain cubic orders". Publicationes Mathematicae Debrecen 37.3-4 (1990): 245-265. Print.
• Mollin, Richard and Williams, Hugh. "Continued fractions of period five and real quadratic fields of class number one". Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica 56.1 (1990): 55-63. Print.
• Williams, Hugh. "Eisenstein's problem and continued fractions". Utilitas Mathematica. An International Journal of Discrete and Combinatorial Mathematics, and Statistical Design 37. (1990): 145-157. Print.
• Mollin, Richard and Williams, Hugh. "Quadratic nonresidues and prime-producing polynomials". Canadian Mathematical Bulletin. Bulletin Canadien de Math\'ematiques 32.4 (1989): 474-478. Print.
• Buchmann, Johannes and Williams, Hugh. "On the computation of the class number of an algebraic number field". Mathematics of Computation 53.188 (1989): 679-688. Print.
• Mollin, Richard and Williams, Hugh. "Period four and real quadratic fields of class number one". Japan Academy. Proceedings. Series A. Mathematical Sciences 65.4 (1989): 89-93. Print.
• Mollin, Richard and Williams, Hugh. "Real quadratic fields of class number one and continued fraction period less than six". La Soci\'et\'e Royale du Canada. L'Academie des Sciences. Comptes Rendus Math\'ematiques. (Mathematical Reports) 11.2 (1989): 51-56. Print.
• Mollin, Richard and Williams, Hugh. "A conjecture of {S}. {C}howla via the generalized {R}iemann hypothesis". Proc. Amer. Math. Soc. 102.4 (1988): 794-796. Print.
• Williams, Hugh. "Corrigenda: {C}omputation of the class number and class group of a complex cubic field'' [{M}ath.\ {C}omp.\ {\bf 45} (1985), no.\ 171, 223-231; \refmr {MR}0790655 (86m:11078)\endrefmr] by {G}.\ {W}.\ {D}ueck and {W}illiams". Mathematics of Computation 50.182 (1988): 655-657. Print.
• Stephens, A. J. and Williams, Hugh. "Computation of real quadratic fields with class number one". Mathematics of Computation 51.184 (1988): 809-824. Print.
• Stephens, A. J. and Williams, Hugh. "Some computational results on a problem concerning powerful numbers". Mathematics of Computation 50.182 (1988): 619-632. Print.
• Buchmann, Johannes and Williams, Hugh. "On the infrastructure of the principal ideal class of an algebraic number field of unit rank one". Mathematics of Computation 50.182 (1988): 569-579. Print.
• Mollin, Richard and Williams, Hugh. "On prime valued polynomials and class numbers of real quadratic fields". Nagoya Mathematical Journal 112. (1988): 143-151. Print.
• Williams, Hugh. "A note on the primality of {$6\sp {2\sp n}+1$} and {$10\sp {2\sp n}+1$}". The Fibonacci Quarterly. The Official Journal of the Fibonacci Association 26.4 (1988): 296-305. Print.
• Buchmann, Johannes and Williams, Hugh. "A key-exchange system based on imaginary quadratic fields". Journal of Cryptology. The Journal of the International Association for Cryptologic Research 1.2 (1988): 107-118. Print.
• Barrucand, Pierre, Loxton, John and Williams, Hugh. "Some explicit upper bounds on the class number and regulator of a cubic field with negative discriminant". Pacific Journal of Mathematics 128.2 (1987): 209-222. Print.
• Williams, Hugh and Wunderlich, M. C.. "On the parallel generation of the residues for the continued fraction factoring algorithm". Mathematics of Computation 48.177 (1987): 405-423. Print.
• Williams, Hugh. "Effective primality tests for some integers of the forms {$A5\sp n-1$} and {$A7\sp n-1$}". Mathematics of Computation 48.177 (1987): 385-403. Print.
• Buchmann, Johannes and Williams, Hugh. "On principal ideal testing in totally complex quartic fields and the determination of certain cyclotomic constants". Mathematics of Computation 48.177 (1987): 55-66. Print.
• Buchmann, Johannes and Williams, Hugh. "On principal ideal testing in algebraic number fields". Journal of Symbolic Computation 4.1 (1987): 11-19. Print.
• Williams, Hugh. "The spacing of the minima in certain cubic lattices". Pacific Journal of Mathematics 124.2 (1986): 483-496. Print.
• Williams, Hugh and Dubner, Harvey. "The primality of {$R1031$}". Mathematics of Computation 47.176 (1986): 703-711. Print.
• Kurtz, G. C., Shanks, Daniel and Williams, Hugh. "Fast primality tests for numbers less than $50cdot 10sp 9$". Mathematics of Computation 46.174 (1986): 691-701. Print.
• Tennenhouse, M. and Williams, Hugh. "A note on class-number one in certain real quadratic and pure cubic fields". Mathematics of Computation 46.173 (1986): 333-336. Print.
• Dueck, G. and Williams, Hugh. "Computation of the class number and class group of a complex cubic field". Mathematics of Computation 45.171 (1985): 223-231. Print.
• Patterson, C. D. and Williams, Hugh. "Some periodic continued fractions with long periods". Mathematics of Computation 44.170 (1985): 523-532. Print.
• Williams, Hugh. "Continued fractions and number-theoretic computations". The Rocky Mountain Journal of Mathematics 15.2 (1985): 621-655. Print.
• Williams, Hugh. "Some public-key crypto-functions as intractable as factorization". Cryptologia 9.3 (1985): 223-237. Print.
• Williams, Hugh. "A note on the period length of the continued fraction expansion of certain {$\sqrt D$}". Utilitas Mathematica. An International Journal of Discrete and Combinatorial Mathematics, and Statistical Design 28. (1985): 201-209. Print.
• Williams, Hugh. "On mid-period criteria for the nearest integer continued fraction expansion of {$\sqrt D$}". Utilitas Mathematica. An International Journal of Discrete and Combinatorial Mathematics, and Statistical Design 27. (1985): 169-185. Print.
• Williams, Hugh and Dueck, G. W.. "An analogue of the nearest integer continued fraction for certain cubic irrationalities". Mathematics of Computation 42.166 (1984): 683-705. Print.
• Williams, Hugh. "Factoring on a computer". The Mathematical Intelligencer 6.3 (1984): 29-36. Print.
• Seah, Eric, Washington, Lawrence C. and Williams, Hugh. "The calculation of a large cubic class number with an application to real cyclotomic fields". Mathematics of Computation 41.163 (1983): 303-305. Print.
• Williams, Hugh, Dueck, G. W. and Schmid, B. K.. "A rapid method of evaluating the regulator and class number of a pure cubic field". Mathematics of Computation 41.163 (1983): 235-286. Print.
• Patterson, C. D. and Williams, Hugh. "A report on the {U}niversity of {M}anitoba {S}ieve {U}nit". Congressus Numerantium 37. (1983): 85-98. Print.
• Williams, Hugh. "A note on the {F}ibonacci quotient {$F\sb{p-\varepsilon }/p$}". Canadian Mathematical Bulletin 25.3 (1982): 366-370. Print.
• Williams, Hugh. "A class of primality tests for trinomials which includes the {L}ucas-{L}ehmer test". Pacific Journal of Mathematics 98.2 (1982): 477-494. Print.
• Williams, Hugh. "Corrigendum: {S}ome primes with interesting digit patterns'' [{M}ath. {C}omp. {\bf 32} (1978), no. 144, 1306-1310;\ {MR} {\bf 58} \#484]". Mathematics of Computation 39.160 (1982): 759. Print.
• Baillie, Robert, Cormack, G. and Williams, Hugh. "Corrigenda: {T}he problem of {S}ierpi\'nski concerning {$k\cdot 2\sp{n}+1$}'' [{M}ath. {C}omp. {\bf 37} (1981), no. 155, 229-231]". Mathematics of Computation 39.159 (1982): 308. Print.
• Williams, Hugh. "A {$p+1$} method of factoring". Mathematics of Computation 39.159 (1982): 225-234. Print.
• Williams, Hugh. "Determination of principal factors in {${\scr Q}(\sqrt{D})$} and {${\scr Q}(\root 3\of D)$}". Mathematics of Computation 38.157 (1982): 261-274. Print.
• Williams, Hugh. "The influence of computers in the development of number theory". Computers \& Mathematics with Applications. An International Journal 8.2 (1982): 75-93. Print.
• Shanks, Daniel and Williams, Hugh. "Gunderson's function in {F}ermat's last theorem". Mathematics of Computation 36.153 (1981): 291-295. Print.
• Baillie, Robert, Cormack, G. and Williams, Hugh. "The problem of {S}ierpi\'nski concerning {$k\cdot 2\sp{n}+1$}". Mathematics of Computation 37.155 (1981): 229-231. Print.
• Williams, Hugh. "Some results concerning {V}orono\u\i 's continued fraction over {${\bf Q}(\root 3\of{D})$}". Mathematics of Computation 36.154 (1981): 631-652. Print.
• Williams, Hugh. "A numerical investigation into the length of the period of the continued fraction expansion of {$\sqrt{D}$}". Mathematics of Computation 36.154 (1981): 593-601. Print.
• Williams, Hugh. "The primality of certain integers of the form {$2Ar\sp{n}-1$}". Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica 39.1 (1981): 7-17. Print.
• Newman, Morris, Shanks, Daniel and Williams, Hugh. "Simple groups of square order and an interesting sequence of primes". Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica 38.2 (1980/81): 129-140. Print.
• Williams, Hugh. "Improving the speed of calculating the regulator of certain pure cubic fields". Mathematics of Computation 35.152 (1980): 1423-1434. Print.
• Cormack, G. V. and Williams, Hugh. "Some very large primes of the form {$k\cdot 2\sp{m}+1$}". Mathematics of Computation 35.152 (1980): 1419-1421. Print.
• Williams, Hugh, Cormack, G. and Seah, E.. "Calculation of the regulator of a pure cubic field". Mathematics of Computation 34.150 (1980): 567-611. Print.
• Williams, Hugh. "Some results concerning the nearest integer continued fraction expansion of {$\surd D$}". Journal f\"ur die Reine und Angewandte Mathematik 315. (1980): 1-15. Print.
• Williams, Hugh. "A modification of the {RSA} public-key encryption procedure". Institute of Electrical and Electronics Engineers. Transactions on Information Theory 26.6 (1980): 726-729. Print.
• Purdy, G., Terras, R., Terras, A. and Williams, Hugh. "Graphing {$L$}-functions of {K}ronecker symbols in the real part of the critical strip". Math. Student 47.2-4 (1979): 101-131 (1985). Print.
• Williams, Hugh and Shanks, Daniel. "A note on class-number one in pure cubic fields". Mathematics of Computation 33.148 (1979): 1317-1320. Print.
• Williams, Hugh and Buhr, P. A.. "Calculation of the regulator of {${\bf Q}(\surd D)$} by use of the nearest integer continued fraction algorithm". Mathematics of Computation 33.145 (1979): 369-381. Print.
• Williams, Hugh. "Errata: {C}ertain pure cubic fields with class-number one'' [{M}ath. {C}omp. {\bf 31} (1977), no. 138, 578-580;\ {MR} {\bf 55} \#5578]". Mathematics of Computation 33.146 (1979): 847-848. Print.
• Diaz, F., Shanks, Daniel and Williams, Hugh. "Quadratic fields with {$3$}-rank equal to {$4$}". Mathematics of Computation 33.146 (1979): 836-840. Print.
• Williams, Hugh and Seah, E.. "Some primes of the form {$(a\sp{n}-1)/(a-1)$}". Mathematics of Computation 33.148 (1979): 1337-1342. Print.
• Williams, Hugh and Schmid, B.. "Some remarks concerning the {M}.{I}.{T}. public-key cryptosystem". BIT; Nordisk Tidskrift for Informationsbehandling (BIT) 19.4 (1979): 525-538. Print.
• Williams, Hugh. "Some properties of a special set of recurring sequences". Pacific Journal of Mathematics 77.1 (1978): 273-285. Print.
• Williams, Hugh. "Some primes with interesting digit patterns". Mathematics of Computation 32.144 (1978): 1306-1310. Print.
• Williams, Hugh and Holte, R.. "Some observations on primality testing". Mathematics of Computation 32.143 (1978): 905-917. Print.
• Williams, Hugh. "Primality testing on a computer". Ars Combinatoria 5. (1978): 127-185. Print.
• Williams, Hugh. "On numbers analogous to the {C}armichael numbers". Canadian Mathematical Bulletin. Bulletin Canadien de Math\'ematiques 20.1 (1977): 133-143. Print.
• Williams, Hugh. "Certain pure cubic fields with class-number one". Mathematics of Computation 31.138 (1977): 578-580. Print.
• Buell, D. A., Williams, Hugh and Williams, K. S.. "On the imaginary bicyclic biquadratic fields with class-number {$2$}". Mathematics of Computation 31.140 (1977): 1034-1042. Print.
• Matthew, G. and Williams, Hugh. "Some new primes of the form {$k\cdot 2\sp{n}+1$}". Mathematics of Computation 31.139 (1977): 797-798. Print.
• Williams, Hugh and Holte, R.. "Computation of the solution of {$x\sp{3}+Dy\sp{3}=1$}". Mathematics of Computation 31.139 (1977): 778-785. Print.
• Williams, Hugh. "Properties of some functions similar to {L}ucas functions". The Fibonacci Quarterly. The Official Journal of the Fibonacci Association 15.2 (1977): 97-112. Print.
• Williams, Hugh. "Some results on fundamental units in cubic fields". Journal fur die Reine und Angewandte Mathematik 286/287. (1976): 75-85. Print.
• Stanton, R. G., Sudler, Jr. and Williams, Hugh. "An upper bound for the period of the simple continued fraction for {$\sqrt{D}$}". Pacific Journal of Mathematics 67.2 (1976): 525-536. Print.
• Williams, Hugh and Broere, J.. "A computational technique for evaluating {$L(1,\chi )$} and the class number of a real quadratic field". Mathematics of Computation 30.136 (1976): 887-893. Print.
• Williams, Hugh and Judd, J. S.. "Some algorithms for prime testing using generalized {L}ehmer functions". Mathematics of Computation 30.136 (1976): 867-886. Print.
• Williams, Hugh and Judd, J. S.. "Determination of the primality of {$N$} by using factors of {$N\sp{2}\pm 1$}". Mathematics of Computation 30.133 (1976): 157-172. Print.
• Barrucand, Pierre, Williams, Hugh and Baniuk, L.. "A computational technique for determining the class number of a pure cubic field". Mathematics of Computation 30.134 (1976): 312-323. Print.
• Williams, Hugh. "A generalization of {L}ehmer's functions". Acta Arithmetica 29.4 (1976): 315-341. Print.
• Stanton, R. G. and Williams, Hugh. "An application of combinatorics in number theory". Ars Combinatoria 1.1 (1976): 321-330. Print.
• Williams, Hugh. "The rank of apparition of a generalized {F}ibonacci sequence". The Fibonacci Quarterly. The Official Journal of the Fibonacci Association 13.3 (1975): 240-242. Print.
• Williams, Hugh. "On {F}ibonacci numbers of the form {$k\sp{2}+1$}". The Fibonacci Quarterly. The Official Journal of the Fibonacci Association 13.3 (1975): 213-214. Print.
• Williams, Hugh and Zarnke, C. R.. "Some algorithms for solving a cubic congruence modulo {$p$}". Utilitas Mathematica. A Canadian Journal of Applied Mathematics, Computer Science, and Statistics 6. (1974): 285-306. Print.
• Williams, Hugh. "The quadratic character of a certain quadratic surd". Utilitas Mathematica. A Canadian Journal of Applied Mathematics, Computer Science, and Statistics 5. (1974): 49-55. Print.
• Williams, Hugh and Zarnke, C. R.. "Computer calculation of units in cubic fields". , (1973): 433-468. Congressus Numerantium, No. VII. Print.
• Williams, Hugh and Zarnke, C. R.. "Computer solution of the {D}iophantine equation {$x\sp{2}-dy\sp{4}=-1$}". , (1973): 405-416. Congressus Numerantium, No. VII. Print.
• Williams, Hugh and Zarnke, C. R.. "Computation of the solutions of the {D}iophantine equation {$x\sp{2}-dy\sp{4}=1$}". , (1972): 463-483. Print.
• Beach, B. D. and Williams, Hugh. "A numerical investigation of the {D}iophantine equation {$x\sp{2}-dy\sp{2}=-1$}". , (1972): 37-68. Print.
• Williams, Hugh. "Some algorithms for solving {$x\sp{q}\equiv N$} {$({\rm mod}\ p)$}". , (1972): 451-462. Print.
• Williams, Hugh. "The primality of {$N=2A3\sp{n}-1$}". Canadian Mathematical Bulletin. Bulletin Canadien de Math\'ematiques 15. (1972): 585-589. Print.
• Williams, Hugh and Zarnke, C. R.. "Some prime numbers of the forms {$2A3\sp{n}+1$} and {$2A3\sp{n}-1$}". Mathematics of Computation 26. (1972): 995-998. Print.
• Williams, Hugh. "On a generalization of the {L}ucas functions". Acta Arithmetica 20. (1972): 33-51. Print.
• Williams, Hugh. "Fibonacci numbers obtained from {P}ascal's triangle with generalizations". The Fibonacci Quarterly. The Official Journal of the Fibonacci Association 10.4 (1972): 405-412. Print.
• Williams, Hugh. "An algorithm for determining certain large primes". , (1971): 533-556. Print.
• Gryte, D. G., Kingsley, R. A. and Williams, Hugh. "On certain forms of {F}ibonacci numbers". , (1971): 339-344. Print.
• Beach, B. D. and Williams, Hugh. "Some computer results on periodic continued fractions". , (1971): 133-146. Print.
• Beach, B. D., Williams, Hugh and Zarnke, C. R.. "Some computer results on units in quadratic and cubic fields". , (1971): 609-648. Print.
• Williams, Hugh and Zarnke, C. R.. "Computation of the solutions of the {D}iophantine equation {$x\sp{3}+dy\sp{3}=1$}". , (1971): 671-676. Print.
• Beach, B. D. and Williams, Hugh. "A computer algorithm for determining the solution of the {D}iophantine equation {$x\sp{4}-dy\sp{4}=1$}". , (1971): 663-670. Dept. Comput. Sci., Univ. Print.
• Williams, Hugh. "Some properties of the general {L}ucas polynomials". Matrix Tensor Quart. 21. (1971): 91-93, 82. Print.
• Williams, Hugh. "Note on a diophantine equation". Elemente der Mathematik. Eine Zeitschrift der Schweizerischen Mathematischen Gesellschaft. Une Revue de la Soci\'et\'e Math\'ematique Suisse. Una Rivista della Societ\`a Matematica Svizzera 25. (1970): 123-125. Print.
• Williams, Hugh and Zarnke, C. R.. "A report on prime numbers of the forms {$M=(6a+1)2\sp{2m-1}-1$} and {$M\sp{\prime} =(6a-1)2\sp{2m}-1$}". Mathematics of Computation 22. (1968): 420-422. Print.
• Stanton, R. G., Williams, Hugh and Zarnke, C. R.. "A packing problem". Canadian Mathematical Bulletin. Bulletin Canadien de Math\'ematiques 10. (1967): 287-297. Print.