Wilson’s theorem | Number Theory, Prime Numbers & Congruence

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Hosch</div> <div class="editor-description font-12 font-serif mt-5 clamp-description text-black">William L. Hosch was an editor at Encyclopædia Britannica.</div> </a> <div data-popper-arrow></div> </div> <span class="btn btn-link editor-link p-0 qa-byline-link gtm-byline font-12 byline-contributor text-decoration-underline"> William L. Hosch</span></div> <div class="font-serif font-12 text-gray-700"> <span class="qa-fact-checked-by">Fact-checked by</span> <div class="editor-popover popover p-0"> <a class="d-block p-20 font-12" href="/editor/The-Editors-of-Encyclopaedia-Britannica/4419" > <div class="editor-title font-16 font-weight-bold">The Editors of Encyclopaedia Britannica</div> <div class="editor-description font-12 font-serif mt-5 text-black">Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree. They write new content and verify and edit content received from contributors.</div> </a> <div data-popper-arrow></div> </div> <span class="btn btn-link editor-link p-0 qa-byline-link font-12 "> The Editors of Encyclopaedia Britannica</span></div> <div class="last-updated font-12 font-serif"> <a class="byline-edit-history" href="https://www.britannica.com/science/Wilsons-theorem/additional-info#history" rel="nofollow">Article History</a> </div></div> </div> <button class="d-flex d-lg-none btn btn-outline-blue border rounded-sm shadow-sm mobile-toc-button gtm-mobile-toc-inline-button d-none d-sm-block js-sections-inline-button module-spacing btn d-lg-none"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="toc"></em> Table of Contents </button> <div class="d-flex d-sm-none flex-row"> <button class="d-flex d-lg-none btn btn-outline-blue border rounded-sm shadow-sm mobile-toc-button gtm-mobile-toc-inline-button js-sections-inline-button module-spacing"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="toc"></em> Table of Contents </button> <button class="ai-ask-button btn border-2 ai-ask-button btn border-2 module-spacing btn-sm js-inline-ai-ask-button btn-outline-red-400 border-red-400 p-10 ml-5"> Ask the Chatbot a Question </button> </div> <div class="js-qf-module qf-module px-40 px-sm-20 py-15 mx-auto module-spacing font-14 bg-gray-50 rounded"> <div class="facts-list mt-10"> <div class=""> <div class="js-fact mb-10 line-clamp clamp-3"> <dl> <dt>Related Topics: </dt> <dd><a href="/science/number-theory" topicid="422325">number theory</a></dd> <dd><a href="/science/prime-number" topicid="476309">prime</a></dd> </dl> <button class="js-more-btn d-none btn btn-unstyled font-12 bg-gray-50" aria-label="Toggle more/less fact data"> <em class="js-content link-blue">(Show more)</em> </button> </div> <div class="text-center"> <a class="btn btn-sm btn-link p-0" href="/facts/Wilsons-theorem"> See all related content </a> </div> </div> </div> </div><span class="marker before-article"></span><section data-level="1" id="ref1"><span class="marker PREMOD1 mod-inline"></span><p class="topic-paragraph"><strong><span id="ref1017129"></span>Wilson’s theorem</strong>, in <a href="https://www.britannica.com/science/number-theory" class="md-crosslink " data-show-preview="true">number theory</a>, <a href="https://www.britannica.com/topic/theorem" class="md-crosslink " data-show-preview="true">theorem</a> that any <a href="https://www.britannica.com/science/prime-number" class="md-crosslink autoxref " data-show-preview="true">prime</a> <em>p</em> divides (<em>p</em> − 1)! + 1, where <em>n</em>! is the <a href="https://www.britannica.com/science/factorial" class="md-crosslink " data-show-preview="true">factorial</a> notation for 1 × 2 × 3 × 4 × ⋯ × <em>n</em>. For example, 5 divides (5 − 1)! + 1 = 4! + 1 = 25. The <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="conjecture" href="https://www.britannica.com/dictionary/conjecture" data-type="EB">conjecture</a> was first published by the English mathematician <span id="ref1107589"></span><a href="https://www.britannica.com/biography/Edward-Waring" class="md-crosslink " data-show-preview="true">Edward Waring</a> in <em>Meditationes Algebraicae</em> (1770; “Thoughts on Algebra”), where he ascribed it to the English mathematician John Wilson.</p><div class="one-good-fact-module"> </div><span class="marker MOD1 mod-inline"></span><span class="marker PREMOD2 mod-inline"></span><p class="topic-paragraph">The theorem was proved by the French mathematician <span id="ref1107590"></span><a href="https://www.britannica.com/biography/Joseph-Louis-Lagrange-comte-de-lEmpire" class="md-crosslink " data-show-preview="true">Joseph-Louis Lagrange</a> in 1771. The converse of the theorem is also true; that is, (<em>n</em> − 1)! + 1 is not divisible by a composite number <em>n</em>. In theory, these theorems provide a test for <a href="https://www.britannica.com/science/prime-number" class="md-crosslink " data-show-preview="true">primes</a>; in practice, the calculations are impractical for large numbers.</p><span class="marker MOD2 mod-inline"></span></section> <span class="md-signature"><a href="/editor/William-L-Hosch/6481">William L. Hosch</a></span> <span class="marker end-of-content"></span><span class="marker after-article"></span></div> <div id="chatbot-root"></div> </div> </div> </div> <div class="ai-dialog-placeholder"></div> </div> </div> <aside class="col-md-da-320"></aside> </div> </div> </div> </div> </article></div> </div></div> </div> </main> <div id="md-footer"></div> <noscript><iframe src="//www.googletagmanager.com/ns.html?id=GTM-5W6NC8" height="0" width="0" style="display:none;visibility:hidden"></iframe></noscript> <script type="text/javascript" id="_informizely_script_tag"> var IzWidget = IzWidget || {}; (function (d) { var scriptElement = d.createElement('script'); scriptElement.type = 'text/javascript'; scriptElement.async = true; scriptElement.src = "https://insitez.blob.core.windows.net/site/f780f33e-a610-4ac2-af81-3eb184037547.js"; var node = d.getElementById('_informizely_script_tag'); node.parentNode.insertBefore(scriptElement, node); } )(document); </script>  <script> window.ap3c = window.ap3c || {}; var ap3c = window.ap3c; ap3c.cmd = ap3c.cmd || []; ap3c.cmd.push(function() { ap3c.init('ZO4siT4cLwnykPnzZWJtd3Byb2Q', 'https://engage.email.britannica.com/'); ap3c.track({v: 0}); }); ap3c.activity = function(act) { ap3c.act = (ap3c.act || []); ap3c.act.push(act); }; var s, t; s = document.createElement('script'); s.type = 'text/javascript'; s.src = "https://engage.email.britannica.com/app.js"; t = document.getElementsByTagName('script')[0]; t.parentNode.insertBefore(s, t); </script> <script class="marketing-page-info" type="application/json"> {"pageType":"Topic","templateName":"DESKTOP","pageNumber":1,"pagesTotal":1,"pageId":644847,"pageLength":130,"initialLoad":true,"lastPageOfScroll":false} </script> <script class="marketing-content-info" type="application/json"> [] </script> <script src="https://cdn.britannica.com/mendel-resources/3-130/js/libs/jquery-3.5.0.min.js?v=3.130.14"></script> <script type="text/javascript" data-type="Init Mendel Code Splitting"> (function() { $.ajax({ dataType: 'script', cache: true, url: 'https://cdn.britannica.com/mendel-resources/3-130/dist/topic-page.js?v=3.130.14' }); })(); </script> <script class="analytics-metadata" type="application/json"> {"leg":"D","adLeg":"D","userType":"ANONYMOUS","pageType":"Topic","pageSubtype":null,"articleTemplateType":"SMALL","gisted":false,"pageNumber":1,"hasSummarizeButton":false,"hasAskButton":false} </script> <script type="text/javascript"> EBStat={accountId:-1,hostnameOverride:'webstats.eb.com',domain:'www.britannica.com', json:''}; </script> <script type="text/javascript"> ( function() { $.ajax( { dataType: 'script', cache: true, url: '//www.britannica.com/webstats/mendelstats.js?v=1' } ) .done( function() { try {writeStat(null,EBStat);} catch(err){} } ); })(); </script> <div id="bc-fixed-dialogue"></div> </body> </html>

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Wilson’s theorem | Number Theory, Prime Numbers & Congruence | Britannica