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<title>Square triangular number</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Square triangular number</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">For squares of triangular numbers, see <a href="Squared_triangular_number" title="Squared triangular number">squared triangular number</a>.</div>

<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>square triangular number</b> (or <b>triangular square number</b>) is a number which is both a <a href="Triangular_number" title="Triangular number">triangular number</a> and a <a href="Square_number" title="Square number">square number</a>, in other words, the sum of all integers from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

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<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

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</math></span><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></span> to <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> has a square root that is an integer. There are <a href="Infinity" title="Infinity">infinitely many</a> square triangular numbers; the first few are:

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</style><div class="block-indent" style="padding-left: 1.6em;">0, 1, 36, <span class="nowrap">1225</span>, <span class="nowrap">41<span style="margin-left:.25em;">616</span></span>, <span class="nowrap">1<span style="margin-left:.25em;">413</span><span style="margin-left:.25em;">721</span></span>, <span class="nowrap">48<span style="margin-left:.25em;">024</span><span style="margin-left:.25em;">900</span></span>, <span class="nowrap">1<span style="margin-left:.25em;">631</span><span style="margin-left:.25em;">432</span><span style="margin-left:.25em;">881</span></span>, <span class="nowrap">55<span style="margin-left:.25em;">420</span><span style="margin-left:.25em;">693</span><span style="margin-left:.25em;">056</span></span>, <span class="nowrap">1<span style="margin-left:.25em;">882</span><span style="margin-left:.25em;">672</span><span style="margin-left:.25em;">131</span><span style="margin-left:.25em;">025</span></span> (sequence <span class="nowrap external"><a href="https://oeis.org/A001110" class="extiw external" title="oeis:A001110">A001110</a></span> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</div>

<div class="mw-heading mw-heading2"><h2 id="Solution_as_a_Pell_equation">Solution as a Pell equation</h2></div>

<p>Write <span class="mwe-math-element mwe-math-element-inline"><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_{k}}">

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<annotation encoding="application/x-tex">{\displaystyle N_{k}}</annotation>

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</math></span><img src="./7698f968a98115830bcc378e0f849e0375c858c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.955ex; height:2.509ex;" alt="{\displaystyle N_{k}}" loading="lazy"></span> for the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>th square triangular number, and write <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{k}}">

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<annotation encoding="application/x-tex">{\displaystyle s_{k}}</annotation>

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</math></span><img src="./04f159343172781e7666dbc88280c91f34117c30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.179ex; height:2.009ex;" alt="{\displaystyle s_{k}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{k}}">

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<annotation encoding="application/x-tex">{\displaystyle t_{k}}</annotation>

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</math></span><img src="./79a3574d49db62b5bb645a1cedf19c19d342eb24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.928ex; height:2.343ex;" alt="{\displaystyle t_{k}}" loading="lazy"></span> for the sides of the corresponding square and triangle, so that

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle N_{k}=s_{k}^{2}={\frac {t_{k}(t_{k}+1)}{2}}.}</annotation>

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</math></span><img src="./e5d2810fe2b5d9d5566a85b19cf06db66ae266ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.483ex; height:5.676ex;" alt="{\displaystyle \displaystyle N_{k}=s_{k}^{2}={\frac {t_{k}(t_{k}+1)}{2}}.}" loading="lazy"></span></div>

<p>Define the <i>triangular root</i> of a triangular number <span class="mwe-math-element mwe-math-element-inline"><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={\tfrac {n(n+1)}{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle N={\tfrac {n(n+1)}{2}}}</annotation>

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</math></span><img src="./e57557a7d56a10b04feda001322030e8aa864e84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.351ex; height:4.176ex;" alt="{\displaystyle N={\tfrac {n(n+1)}{2}}}" loading="lazy"></span> to be <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>. From this definition and the quadratic formula,

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle n={\frac {{\sqrt {8N+1}}-1}{2}}.}</annotation>

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</math></span><img src="./954530bc20d20889dbccd92c6b1406668094f89a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.144ex; height:5.843ex;" alt="{\displaystyle \displaystyle n={\frac {{\sqrt {8N+1}}-1}{2}}.}" loading="lazy"></span></div>

<p>Therefore, <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> is triangular (<span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> is an integer) <a href="If_and_only_if" title="If and only if">if and only if</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8N+1}">

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<annotation encoding="application/x-tex">{\displaystyle 8N+1}</annotation>

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</math></span><img src="./0871ecee561e6e5184e5d7586db657e6ad4ba59f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.229ex; height:2.343ex;" alt="{\displaystyle 8N+1}" loading="lazy"></span> is square. Consequently, a square number <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle M^{2}}</annotation>

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</math></span><img src="./ae9c3e3aa1aec0e135b5d342181d9ee7f9814f87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.553ex; height:2.676ex;" alt="{\displaystyle M^{2}}" loading="lazy"></span> is also triangular if and only if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8M^{2}+1}">

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<annotation encoding="application/x-tex">{\displaystyle 8M^{2}+1}</annotation>

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</math></span><img src="./44078a0faf82dd32ba2af72c3cb918d56f99b8f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.718ex; height:2.843ex;" alt="{\displaystyle 8M^{2}+1}" loading="lazy"></span> is square, that is, there are numbers <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math></span><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span> such that <span class="mwe-math-element mwe-math-element-inline"><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}-8y^{2}=1}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle x^{2}-8y^{2}=1}</annotation>

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</math></span><img src="./65c9c93adf8a531ed3eb7fe78a5d95c994b05913.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.862ex; height:3.009ex;" alt="{\displaystyle x^{2}-8y^{2}=1}" loading="lazy"></span>. This is an instance of the <a href="Pell_equation" class="mw-redirect" title="Pell equation">Pell equation</a> <span class="mwe-math-element mwe-math-element-inline"><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}-ny^{2}=1}">

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<annotation encoding="application/x-tex">{\displaystyle x^{2}-ny^{2}=1}</annotation>

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</math></span><img src="./6e647f7460211d103b7a53835e599cbb9d7010ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.095ex; height:3.009ex;" alt="{\displaystyle x^{2}-ny^{2}=1}" loading="lazy"></span> with <span class="mwe-math-element mwe-math-element-inline"><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=8}">

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<annotation encoding="application/x-tex">{\displaystyle n=8}</annotation>

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</math></span><img src="./1344e2c194f016e12bb978493013955fe2f445f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=8}" loading="lazy"></span>. All Pell equations have the trivial solution <span class="mwe-math-element mwe-math-element-inline"><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=1,y=0}">

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<annotation encoding="application/x-tex">{\displaystyle x=1,y=0}</annotation>

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</math></span><img src="./8e699466d6d483f2cce66fc4d29ff9c64895658e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.041ex; height:2.509ex;" alt="{\displaystyle x=1,y=0}" loading="lazy"></span> for any <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>; this is called the zeroth solution, and indexed as <span class="mwe-math-element mwe-math-element-inline"><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_{0},y_{0})=(1,0)}">

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<annotation encoding="application/x-tex">{\displaystyle (x_{0},y_{0})=(1,0)}</annotation>

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</math></span><img src="./ddc1f78f911f6a47fc27cf26788382b07b3e6ae1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.687ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0})=(1,0)}" loading="lazy"></span>. If <span class="mwe-math-element mwe-math-element-inline"><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_{k},y_{k})}">

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<annotation encoding="application/x-tex">{\displaystyle (x_{k},y_{k})}</annotation>

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</math></span><img src="./7fd16782a99566aafcdd4dd63ba649e6fc34af1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.49ex; height:2.843ex;" alt="{\displaystyle (x_{k},y_{k})}" loading="lazy"></span> denotes the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>th nontrivial solution to any Pell equation for a particular <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>, it can be shown by the method of descent that the next solution is

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {\begin{aligned}x_{k+1}&=2x_{k}x_{1}-x_{k-1},\\y_{k+1}&=2y_{k}x_{1}-y_{k-1}.\end{aligned}}}</annotation>

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</math></span><img src="./33522af7d7f4771010f0f379a44f6a466b67f55f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.111ex; margin-bottom: -0.227ex; width:22.34ex; height:5.843ex;" alt="{\displaystyle \displaystyle {\begin{aligned}x_{k+1}&=2x_{k}x_{1}-x_{k-1},\\y_{k+1}&=2y_{k}x_{1}-y_{k-1}.\end{aligned}}}" loading="lazy"></span></div>

<p>Hence there are infinitely many solutions to any Pell equation for which there is one non-trivial one, which is true whenever <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> is not a square. The first non-trivial solution when <span class="mwe-math-element mwe-math-element-inline"><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=8}">

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</math></span><img src="./1344e2c194f016e12bb978493013955fe2f445f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=8}" loading="lazy"></span> is easy to find: it is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (3,1)}">

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<annotation encoding="application/x-tex">{\displaystyle (3,1)}</annotation>

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</math></span><img src="./a8933db1c87b5fefc8d54c6e2d157e4b343bb8b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (3,1)}" loading="lazy"></span>. A solution <span class="mwe-math-element mwe-math-element-inline"><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_{k},y_{k})}">

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</math></span><img src="./7fd16782a99566aafcdd4dd63ba649e6fc34af1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.49ex; height:2.843ex;" alt="{\displaystyle (x_{k},y_{k})}" loading="lazy"></span> to the Pell equation for <span class="mwe-math-element mwe-math-element-inline"><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=8}">

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<annotation encoding="application/x-tex">{\displaystyle n=8}</annotation>

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</math></span><img src="./1344e2c194f016e12bb978493013955fe2f445f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=8}" loading="lazy"></span> yields a square triangular number and its square and triangular roots as follows:

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle s_{k}=y_{k},\quad t_{k}={\frac {x_{k}-1}{2}},\quad N_{k}=y_{k}^{2}.}</annotation>

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</math></span><img src="./18836bb5d416cbdfef5daf0f97abef0a36cc3dfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.431ex; height:5.176ex;" alt="{\displaystyle \displaystyle s_{k}=y_{k},\quad t_{k}={\frac {x_{k}-1}{2}},\quad N_{k}=y_{k}^{2}.}" loading="lazy"></span></div>

<p>Hence, the first square triangular number, derived from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (3,1)}">

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<annotation encoding="application/x-tex">{\displaystyle (3,1)}</annotation>

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</math></span><img src="./a8933db1c87b5fefc8d54c6e2d157e4b343bb8b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (3,1)}" loading="lazy"></span>, is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

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<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

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</math></span><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></span>, and the next, derived from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6\cdot (3,1)-(1,0)=(17,6)}">

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</math></span><img src="./e192af9bdc1486d64b70ecd35640ad60fa477afc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.447ex; height:2.843ex;" alt="{\displaystyle 6\cdot (3,1)-(1,0)=(17,6)}" loading="lazy"></span>, is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 36}">

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</math></span><img src="./a162192c8f313a9ae7a2d558fe8af7746043d845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 36}" loading="lazy"></span>.

</p><p>The sequences <span class="mwe-math-element mwe-math-element-inline"><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_{k}}">

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</math></span><img src="./7698f968a98115830bcc378e0f849e0375c858c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.955ex; height:2.509ex;" alt="{\displaystyle N_{k}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{k}}">

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</math></span><img src="./04f159343172781e7666dbc88280c91f34117c30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.179ex; height:2.009ex;" alt="{\displaystyle s_{k}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{k}}">

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</math></span><img src="./79a3574d49db62b5bb645a1cedf19c19d342eb24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.928ex; height:2.343ex;" alt="{\displaystyle t_{k}}" loading="lazy"></span> are the <a href="OEIS" class="mw-redirect" title="OEIS">OEIS</a> sequences <span class="nowrap external"><a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A001110" class="extiw external" title="oeis:A001110">A001110</a></span>, <span class="nowrap external"><a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A001109" class="extiw external" title="oeis:A001109">A001109</a></span>, and <span class="nowrap external"><a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A001108" class="extiw external" title="oeis:A001108">A001108</a></span> respectively.

</p>

<div class="mw-heading mw-heading2"><h2 id="Explicit_formula">Explicit formula</h2></div>

<p>In 1778 <a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> determined the explicit formula<sup id="cite_ref-Dickson_1-0" class="reference"><a href="#cite_note-Dickson-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Euler_2-0" class="reference"><a href="#cite_note-Euler-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 12–13">: 12–13 </span></sup>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle N_{k}=\left({\frac {\left(3+2{\sqrt {2}}\right)^{k}-\left(3-2{\sqrt {2}}\right)^{k}}{4{\sqrt {2}}}}\right)^{2}.}</annotation>

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</math></span><img src="./5e4c0ebfee2307ff8be5d6218a320b8fbc5d9473.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:38.462ex; height:9.009ex;" alt="{\displaystyle \displaystyle N_{k}=\left({\frac {\left(3+2{\sqrt {2}}\right)^{k}-\left(3-2{\sqrt {2}}\right)^{k}}{4{\sqrt {2}}}}\right)^{2}.}" loading="lazy"></span></div>

<p>Other equivalent formulas (obtained by expanding this formula) that may be convenient include

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {\begin{aligned}N_{k}&={\tfrac {1}{32}}\left(\left(1+{\sqrt {2}}\right)^{2k}-\left(1-{\sqrt {2}}\right)^{2k}\right)^{2}\\&={\tfrac {1}{32}}\left(\left(1+{\sqrt {2}}\right)^{4k}-2+\left(1-{\sqrt {2}}\right)^{4k}\right)\\&={\tfrac {1}{32}}\left(\left(17+12{\sqrt {2}}\right)^{k}-2+\left(17-12{\sqrt {2}}\right)^{k}\right).\end{aligned}}}</annotation>

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</math></span><img src="./6f72d28fe21f2f5c0bfe799ef0782c3e70e5ff52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:47.939ex; height:14.843ex;" alt="{\displaystyle \displaystyle {\begin{aligned}N_{k}&={\tfrac {1}{32}}\left(\left(1+{\sqrt {2}}\right)^{2k}-\left(1-{\sqrt {2}}\right)^{2k}\right)^{2}\\&={\tfrac {1}{32}}\left(\left(1+{\sqrt {2}}\right)^{4k}-2+\left(1-{\sqrt {2}}\right)^{4k}\right)\\&={\tfrac {1}{32}}\left(\left(17+12{\sqrt {2}}\right)^{k}-2+\left(17-12{\sqrt {2}}\right)^{k}\right).\end{aligned}}}" loading="lazy"></span></div>

<p>The corresponding explicit formulas for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{k}}">

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<annotation encoding="application/x-tex">{\displaystyle s_{k}}</annotation>

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</math></span><img src="./04f159343172781e7666dbc88280c91f34117c30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.179ex; height:2.009ex;" alt="{\displaystyle s_{k}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{k}}">

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</math></span><img src="./79a3574d49db62b5bb645a1cedf19c19d342eb24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.928ex; height:2.343ex;" alt="{\displaystyle t_{k}}" loading="lazy"></span> are:<sup id="cite_ref-Euler_2-1" class="reference"><a href="#cite_note-Euler-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 13">: 13 </span></sup>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {\begin{aligned}s_{k}&={\frac {\left(3+2{\sqrt {2}}\right)^{k}-\left(3-2{\sqrt {2}}\right)^{k}}{4{\sqrt {2}}}},\\t_{k}&={\frac {\left(3+2{\sqrt {2}}\right)^{k}+\left(3-2{\sqrt {2}}\right)^{k}-2}{4}}.\end{aligned}}}</annotation>

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</math></span><img src="./c8d9424754cf221404c88a643233e32b73cfa8cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:37.32ex; height:14.509ex;" alt="{\displaystyle \displaystyle {\begin{aligned}s_{k}&={\frac {\left(3+2{\sqrt {2}}\right)^{k}-\left(3-2{\sqrt {2}}\right)^{k}}{4{\sqrt {2}}}},\\t_{k}&={\frac {\left(3+2{\sqrt {2}}\right)^{k}+\left(3-2{\sqrt {2}}\right)^{k}-2}{4}}.\end{aligned}}}" loading="lazy"></span></div>

<div class="mw-heading mw-heading2"><h2 id="Recurrence_relations">Recurrence relations</h2></div>

<p>The solution to the Pell equation can be expressed as a <a href="Recurrence_relation" title="Recurrence relation">recurrence relation</a> for the equation's solutions. This can be translated into recurrence equations that directly express the square triangular numbers, as well as the sides of the square and triangle involved. We have<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><sup class="reference nowrap"><span title="Page / location: (12)">: (12) </span></sup>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {\begin{aligned}N_{k}&=34N_{k-1}-N_{k-2}+2,&{\text{with }}N_{0}&=0{\text{ and }}N_{1}=1;\\N_{k}&=\left(6{\sqrt {N_{k-1}}}-{\sqrt {N_{k-2}}}\right)^{2},&{\text{with }}N_{0}&=0{\text{ and }}N_{1}=1.\end{aligned}}}</annotation>

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</math></span><img src="./85d86016a39aefcac9afba8858100ad54983ae9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:59.063ex; height:6.843ex;" alt="{\displaystyle \displaystyle {\begin{aligned}N_{k}&=34N_{k-1}-N_{k-2}+2,&{\text{with }}N_{0}&=0{\text{ and }}N_{1}=1;\\N_{k}&=\left(6{\sqrt {N_{k-1}}}-{\sqrt {N_{k-2}}}\right)^{2},&{\text{with }}N_{0}&=0{\text{ and }}N_{1}=1.\end{aligned}}}" loading="lazy"></span></div>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {\begin{aligned}s_{k}&=6s_{k-1}-s_{k-2},&{\text{with }}s_{0}&=0{\text{ and }}s_{1}=1;\\t_{k}&=6t_{k-1}-t_{k-2}+2,&{\text{with }}t_{0}&=0{\text{ and }}t_{1}=1.\end{aligned}}}</annotation>

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</math></span><img src="./b07f40b6948199b6f1d5cee9aca643fc67321a41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.111ex; margin-bottom: -0.227ex; width:50.854ex; height:5.843ex;" alt="{\displaystyle \displaystyle {\begin{aligned}s_{k}&=6s_{k-1}-s_{k-2},&{\text{with }}s_{0}&=0{\text{ and }}s_{1}=1;\\t_{k}&=6t_{k-1}-t_{k-2}+2,&{\text{with }}t_{0}&=0{\text{ and }}t_{1}=1.\end{aligned}}}" loading="lazy"></span></div>

<div class="mw-heading mw-heading2"><h2 id="Other_characterizations">Other characterizations</h2></div>

<p>All square triangular numbers have the form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{2}c^{2}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b^{2}c^{2}}</annotation>

</semantics>

</math></span><img src="./1d479c94ea24bf0d97416f91ae96e0aad1a82d8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.113ex; height:2.676ex;" alt="{\displaystyle b^{2}c^{2}}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {b}{c}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {b}{c}}}</annotation>

</semantics>

</math></span><img src="./a920272e7af7800eb943b8e1084a24c69d3d67ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.548ex; height:3.509ex;" alt="{\displaystyle {\tfrac {b}{c}}}" loading="lazy"></span> is a <a href="Convergent_(continued_fraction)" class="mw-redirect" title="Convergent (continued fraction)">convergent</a> to the <a href="Simple_continued_fraction" title="Simple continued fraction">continued fraction expansion</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation>

</semantics>

</math></span><img src="./b4afc1e27d418021bf10898eb44a7f5f315735ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}" loading="lazy"></span>, the <a href="Square_root_of_2" title="Square root of 2">square root of 2</a>.<sup id="cite_ref-Ball_4-0" class="reference"><a href="#cite_note-Ball-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p><p>A. V. Sylwester gave a short proof that there are infinitely many square triangular numbers: If the <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>th triangular number <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {n(n+1)}{2}}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {n(n+1)}{2}}}</annotation>

</semantics>

</math></span><img src="./e1dc30e5c6b7a621f43c768266851d4cf953124c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.188ex; height:4.176ex;" alt="{\displaystyle {\tfrac {n(n+1)}{2}}}" loading="lazy"></span> is square, then so is the larger <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4n(n+1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4n(n+1)}</annotation>

</semantics>

</math></span><img src="./4f1be081d7f0ab29787ecc8b101cfbb987b5d1f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.764ex; height:2.843ex;" alt="{\displaystyle 4n(n+1)}" loading="lazy"></span>th triangular number, since:

</p>

<mfrac>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \displaystyle {\frac {{\bigl (}4n(n+1){\bigr )}{\bigl (}4n(n+1)+1{\bigr )}}{2}}=4\,{\frac {n(n+1)}{2}}\,\left(2n+1\right)^{2}.}</annotation>

</semantics>

</math></span><img src="./ed8d1f4e607d7b4777b5d0ed9c927e02226f38fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:53.17ex; height:6.009ex;" alt="{\displaystyle \displaystyle {\frac {{\bigl (}4n(n+1){\bigr )}{\bigl (}4n(n+1)+1{\bigr )}}{2}}=4\,{\frac {n(n+1)}{2}}\,\left(2n+1\right)^{2}.}" loading="lazy"></span></div>

<p>The left hand side of this equation is in the form of a triangular number, and as the product of three squares, the right hand side is square.<sup id="cite_ref-Sylwester_5-0" class="reference"><a href="#cite_note-Sylwester-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p><p>The <a href="Generating_function" title="Generating function">generating function</a> for the square triangular numbers is:<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>

<mfrac>

<mrow>

</mrow>

<mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1+z}{(1-z)\left(z^{2}-34z+1\right)}}=1+36z+1225z^{2}+\cdots }</annotation>

</semantics>

</math></span><img src="./e9e8fed671642cf725aeeb01b9b639e3ef9bc482.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:48.046ex; height:6.009ex;" alt="{\displaystyle {\frac {1+z}{(1-z)\left(z^{2}-34z+1\right)}}=1+36z+1225z^{2}+\cdots }" loading="lazy"></span></dd></dl>

<ul><li><a href="Cannonball_problem" title="Cannonball problem">Cannonball problem</a>, on numbers that are simultaneously square and square pyramidal</li>

<li><a href="Sixth_power" title="Sixth power">Sixth power</a>, numbers that are simultaneously square and cubical</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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.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("./mw/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("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/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}}

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</style><cite id="CITEREFDickson1999" class="citation book cs1"><a href="Leonard_Eugene_Dickson" title="Leonard Eugene Dickson">Dickson, Leonard Eugene</a> (1999) [1920]. <i><a href="History_of_the_Theory_of_Numbers" title="History of the Theory of Numbers">History of the Theory of Numbers</a></i>. Vol. 2. Providence: American Mathematical Society. p. 16. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-1935-7</bdi>.</cite></span>

</li>

<cite id="CITEREFEuler1813" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="Leonhard_Euler" title="Leonhard Euler">Euler, Leonhard</a> (1813). <a rel="nofollow" class="external text" href="http://math.dartmouth.edu/~euler/pages/E739.html">"Regula facilis problemata Diophantea per numeros integros expedite resolvendi (An easy rule for Diophantine problems which are to be resolved quickly by integral numbers)"</a>. <i>Mémoires de l'Académie des Sciences de St.-Pétersbourg</i> (in Latin). <b>4</b>: <span class="nowrap">3–</span>17<span class="reference-accessdate">. Retrieved <span class="nowrap">2009-05-11</span></span>. <q>According to the records, it was presented to the St. Petersburg Academy on May 4, 1778.</q></cite></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"><span class="citation mathworld" id="Reference-Mathworld-Square_Triangular_Number"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/SquareTriangularNumber.html">"Square Triangular Number"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span></span>

</li>

<cite id="CITEREFBallCoxeter1987" class="citation book cs1"><a href="W._W._Rouse_Ball" title="W. W. Rouse Ball">Ball, W. W. Rouse</a>; <a href="Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, H. S. M.</a> (1987). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalrecr00coxe"><i>Mathematical Recreations and Essays</i></a></span>. New York: Dover Publications. p. <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalrecr00coxe/page/n72">59</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-25357-2</bdi>.</cite></span>

</li>

<cite id="CITEREFPietenpolSylwesterJustWarten1962" class="citation journal cs1">Pietenpol, J. L.; Sylwester, A. V.; Just, Erwin; Warten, R. M. (February 1962). "Elementary Problems and Solutions: E 1473, Square Triangular Numbers". <i>American Mathematical Monthly</i>. <b>69</b> (2). Mathematical Association of America: <span class="nowrap">168–</span>169. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2312558">10.2307/2312558</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9890">0002-9890</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2312558">2312558</a>.</cite></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"><cite id="CITEREFPlouffe1992" class="citation web cs1"><a href="Simon_Plouffe" title="Simon Plouffe">Plouffe, Simon</a> (August 1992). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120820012535/http://www.plouffe.fr/simon/articles/FonctionsGeneratrices.pdf">"1031 Generating Functions"</a> <span class="cs1-format">(PDF)</span>. University of Quebec, Laboratoire de combinatoire et d'informatique mathématique. p. A.129. Archived from <a rel="nofollow" class="external text" href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2012-08-20<span class="reference-accessdate">. Retrieved <span class="nowrap">2009-05-11</span></span>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/do_you_know/triSquare.shtml">Triangular numbers that are also square</a> at <a href="Cut-the-knot" class="mw-redirect" title="Cut-the-knot">cut-the-knot</a></li>

<li><span class="citation mathworld" id="Reference-Mathworld-Square_Triangular_Number"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/SquareTriangularNumber.html">"Square Triangular Number"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span></li>

<li><a rel="nofollow" class="external text" href="https://opinionator.blogs.nytimes.com/2012/01/04/remembering-michael-dummett/">Michael Dummett's solution</a></li></ul>

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</style><div id="Classes_of_natural_numbers743" style="font-size:114%;margin:0 4em">Classes of <a href="Natural_number" title="Natural number">natural numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Powers_and_related_numbers743" style="font-size:114%;margin:0 4em"><a href="Exponentiation" title="Exponentiation">Powers</a> and related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Achilles_number" title="Achilles number">Achilles</a></li>

<li><a href="Power_of_two" title="Power of two">Power of 2</a></li>

<li><a href="Power_of_three" title="Power of three">Power of 3</a></li>

<li><a href="Power_of_10" title="Power of 10">Power of 10</a></li>

<li><a href="Square_number" title="Square number">Square</a></li>

<li><a href="Fourth_power" title="Fourth power">Fourth power</a></li>

<li><a href="Fifth_power_(algebra)" title="Fifth power (algebra)">Fifth power</a></li>

<li><a href="Sixth_power" title="Sixth power">Sixth power</a></li>

<li><a href="Seventh_power" title="Seventh power">Seventh power</a></li>

<li><a href="Eighth_power" title="Eighth power">Eighth power</a></li>

<li><a href="Perfect_power" title="Perfect power">Perfect power</a></li>

<li><a href="Powerful_number" title="Powerful number">Powerful</a></li>

<li><a href="Prime_power" title="Prime power">Prime power</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Of_the_form_a_×_2b_±_1743" style="font-size:114%;margin:0 4em">Of the form <i>a</i> × 2<sup><i>b</i></sup> ± 1</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cullen_number" title="Cullen number">Cullen</a></li>

<li><a href="Double_Mersenne_number" title="Double Mersenne number">Double Mersenne</a></li>

<li><a href="Fermat_number" title="Fermat number">Fermat</a></li>

<li><a href="Mersenne_prime" title="Mersenne prime">Mersenne</a></li>

<li><a href="Proth_number" class="mw-redirect" title="Proth number">Proth</a></li>

<li><a href="Thabit_number" title="Thabit number">Thabit</a></li>

<li><a href="Woodall_number" title="Woodall number">Woodall</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_polynomial_numbers743" style="font-size:114%;margin:0 4em">Other polynomial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hilbert_number" title="Hilbert number">Hilbert</a></li>

<li><a href="Idoneal_number" title="Idoneal number">Idoneal</a></li>

<li><a href="Leyland_number" title="Leyland number">Leyland</a></li>

<li><a href="Loeschian_number" class="mw-redirect" title="Loeschian number">Loeschian</a></li>

<li><a href="Lucky_numbers_of_Euler" title="Lucky numbers of Euler">Lucky numbers of Euler</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Recursively_defined_numbers743" style="font-size:114%;margin:0 4em"><a href="Recursion" title="Recursion">Recursively</a> defined numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Fibonacci_sequence" title="Fibonacci sequence">Fibonacci</a></li>

<li><a href="Jacobsthal_number" title="Jacobsthal number">Jacobsthal</a></li>

<li><a href="Leonardo_number" title="Leonardo number">Leonardo</a></li>

<li><a href="Lucas_number" title="Lucas number">Lucas</a></li>

<li><a href="Supergolden_ratio#Narayana_sequence" title="Supergolden ratio">Narayana</a></li>

<li><a href="Padovan_sequence" title="Padovan sequence">Padovan</a></li>

<li><a href="Perrin_number" title="Perrin number">Perrin</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Possessing_a_specific_set_of_other_numbers743" style="font-size:114%;margin:0 4em">Possessing a specific set of other numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Amenable_number" title="Amenable number">Amenable</a></li>

<li><a href="Congruent_number" title="Congruent number">Congruent</a></li>

<li><a href="Kn%C3%B6del_number" title="Knödel number">Knödel</a></li>

<li><a href="Riesel_number" title="Riesel number">Riesel</a></li>

<li><a href="Sierpi%C5%84ski_number" title="Sierpiński number">Sierpiński</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Expressible_via_specific_sums743" style="font-size:114%;margin:0 4em">Expressible via specific sums</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Nonhypotenuse_number" title="Nonhypotenuse number">Nonhypotenuse</a></li>

<li><a href="Polite_number" title="Polite number">Polite</a></li>

<li><a href="Practical_number" title="Practical number">Practical</a></li>

<li><a href="Primary_pseudoperfect_number" title="Primary pseudoperfect number">Primary pseudoperfect</a></li>

<li><a href="Wolstenholme_number" title="Wolstenholme number">Wolstenholme</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Figurate_numbers743" style="font-size:114%;margin:0 4em"><a href="Figurate_number" title="Figurate number">Figurate numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Plane_(mathematics)" title="Plane (mathematics)">2-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Centered_polygonal_number" title="Centered polygonal number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Centered_triangular_number" title="Centered triangular number">Centered triangular</a></li>

<li><a href="Centered_square_number" title="Centered square number">Centered square</a></li>

<li><a href="Centered_pentagonal_number" title="Centered pentagonal number">Centered pentagonal</a></li>

<li><a href="Centered_hexagonal_number" title="Centered hexagonal number">Centered hexagonal</a></li>

<li><a href="Centered_heptagonal_number" title="Centered heptagonal number">Centered heptagonal</a></li>

<li><a href="Centered_octagonal_number" title="Centered octagonal number">Centered octagonal</a></li>

<li><a href="Centered_nonagonal_number" title="Centered nonagonal number">Centered nonagonal</a></li>

<li><a href="Centered_decagonal_number" title="Centered decagonal number">Centered decagonal</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Polygonal_number" title="Polygonal number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Triangular_number" title="Triangular number">Triangular</a></li>

<li><a href="Square_number" title="Square number">Square</a></li>

<li><a href="Pentagonal_number" title="Pentagonal number">Pentagonal</a></li>

<li><a href="Hexagonal_number" title="Hexagonal number">Hexagonal</a></li>

<li><a href="Heptagonal_number" title="Heptagonal number">Heptagonal</a></li>

<li><a href="Octagonal_number" title="Octagonal number">Octagonal</a></li>

<li><a href="Nonagonal_number" title="Nonagonal number">Nonagonal</a></li>

<li><a href="Decagonal_number" title="Decagonal number">Decagonal</a></li>

<li><a href="Dodecagonal_number" title="Dodecagonal number">Dodecagonal</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Three-dimensional_space" title="Three-dimensional space">3-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Centered_polyhedral_number" title="Centered polyhedral number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Centered_tetrahedral_number" title="Centered tetrahedral number">Centered tetrahedral</a></li>

<li><a href="Centered_cube_number" title="Centered cube number">Centered cube</a></li>

<li><a href="Centered_octahedral_number" title="Centered octahedral number">Centered octahedral</a></li>

<li><a href="Centered_dodecahedral_number" title="Centered dodecahedral number">Centered dodecahedral</a></li>

<li><a href="Centered_icosahedral_number" title="Centered icosahedral number">Centered icosahedral</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Polyhedral_number" class="mw-redirect" title="Polyhedral number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Tetrahedral_number" title="Tetrahedral number">Tetrahedral</a></li>

<li><a href="Cube_(algebra)" title="Cube (algebra)">Cubic</a></li>

<li><a href="Octahedral_number" title="Octahedral number">Octahedral</a></li>

<li><a href="Dodecahedral_number" title="Dodecahedral number">Dodecahedral</a></li>

<li><a href="Icosahedral_number" title="Icosahedral number">Icosahedral</a></li>

<li><a href="Stella_octangula_number" title="Stella octangula number">Stella octangula</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Pyramidal_number" title="Pyramidal number">pyramidal</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Square_pyramidal_number" title="Square pyramidal number">Square pyramidal</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Four-dimensional_space" title="Four-dimensional space">4-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">non-centered</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Pentatope_number" title="Pentatope number">Pentatope</a></li>

<li><a href="Squared_triangular_number" title="Squared triangular number">Squared triangular</a></li>

<li><a href="Fourth_power" title="Fourth power">Tesseractic</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial_numbers743" style="font-size:114%;margin:0 4em">Combinatorial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Catalan_number" title="Catalan number">Catalan</a></li>

<li><a href="Dedekind_number" title="Dedekind number">Dedekind</a></li>

<li><a href="Delannoy_number" title="Delannoy number">Delannoy</a></li>

<li><a href="Euler_number" class="mw-redirect" title="Euler number">Euler</a></li>

<li><a href="Eulerian_number" title="Eulerian number">Eulerian</a></li>

<li><a href="Fuss%E2%80%93Catalan_number" title="Fuss–Catalan number">Fuss–Catalan</a></li>

<li><a href="Lazy_caterer's_sequence" title="Lazy caterer's sequence">Lazy caterer's sequence</a></li>

<li><a href="Motzkin_number" title="Motzkin number">Motzkin</a></li>

<li><a href="Narayana_number" title="Narayana number">Narayana</a></li>

<li><a href="Ordered_Bell_number" title="Ordered Bell number">Ordered Bell</a></li>

<li><a href="Schr%C3%B6der_number" title="Schröder number">Schröder</a></li>

<li><a href="Schr%C3%B6der%E2%80%93Hipparchus_number" title="Schröder–Hipparchus number">Schröder–Hipparchus</a></li>

<li><a href="Stirling_numbers_of_the_first_kind" title="Stirling numbers of the first kind">Stirling first</a></li>

<li><a href="Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling second</a></li>

<li><a href="Telephone_number_(mathematics)" title="Telephone number (mathematics)">Telephone number</a></li>

<li><a href="Wedderburn%E2%80%93Etherington_number" title="Wedderburn–Etherington number">Wedderburn–Etherington</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Primes743" style="font-size:114%;margin:0 4em"><a href="Prime_number" title="Prime number">Primes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Wieferich_prime#Wieferich_numbers" title="Wieferich prime">Wieferich</a></li>

<li><a href="Wolstenholme_prime" title="Wolstenholme prime">Wolstenholme prime</a></li>

<li><a href="Wilson_prime#Wilson_numbers" title="Wilson prime">Wilson</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Pseudoprimes743" style="font-size:114%;margin:0 4em"><a href="Pseudoprime" title="Pseudoprime">Pseudoprimes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Carmichael_number" title="Carmichael number">Carmichael number</a></li>

<li><a href="Catalan_pseudoprime" title="Catalan pseudoprime">Catalan pseudoprime</a></li>

<li><a href="Elliptic_pseudoprime" title="Elliptic pseudoprime">Elliptic pseudoprime</a></li>

<li><a href="Euler_pseudoprime" title="Euler pseudoprime">Euler pseudoprime</a></li>

<li><a href="Euler%E2%80%93Jacobi_pseudoprime" title="Euler–Jacobi pseudoprime">Euler–Jacobi pseudoprime</a></li>

<li><a href="Fermat_pseudoprime" title="Fermat pseudoprime">Fermat pseudoprime</a></li>

<li><a href="Frobenius_pseudoprime" title="Frobenius pseudoprime">Frobenius pseudoprime</a></li>

<li><a href="Lucas_pseudoprime" title="Lucas pseudoprime">Lucas pseudoprime</a></li>

<li><a href="Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael number</a></li>

<li><a href="Perrin_number#Perrin_primality_test" title="Perrin number">Perrin pseudoprime</a></li>

<li><a href="Somer%E2%80%93Lucas_pseudoprime" title="Somer–Lucas pseudoprime">Somer–Lucas pseudoprime</a></li>

<li><a href="Strong_pseudoprime" title="Strong pseudoprime">Strong pseudoprime</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Arithmetic_functions_and_dynamics743" style="font-size:114%;margin:0 4em"><a href="Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> and <a href="Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Divisor_function" title="Divisor function">Divisor functions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Abundant_number" title="Abundant number">Abundant</a></li>

<li><a href="Almost_perfect_number" title="Almost perfect number">Almost perfect</a></li>

<li><a href="Arithmetic_number" title="Arithmetic number">Arithmetic</a></li>

<li><a href="Betrothed_numbers" title="Betrothed numbers">Betrothed</a></li>

<li><a href="Colossally_abundant_number" title="Colossally abundant number">Colossally abundant</a></li>

<li><a href="Deficient_number" title="Deficient number">Deficient</a></li>

<li><a href="Descartes_number" title="Descartes number">Descartes</a></li>

<li><a href="Hemiperfect_number" title="Hemiperfect number">Hemiperfect</a></li>

<li><a href="Highly_abundant_number" title="Highly abundant number">Highly abundant</a></li>

<li><a href="Highly_composite_number" title="Highly composite number">Highly composite</a></li>

<li><a href="Hyperperfect_number" title="Hyperperfect number">Hyperperfect</a></li>

<li><a href="Multiply_perfect_number" title="Multiply perfect number">Multiply perfect</a></li>

<li><a href="Perfect_number" title="Perfect number">Perfect</a></li>

<li><a href="Practical_number" title="Practical number">Practical</a></li>

<li><a href="Primitive_abundant_number" title="Primitive abundant number">Primitive abundant</a></li>

<li><a href="Quasiperfect_number" title="Quasiperfect number">Quasiperfect</a></li>

<li><a href="Refactorable_number" title="Refactorable number">Refactorable</a></li>

<li><a href="Semiperfect_number" title="Semiperfect number">Semiperfect</a></li>

<li><a href="Sublime_number" title="Sublime number">Sublime</a></li>

<li><a href="Superabundant_number" title="Superabundant number">Superabundant</a></li>

<li><a href="Superior_highly_composite_number" title="Superior highly composite number">Superior highly composite</a></li>

<li><a href="Superperfect_number" title="Superperfect number">Superperfect</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Prime_omega_function" title="Prime omega function">Prime omega functions</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Almost_prime" title="Almost prime">Almost prime</a></li>

<li><a href="Semiprime" title="Semiprime">Semiprime</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Euler's_totient_function" title="Euler's totient function">Euler's totient function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Highly_cototient_number" title="Highly cototient number">Highly cototient</a></li>

<li><a href="Highly_totient_number" title="Highly totient number">Highly totient</a></li>

<li><a href="Noncototient" title="Noncototient">Noncototient</a></li>

<li><a href="Nontotient" title="Nontotient">Nontotient</a></li>

<li><a href="Perfect_totient_number" title="Perfect totient number">Perfect totient</a></li>

<li><a href="Sparsely_totient_number" title="Sparsely totient number">Sparsely totient</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Aliquot_sequence" title="Aliquot sequence">Aliquot sequences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Amicable_numbers" title="Amicable numbers">Amicable</a></li>

<li><a href="Perfect_number" title="Perfect number">Perfect</a></li>

<li><a href="Sociable_numbers" class="mw-redirect" title="Sociable numbers">Sociable</a></li>

<li><a href="Untouchable_number" title="Untouchable number">Untouchable</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Primorial" title="Primorial">Primorial</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Euclid_number" title="Euclid number">Euclid</a></li>

<li><a href="Fortunate_number" title="Fortunate number">Fortunate</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_prime_factor_or_divisor_related_numbers743" style="font-size:114%;margin:0 4em">Other <a href="Prime_factor" class="mw-redirect" title="Prime factor">prime factor</a> or <a href="Divisor" title="Divisor">divisor</a> related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Cyclic_number_(group_theory)" title="Cyclic number (group theory)">Cyclic</a></li>

<li><a href="Erd%C5%91s%E2%80%93Nicolas_number" title="Erdős–Nicolas number">Erdős–Nicolas</a></li>

<li><a href="Erd%C5%91s%E2%80%93Woods_number" title="Erdős–Woods number">Erdős–Woods</a></li>

<li><a href="Friendly_number" title="Friendly number">Friendly</a></li>

<li><a href="Giuga_number" title="Giuga number">Giuga</a></li>

<li><a href="Harmonic_divisor_number" title="Harmonic divisor number">Harmonic divisor</a></li>

<li><a href="Jordan%E2%80%93P%C3%B3lya_number" title="Jordan–Pólya number">Jordan–Pólya</a></li>

<li><a href="Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael</a></li>

<li><a href="Pronic_number" title="Pronic number">Pronic</a></li>

<li><a href="Regular_number" title="Regular number">Regular</a></li>

<li><a href="Rough_number" title="Rough number">Rough</a></li>

<li><a href="Smooth_number" title="Smooth number">Smooth</a></li>

<li><a href="Sphenic_number" title="Sphenic number">Sphenic</a></li>

<li><a href="St%C3%B8rmer_number" title="Størmer number">Størmer</a></li>

<li><a href="Super-Poulet_number" title="Super-Poulet number">Super-Poulet</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Numeral_system-dependent_numbers743" style="font-size:114%;margin:0 4em"><a href="Numeral_system" title="Numeral system">Numeral system</a>-dependent numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> <br>and <a href="Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Persistence_of_a_number" title="Persistence of a number">Persistence</a>

<ul><li><a href="Additive_persistence" class="mw-redirect" title="Additive persistence">Additive</a></li>

<li><a href="Multiplicative_persistence" class="mw-redirect" title="Multiplicative persistence">Multiplicative</a></li></ul></li></ul>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Digit_sum" title="Digit sum">Digit sum</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Digit_sum" title="Digit sum">Digit sum</a></li>

<li><a href="Digital_root" title="Digital root">Digital root</a></li>

<li><a href="Sum-product_number" title="Sum-product number">Sum-product</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit product</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Multiplicative_digital_root" title="Multiplicative digital root">Multiplicative digital root</a></li>

<li><a href="Sum-product_number" title="Sum-product number">Sum-product</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Coding-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Meertens_number" title="Meertens number">Meertens</a></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-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dudeney_number" title="Dudeney number">Dudeney</a></li>

<li><a href="Factorion" title="Factorion">Factorion</a></li>

<li><a href="Kaprekar_number" title="Kaprekar number">Kaprekar</a></li>

<li><a href="Kaprekar's_routine" title="Kaprekar's routine">Kaprekar's constant</a></li>

<li><a href="Keith_number" title="Keith number">Keith</a></li>

<li><a href="Lychrel_number" title="Lychrel number">Lychrel</a></li>

<li><a href="Narcissistic_number" title="Narcissistic number">Narcissistic</a></li>

<li><a href="Perfect_digit-to-digit_invariant" title="Perfect digit-to-digit invariant">Perfect digit-to-digit invariant</a></li>

<li><a href="Perfect_digital_invariant" title="Perfect digital invariant">Perfect digital invariant</a>

<ul><li><a href="Happy_number" title="Happy number">Happy</a></li></ul></li></ul>

</div></td></tr></tbody></table><div>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="P-adic_numbers" class="mw-redirect" title="P-adic numbers">P-adic numbers</a>-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Automorphic_number" title="Automorphic number">Automorphic</a>

<ul><li><a href="Trimorphic_number" class="mw-redirect" title="Trimorphic number">Trimorphic</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Numerical_digit" title="Numerical digit">Digit</a>-composition related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Palindromic_number" title="Palindromic number">Palindromic</a></li>

<li><a href="Pandigital_number" title="Pandigital number">Pandigital</a></li>

<li><a href="Repdigit" title="Repdigit">Repdigit</a></li>

<li><a href="Repunit" title="Repunit">Repunit</a></li>

<li><a href="Self-descriptive_number" title="Self-descriptive number">Self-descriptive</a></li>

<li><a href="Smarandache%E2%80%93Wellin_number" title="Smarandache–Wellin number">Smarandache–Wellin</a></li>

<li><a href="Undulating_number" title="Undulating number">Undulating</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit-<a href="Permutation" title="Permutation">permutation</a> related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cyclic_number" title="Cyclic number">Cyclic</a></li>

<li><a href="Digit-reassembly_number" title="Digit-reassembly number">Digit-reassembly</a></li>

<li><a href="Parasitic_number" title="Parasitic number">Parasitic</a></li>

<li><a href="Primeval_number" title="Primeval number">Primeval</a></li>

<li><a href="Transposable_integer" title="Transposable integer">Transposable</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Divisor-related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Equidigital_number" title="Equidigital number">Equidigital</a></li>

<li><a href="Extravagant_number" title="Extravagant number">Extravagant</a></li>

<li><a href="Frugal_number" title="Frugal number">Frugal</a></li>

<li><a href="Harshad_number" title="Harshad number">Harshad</a></li>

<li><a href="Polydivisible_number" title="Polydivisible number">Polydivisible</a></li>

<li><a href="Smith_number" title="Smith number">Smith</a></li>

<li><a href="Vampire_number" title="Vampire number">Vampire</a></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><a href="Friedman_number" title="Friedman number">Friedman</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Binary_numbers743" style="font-size:114%;margin:0 4em"><a href="Binary_number" title="Binary number">Binary numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Odious_number" title="Odious number">Odious</a></li>

<li><a href="Pernicious_number" title="Pernicious number">Pernicious</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Generated_via_a_sieve743" style="font-size:114%;margin:0 4em">Generated via a <a href="Sieve_theory" title="Sieve theory">sieve</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Lucky_number" title="Lucky number">Lucky</a></li>

<li><a href="Generation_of_primes" title="Generation of primes">Prime</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Sorting_related743" style="font-size:114%;margin:0 4em"><a href="Sorting_algorithm" title="Sorting algorithm">Sorting</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Pancake_sorting" title="Pancake sorting">Pancake number</a></li>

<li><a href="Sorting_number" title="Sorting number">Sorting number</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Natural_language_related743" style="font-size:114%;margin:0 4em"><a href="Natural_language" title="Natural language">Natural language</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Aronson's_sequence" title="Aronson's sequence">Aronson's sequence</a></li>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Graphemics_related743" style="font-size:114%;margin:0 4em"><a href="Graphemics" title="Graphemics">Graphemics</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Strobogrammatic_number" title="Strobogrammatic number">Strobogrammatic</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2" style="font-weight:bold;"><div>

<ul><li><span class="noviewer" typeof="mw:File"></span> <a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul>

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This article is issued from <a class="external text" title="Last edited on 2025-07-22" href="https://en.wikipedia.org/wiki/?title=Square_triangular_number&oldid=1302019089">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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