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dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Canadian mathematician (born 1943)</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid 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If possible, verify the text with references provided in the foreign-language article.</li> <li>You <b>must</b> provide <a href="/wiki/Wikipedia:Copying_within_Wikipedia" title="Wikipedia:Copying within Wikipedia">copyright attribution</a> in the <a href="/wiki/Help:Edit_summary" title="Help:Edit summary">edit summary</a> accompanying your translation by providing an <a href="/wiki/Help:Interlanguage_links" title="Help:Interlanguage links">interlanguage link</a> to the source of your translation. A model attribution edit summary is <code>Content in this edit is translated from the existing German Wikipedia article at [[:de:Hugh C. Williams]]; see its history for attribution.</code></li> <li>You may also add the template <code>{{Translated|de|Hugh C. Williams}}</code> to the <a href="/wiki/Talk:Hugh_C._Williams" title="Talk:Hugh C. 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Williams</div></th></tr><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Hugh_C._Williams.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Hugh_C._Williams.jpg/220px-Hugh_C._Williams.jpg" decoding="async" width="220" height="155" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Hugh_C._Williams.jpg/330px-Hugh_C._Williams.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/8/84/Hugh_C._Williams.jpg 2x" data-file-width="400" data-file-height="282" /></a></span><div class="infobox-caption">Williams in 1984</div></td></tr><tr><th scope="row" class="infobox-label">Born</th><td class="infobox-data"><span style="display:none"> (<span class="bday">1943-07-23</span>) </span>23 July 1943<span class="noprint ForceAgeToShow"> (age 81)</span><br /><div style="display:inline" class="birthplace"><a href="/wiki/London,_Ontario" title="London, Ontario">London, Ontario</a>, Canada</div></td></tr><tr><th scope="row" class="infobox-label">Education</th><td class="infobox-data"><a href="/wiki/University_of_Waterloo" title="University of Waterloo">University of Waterloo</a></td></tr><tr><th scope="row" class="infobox-label">Occupation</th><td class="infobox-data role">Mathematician</td></tr><tr><td colspan="2" class="infobox-full-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546"><b>Scientific career</b></td></tr><tr><th scope="row" class="infobox-label">Fields</th><td class="infobox-data category"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="plainlist"> <ul><li><a href="/wiki/Number_theory" title="Number theory">Number theory</a></li> <li><a href="/wiki/Cryptography" title="Cryptography">Cryptography</a></li></ul> </div></td></tr><tr><th scope="row" class="infobox-label">Institutions</th><td class="infobox-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"> <ul><li><a href="/wiki/York_University" title="York University">York University</a></li> <li><a href="/wiki/University_of_Manitoba" title="University of Manitoba">University of Manitoba</a></li> <li><a href="/wiki/University_of_Calgary" title="University of Calgary">University of Calgary</a></li></ul> </div></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Thesis" title="Thesis">Thesis</a></th><td class="infobox-data"><i> A Generalization of the Lucas Functions </i> <span style="font-size:97%;">(1969)</span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Doctoral_advisor" title="Doctoral advisor">Doctoral advisors</a></th><td class="infobox-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"> <ul><li>Ronald C. Mullin</li> <li><a href="/wiki/Ralph_Gordon_Stanton" title="Ralph Gordon Stanton">Ralph Gordon Stanton</a></li></ul> </div></td></tr><tr><th scope="row" class="infobox-label">Doctoral students</th><td class="infobox-data"><a href="/wiki/Renate_Scheidler" title="Renate Scheidler">Renate Scheidler</a></td></tr><tr style="display:none"><td colspan="2"> </td></tr></tbody></table> <p><b>Hugh Cowie Williams</b> (born 23 July 1943) is a Canadian mathematician. He deals with <a href="/wiki/Number_theory" title="Number theory">number theory</a> and <a href="/wiki/Cryptography" title="Cryptography">cryptography</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Early_life">Early life</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hugh_C._Williams&action=edit&section=1" title="Edit section: Early life"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Williams studied mathematics at the <a href="/wiki/University_of_Waterloo" title="University of Waterloo">University of Waterloo</a> (bachelor's degree 1966, master's degree 1967), where he received his doctorate in 1969 in computer science under Ronald C. Mullin and <a href="/wiki/Ralph_Gordon_Stanton" title="Ralph Gordon Stanton">Ralph Gordon Stanton</a> (<i>A generalization of the Lucas functions</i>).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> He was a post-doctoral student at <a href="/wiki/York_University" title="York University">York University</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Career">Career</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hugh_C._Williams&action=edit&section=2" title="Edit section: Career"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1970 he became assistant professor at the <a href="/wiki/University_of_Manitoba" title="University of Manitoba">University of Manitoba</a>, where in 1972 he attained associate professor status and professor in 1979. </p><p>In 2001 he became a professor at the <a href="/wiki/University_of_Calgary" title="University of Calgary">University of Calgary</a>, and professor emeritus since 2004. Since 2001 he has held the "iCore Chair" in Algorithmic Number Theory and Cryptography. </p><p>Together with <a href="/wiki/Rei_Safavi-Naini" class="mw-redirect" title="Rei Safavi-Naini">Rei Safavi-Naini</a> he heads the Institute for Security, Privacy and Information Assurance (ISPIA) - formerly Centre for Information Security and Cryptography - at Calgary.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Between 1998 and 2001 he was an adjunct professor at the University of Waterloo. He was a visiting scholar at the <a href="/wiki/University_of_Bordeaux" title="University of Bordeaux">University of Bordeaux</a>, at <a href="/wiki/Macquarie_University" title="Macquarie University">Macquarie University</a> and at <a href="/wiki/University_of_Leiden" class="mw-redirect" title="University of Leiden">University of Leiden</a>. From 1978 to January 2007 he was associate editor of the journal <i><a href="/wiki/Mathematics_of_Computation" title="Mathematics of Computation">Mathematics of Computation</a></i>. </p><p>Among other things Williams dealt with primality tests;<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Williams_prime" class="mw-redirect" title="Williams prime">Williams primes</a> were named for him. He developed custom hardware for number-theoretical calculations, for example the MSSU in 1995.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> In cryptography, he developed in 1994 with Renate Scheidler and Johannes Buchmann a method of <a href="/wiki/Public_key_cryptography" class="mw-redirect" title="Public key cryptography">public key cryptography</a> based on real quadratic number fields.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Williams developed algorithms for calculating invariants of algebraic number fields such as class numbers and regulators. </p><p>Williams deals with math history and wrote a book about the history of primality tests. In it, he showed among other things that <a href="/wiki/%C3%89douard_Lucas" title="Édouard Lucas">Édouard Lucas</a> worked shortly before his early death on a test similar to today's <a href="/wiki/Elliptic_curve" title="Elliptic curve">elliptic curve</a> method. He reconstructed the method that Fortuné Landry used in 1880 (at the age of 82) to factor the sixth <a href="/wiki/Fermat_number" title="Fermat number">Fermat number</a> (a 20-digit number).<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Together with <a href="/wiki/Jeffrey_Shallit" title="Jeffrey Shallit">Jeffrey Shallit</a> and François Morain he discovered a forgotten mechanical number sieve created by Eugène Olivier Carissan, the first such device from the beginning of the 20th century (1912), and described it in detail.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Publications">Publications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hugh_C._Williams&action=edit&section=3" title="Edit section: Publications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><i>The influence of computers in the development of number theory.</i> In: <i>Computational Mathematics with Applications.</i> Band 8, 1982, S. 75–93.</li> <li><i>Factoring on a computer.</i> Mathematical Intelligencer, 1984, Nr. 3.</li> <li>with Attila Pethö, Horst-Günter Zimmer, Michael Pohst (Hrsg.): <i>Computational Number Theory.</i> de Gruyter 1991.</li> <li>with J. O. Shallit: <i>Factoring integers before computers.</i> In: W. Gautschi (Hrsg.): <i>Mathematics of computation – 50 years of computational mathematics 1943–1993.</i> Proc. Symposium Applied Math., Band 48. American Mathematical Society, 1994, S. 481–531.</li> <li><i><a href="/wiki/%C3%89douard_Lucas" title="Édouard Lucas">Édouard Lucas</a> and primality testing.</i> Wiley 1998. (Canadian Mathematical Society Series of Monographs and Advanced Texts. Band 22.)</li> <li>with M. J. Jacobson: <i>Solving the Pell Equation.</i> Springer 2008.</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hugh_C._Williams&action=edit&section=4" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://mathgenealogy.org/id.php?id=67113">Hugh C. Williams</a> at the <a href="/wiki/Mathematics_Genealogy_Project" title="Mathematics Genealogy Project">Mathematics Genealogy Project</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20171002084118/http://cisac.ucalgary.ca/">"Website of ISPIA"</a>. Archived from <a rel="nofollow" class="external text" href="http://cisac.ucalgary.ca/">the original</a> on 2017-10-02<span class="reference-accessdate">. Retrieved <span class="nowrap">2018-09-30</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Website+of+ISPIA&rft_id=http%3A%2F%2Fcisac.ucalgary.ca%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHugh+C.+Williams" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Er schrieb in den 1970er Jahren die Übersicht <i>Primality testing on a computer.</i> in <i>Ars Combinatoria.</i> Band 5, 1978, S. 127–185, und entwickelte in den 1970er Jahren dazu neue Methoden.<br />Williams, J. S. Judd: <i>Determination of the primality of N by using prime factors of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe131b76af8a2bc86e01b14a7ba843db69c1a164" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.177ex; height:2.676ex;" alt="{\displaystyle N^{2}}"></span> ± 1.</i> In: <i>Mathematics of Computation.</i> Band 30, 1976, S. 157–172<br /><i>Some algorithms for prime testing using generalized Lehmer functions.</i> In: <i>Mathematics of Computation.</i> Band 30, 1976, S. 867–886</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://archive.today/20120630214036/http://math.ucalgary.ca/~hwilliam/research.html%23HardwareDevices">Hardware Sieves: Function and Applications, and other projects</a></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Buchmann, Williams: <i>Quadratic fields and cryptography.</i> In: Loxton (Hrsg.): <i>Number theory and cryptography.</i> 1989</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Williams: <i>How was <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f760637659fa525945b3fdb906c673089642ed4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{6}}"></span> factored?</i> In: <i>Mathematics of Computation.</i> Band 61, 1993, S. 463. Landry publizierte seine Methode nicht, es fanden sich aber Hinweise im Nachlass.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">J. Shallit, H. C. Williams, F. Morain: <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF03024369"><i>Discovery of a lost factoring machine.</i></a> In: <i>Mathematical Intelligencer.</i> 17, No. 3, 1995, S. 41–47; Ivars Peterson: The brothers E. and Pierre Carissan set up the machine in the observatory of Bordeaux and introduced them to the public in 1920.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hugh_C._Williams&action=edit&section=5" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://portal.dnb.de/opac.htm?method=simpleSearch&cqlMode=true&query=idn%3D124230415">Literature by and about Hugh C. Williams</a> in the <a href="/wiki/German_National_Library" title="German National Library">German National Library</a> catalogue</li> <li><a rel="nofollow" class="external text" href="http://math.ucalgary.ca/~hwilliam/">Hugh C. Williams</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110827113139/http://math.ucalgary.ca/~hwilliam/">Archived</a> 2011-08-27 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> at the website of the University of Calgary</li> <li><a rel="nofollow" class="external text" href="http://math.ucalgary.ca/profiles/hugh-williams">Profile of Hugh C. Williams</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140112161644/http://math.ucalgary.ca/profiles/hugh-williams">Archived</a> 2014-01-12 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> at the faculty with links to publications</li> <li><a rel="nofollow" class="external text" href="https://primes.utm.edu/references/refs.cgi?author=Williams">Williams references</a> at the <a href="/wiki/Prime_Pages" class="mw-redirect" title="Prime Pages">Prime Pages</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl 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href="https://isni.org/isni/0000000116769920">ISNI</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://viaf.org/viaf/77587267">VIAF</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.oclc.org/worldcat/entity/E39PBJyrPBKVkFVqCCCKcgv4MP">WorldCat</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/124230415">Germany</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/n79049331">United States</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb150187614">France</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb150187614">BnF data</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://nla.gov.au/anbd.aut-an35882125">Australia</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=vut2011659520&CON_LNG=ENG">Czech Republic</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="http://data.bibliotheken.nl/id/thes/p14234110X">Netherlands</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://dbn.bn.org.pl/descriptor-details/9810645714205606">Poland</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007452870705171">Israel</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://opac.kbr.be/LIBRARY/doc/AUTHORITY/14741009">Belgium</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Academics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://ci.nii.ac.jp/author/DA00277467?l=en">CiNii</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.mathgenealogy.org/id.php?id=67113">Mathematics Genealogy Project</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://zbmath.org/authors/?q=ai:williams.hugh-cowie">zbMATH</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://dblp.org/pid/92/3922">DBLP</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet/MRAuthorID/183160">MathSciNet</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.idref.fr/139452931">IdRef</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Hugh_C._Williams&oldid=1241944249">https://en.wikipedia.org/w/index.php?title=Hugh_C._Williams&oldid=1241944249</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div 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