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Disponible en 12 idiomas" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-12" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">12 idiomas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Pythagor%C3%A6iske_primtal" title="Pythagoræiske primtal (danés)" lang="da" hreflang="da" data-title="Pythagoræiske primtal" data-language-autonym="Dansk" data-language-local-name="danés" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Pythagoreische_Primzahl" title="Pythagoreische Primzahl (alemán)" lang="de" hreflang="de" data-title="Pythagoreische Primzahl" data-language-autonym="Deutsch" data-language-local-name="alemán" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Pythagorean_prime" title="Pythagorean prime (inglés)" lang="en" hreflang="en" data-title="Pythagorean prime" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Pythagoraan_alkuluku" title="Pythagoraan alkuluku (finés)" lang="fi" hreflang="fi" data-title="Pythagoraan alkuluku" data-language-autonym="Suomi" data-language-local-name="finés" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_premier_pythagoricien" title="Nombre premier pythagoricien (francés)" lang="fr" hreflang="fr" data-title="Nombre premier pythagoricien" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Pitagoraszi_pr%C3%ADmek" title="Pitagoraszi prímek (húngaro)" lang="hu" hreflang="hu" data-title="Pitagoraszi prímek" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%94%E3%82%BF%E3%82%B4%E3%83%A9%E3%82%B9%E7%B4%A0%E6%95%B0" title="ピタゴラス素数 (japonés)" lang="ja" hreflang="ja" data-title="ピタゴラス素数" data-language-autonym="日本語" data-language-local-name="japonés" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%B8%D1%84%D0%B0%D0%B3%D0%BE%D1%80%D0%BE%D0%B2%D0%BE_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Пифагорово простое число (ruso)" lang="ru" hreflang="ru" data-title="Пифагорово простое число" data-language-autonym="Русский" data-language-local-name="ruso" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Pitagorejsko_pra%C5%A1tevilo" title="Pitagorejsko praštevilo (esloveno)" lang="sl" hreflang="sl" data-title="Pitagorejsko praštevilo" data-language-autonym="Slovenščina" data-language-local-name="esloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Pythagoreiska_primtal" title="Pythagoreiska primtal (sueco)" lang="sv" hreflang="sv" data-title="Pythagoreiska primtal" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%BF%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%95%E0%AF%8B%E0%AE%B0%E0%AE%9A%E0%AF%81_%E0%AE%AA%E0%AE%95%E0%AE%BE%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%A9%E0%AE%BF" title="பித்தகோரசு பகாத்தனி (tamil)" lang="ta" hreflang="ta" data-title="பித்தகோரசு பகாத்தனி" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%95%A2%E9%81%94%E5%93%A5%E6%8B%89%E6%96%AF%E8%B3%AA%E6%95%B8" title="畢達哥拉斯質數 (chino)" lang="zh" hreflang="zh" data-title="畢達哥拉斯質數" 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data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Apariencia</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mover a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ocultar</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">De Wikipedia, la enciclopedia libre</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="es" dir="ltr"><p>Un <b>número primo pitagórico</b><sup id="cite_ref-1" class="reference separada"><a href="#cite_note-1"><span class="corchete-llamada">[</span>1<span class="corchete-llamada">]</span></a></sup> es un <a href="/wiki/N%C3%BAmero_primo" title="Número primo">número primo</a> de la forma 4<i>n</i> + 1. El conjunto de los números primos pitagóricos es exactamente el conjunto de los números primos que pueden ser la longitud de la <a href="/wiki/Hipotenusa" title="Hipotenusa">hipotenusa</a> de un <a href="/wiki/Tri%C3%A1ngulo_rect%C3%A1ngulo" title="Triángulo rectángulo">triángulo rectángulo</a> de lados enteros. </p><p>Los primeros números primos pitagóricos son: </p> <dl><dd><a href="/wiki/Cinco" title="Cinco">5</a>, <a href="/wiki/Trece" title="Trece">13</a>, <a href="/wiki/Diecisiete" title="Diecisiete">17</a>, <a href="/wiki/Veintinueve" title="Veintinueve">29</a>, <a href="/wiki/Treinta_y_siete" title="Treinta y siete">37</a>, <a href="/wiki/Cuarenta_y_uno" title="Cuarenta y uno">41</a>, <a href="/wiki/Cincuenta_y_tres" title="Cincuenta y tres">53</a>, <a href="/wiki/Sesenta_y_uno" title="Sesenta y uno">61</a>, <a href="/wiki/Setenta_y_tres" title="Setenta y tres">73</a>, <a href="/wiki/Ochenta_y_nueve" title="Ochenta y nueve">89</a>, <a href="/wiki/Noventa_y_siete" title="Noventa y siete">97</a>, <a href="/wiki/Ciento_uno" title="Ciento uno">101</a>, <a href="/wiki/Ciento_nueve" title="Ciento nueve">109</a>, <a href="/wiki/Ciento_trece" title="Ciento trece">113</a>, … ((sucesión <a rel="nofollow" class="external text" href="//oeis.org/A002144">A002144</a> en <a href="/wiki/OEIS" title="OEIS">OEIS</a>)).</dd></dl> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Propiedades">Propiedades</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmero_primo_pitag%C3%B3rico&action=edit&section=1" title="Editar sección: Propiedades"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>El <a href="/wiki/Teorema_de_la_suma_de_dos_cuadrados" class="mw-redirect" title="Teorema de la suma de dos cuadrados">teorema de la suma de dos cuadrados</a> de Fermat establece que estos números primos admiten representación única (<a href="/wiki/Salvo" class="mw-redirect" title="Salvo">salvo</a> orden) en forma de suma de dos cuadrados, y que ningún otro número primo se puede expresar de esta manera con la excepción de 2=1<sup>2</sup>+1<sup>2</sup>. Por tanto, solo estos números primos (y el 2) pueden ser normas de <a href="/wiki/Entero_gaussiano" title="Entero gaussiano">enteros gaussianos</a>.Pero como entero gaussiano no son primos, pues <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 5=(2+i)(2-i)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5=(2+i)(2-i)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6f25c37f5b638c7199b153540bb301cb55485e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.49ex; height:2.843ex;" alt="{\displaystyle 5=(2+i)(2-i)}"></span> </p><p>La ley de la <a href="/wiki/Reciprocidad_cuadr%C3%A1tica" class="mw-redirect" title="Reciprocidad cuadrática">reciprocidad cuadrática</a> establece que si <i>p</i> y <i>q</i> son primos impares y al menos uno de ellos es pitagórico, entonces <i>p</i> es un <a href="/wiki/Residuo_cuadr%C3%A1tico" title="Residuo cuadrático">residuo cuadrático</a> módulo <i>q</i> <a href="/wiki/Si_y_solo_si" class="mw-redirect" title="Si y solo si">si y solo si</a> <i>q</i> es un residuo cuadrático módulo <i>p</i>. Sin embargo, si ni <i>p</i> ni <i>q</i> son pitagóricos, entonces <i>p</i> es un residuo cuadrático módulo <i>q</i> si y solo si <i>q</i> <b>no</b> es un residuo cuadrático módulo <i>p</i>. −1 es un residuo cuadrático módulo <i>p</i> si y solo si <i>p</i> es un número primo pitagórico (o 2). </p> <div class="mw-heading mw-heading2"><h2 id="Véase_también"><span id="V.C3.A9ase_tambi.C3.A9n"></span>Véase también</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmero_primo_pitag%C3%B3rico&action=edit&section=2" title="Editar sección: Véase también"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Teorema_de_Fermat_sobre_la_suma_de_dos_cuadrados" title="Teorema de Fermat sobre la suma de dos cuadrados">Teorema de Fermat sobre la suma de dos cuadrados</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Enlaces_externos">Enlaces externos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmero_primo_pitag%C3%B3rico&action=edit&section=3" title="Editar sección: Enlaces externos"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://planetmath.org/Pythagoreanprime">Pythagorean prime</a> en <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Referencias">Referencias</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmero_primo_pitag%C3%B3rico&action=edit&section=4" title="Editar sección: Referencias"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="listaref" style="list-style-type: decimal;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">La denominación no significa que estos primos hayan sido estudiados o nombrados por Pitágoras, sino que surge de un estudio de Fermat y un teorema de <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a>, quien, 125 años después de lo expresado por Fermat en sus notas marginales, pudo demostrar la afirmación por el mismo método de decrecimiento indefinido con el que <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Fermat</a> aseguraba haber demostrado el teorema (En una carta a <a href="/wiki/Gilles_Personne_de_Roberval" class="mw-redirect" title="Gilles Personne de Roberval">Roberval</a>). El nombre se usó posteriormente con el sentido de que es un número primo que es hipotenusa de una terna pitagórica. Esto consta en el Capítulo "Temas de los Enteros", Apartado 8, del libro "El Número. Lenguaje de la Ciencia", de <a href="/wiki/Tob%C3%ADas_Dantzig" title="Tobías Dantzig">Tobías Dantzig</a>, según la traducción española de la cuarta edición, en inglés, publicado en Buenos Aires por Editorial Hobbs Sudamericana S.A. (1971), páginas 296 y 297.</span> </li> </ol></div> <p><br /> </p> <style data-mw-deduplicate="TemplateStyles:r161257576">.mw-parser-output .mw-authority-control{margin-top:1.5em}.mw-parser-output .mw-authority-control .navbox table{margin:0}.mw-parser-output .mw-authority-control .navbox hr:last-child{display:none}.mw-parser-output .mw-authority-control .navbox+.mw-mf-linked-projects{display:none}.mw-parser-output .mw-authority-control .mw-mf-linked-projects{display:flex;padding:0.5em;border:1px solid var(--border-color-base,#a2a9b1);background-color:var(--background-color-neutral,#eaecf0);color:var(--color-base,#202122)}.mw-parser-output .mw-authority-control .mw-mf-linked-projects ul li{margin-bottom:0}.mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#a2a9b1);background-color:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .mw-authority-control .navbox-list{border-color:#f8f9fa}.mw-parser-output .mw-authority-control .navbox th{background-color:#eeeeff}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .mw-mf-linked-projects{border:1px solid var(--border-color-base,#72777d);background-color:var(--background-color-neutral,#27292d);color:var(--color-base,#eaecf0)}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral-subtle,#202122)!important}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox-list{border-color:#202122!important}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox th{background-color:#27292d!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .mw-mf-linked-projects{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral,#27292d)!important;color:var(--color-base,#eaecf0)!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral-subtle,#202122)!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox-list{border-color:#202122!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox th{background-color:#27292d!important}}</style><div class="mw-authority-control"><div role="navigation" class="navbox" aria-label="Navbox" style="width: inherit;padding:3px"><table class="hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width: 12%; 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