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0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-One_source plainlinks metadata ambox ambox-content ambox-one_source" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>relies largely or entirely on a <a href="/wiki/Wikipedia:Articles_with_a_single_source" title="Wikipedia:Articles with a single source">single source</a></b>.<span class="hide-when-compact"> Relevant discussion may be found on the <a href="/wiki/Talk:Perron_number##" title="Talk:Perron number">talk page</a>. Please help <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Perron_number&action=edit">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">introducing citations to additional sources</a>.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Perron+number%22">"Perron number"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Perron+number%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Perron+number%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Perron+number%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Perron+number%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Perron+number%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">April 2024</span>)</i></span></div></td></tr></tbody></table> <p> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>Perron number</b> is an <a href="/wiki/Algebraic_integer" title="Algebraic integer">algebraic integer</a> α which is <a href="/wiki/Real_number" title="Real number">real</a> and greater than 1, but such that its <a href="/wiki/Conjugate_element" class="mw-redirect" title="Conjugate element">conjugate elements</a> are all less than α in <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a>. For example, the larger of the two <a href="/wiki/Root_of_a_polynomial" class="mw-redirect" title="Root of a polynomial">roots</a> of the <a href="/wiki/Irreducible_polynomial" title="Irreducible polynomial">irreducible polynomial</a> <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 x^{2}-3x+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}-3x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e08de52158182d7bfacec9bde126ff636d6190a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.719ex; height:2.843ex;" alt="{\displaystyle x^{2}-3x+1}"></span> is a Perron number. </p><p>Perron numbers are named after <a href="/wiki/Oskar_Perron" title="Oskar Perron">Oskar Perron</a>; the <a href="/wiki/Perron%E2%80%93Frobenius_theorem" title="Perron–Frobenius theorem">Perron–Frobenius theorem</a> asserts that, for a real <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a> with positive algebraic entries whose largest <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a> is greater than one, this eigenvalue is a Perron number. As a closely related case, the Perron number of a <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> is defined to be the <a href="/wiki/Spectral_radius" title="Spectral radius">spectral radius</a> of its <a href="/wiki/Adjacency_matrix" title="Adjacency matrix">adjacency matrix</a>. </p><p>Any <a href="/wiki/Pisot_number" class="mw-redirect" title="Pisot number">Pisot number</a> or <a href="/wiki/Salem_number" title="Salem number">Salem number</a> is a Perron number, as is the <a href="/wiki/Mahler_measure" title="Mahler measure">Mahler measure</a> of a <a href="/wiki/Monic_polynomial" title="Monic polynomial">monic</a> <a href="/wiki/Integer" title="Integer">integer</a> polynomial. </p> <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=Perron_number&action=edit&section=1" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><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 id="CITEREFBorwein2007" class="citation book cs1"><a href="/wiki/Peter_Borwein" title="Peter Borwein">Borwein, Peter</a> (2007). <i>Computational Excursions in Analysis and Number Theory</i>. <a href="/wiki/Springer_Verlag" class="mw-redirect" title="Springer Verlag">Springer Verlag</a>. p. 24. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-95444-9" title="Special:BookSources/0-387-95444-9"><bdi>0-387-95444-9</bdi></a>.</cite><span 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