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<p>Implementation of <a href="https://en.wikipedia.org/wiki/Euler%27s_totient_function" target="_blank">Euler's Totient</a> @description Euler Totient Function is also known as phi function.
<a href="#details">More...</a></p>
<div class="textblock"><code>#include &lt;iostream&gt;</code><br />
<code>#include &lt;cassert&gt;</code><br />
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Include dependency graph for eulers_totient_function.cpp:</div>
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<tr class="memdesc:dd/d47/namespacemath"><td class="mdescLeft">&#160;</td><td class="mdescRight">for IO operations <br /></td></tr>
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Functions</h2></td></tr>
<tr class="memitem:ac37d3ba52eb296597d7a024ba8c4a5a5" id="r_ac37d3ba52eb296597d7a024ba8c4a5a5"><td class="memItemLeft" align="right" valign="top">uint64_t&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../dd/d47/namespacemath.html#ac37d3ba52eb296597d7a024ba8c4a5a5">math::phiFunction</a> (uint64_t n)</td></tr>
<tr class="memdesc:ac37d3ba52eb296597d7a024ba8c4a5a5"><td class="mdescLeft">&#160;</td><td class="mdescRight">Function to calculate Euler's Totient. <br /></td></tr>
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<tr class="memitem:aa8dca7b867074164d5f45b0f3851269d" id="r_aa8dca7b867074164d5f45b0f3851269d"><td class="memItemLeft" align="right" valign="top">static void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../da/d23/eulers__totient__function_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a> ()</td></tr>
<tr class="memdesc:aa8dca7b867074164d5f45b0f3851269d"><td class="mdescLeft">&#160;</td><td class="mdescRight">Self-test implementations. <br /></td></tr>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><p>Implementation of <a href="https://en.wikipedia.org/wiki/Euler%27s_totient_function" target="_blank">Euler's Totient</a> @description Euler Totient Function is also known as phi function. </p>
<p class="formulaDsp">
\[\phi(n) =
\phi\left({p_1}^{a_1}\right)\cdot\phi\left({p_2}^{a_2}\right)\ldots\]
</p>
<p> where \(p_1\), \(p_2\), \(\ldots\) are prime factors of n. <br />
3 Euler's properties:</p><ol type="1">
<li>\(\phi(n) = n-1\)</li>
<li>\(\phi(n^k) = n^k - n^{k-1}\)</li>
<li>\(\phi(a,b) = \phi(a)\cdot\phi(b)\) where a and b are relative primes.</li>
</ol>
<p>Applying this 3 properties on the first equation. </p><p class="formulaDsp">
\[\phi(n) =
n\cdot\left(1-\frac{1}{p_1}\right)\cdot\left(1-\frac{1}{p_2}\right)\cdots\]
</p>
<p> where \(p_1\), \(p_2\)... are prime factors. Hence Implementation in \(O\left(\sqrt{n}\right)\). <br />
Some known values are:</p><ul>
<li>\(\phi(100) = 40\)</li>
<li>\(\phi(1) = 1\)</li>
<li>\(\phi(17501) = 15120\)</li>
<li>\(\phi(1420) = 560\) <dl class="section author"><dt>Author</dt><dd><a href="https://github.com/mann2108" target="_blank">Mann Mehta</a> </dd></dl>
</li>
</ul>
</div><h2 class="groupheader">Function Documentation</h2>
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<h2 class="memtitle"><span class="permalink"><a href="#a0ddf1224851353fc92bfbff6f499fa97">&#9670;&#160;</a></span>main()</h2>
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<td class="paramtype">int&#160;</td>
<td class="paramname"><em>argc</em>, </td>
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<p>Main function. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">argc</td><td>commandline argument count (ignored) </td></tr>
<tr><td class="paramname">argv</td><td>commandline array of arguments (ignored) </td></tr>
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</dd>
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<dl class="section return"><dt>Returns</dt><dd>0 on exit </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 75</span> {</div>
<div class="line"><span class="lineno"> 76</span> <a class="code hl_function" href="../../da/d23/eulers__totient__function_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a>();</div>
<div class="line"><span class="lineno"> 77</span> <span class="keywordflow">return</span> 0;</div>
<div class="line"><span class="lineno"> 78</span>}</div>
<div class="ttc" id="aeulers__totient__function_8cpp_html_aa8dca7b867074164d5f45b0f3851269d"><div class="ttname"><a href="../../da/d23/eulers__totient__function_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a></div><div class="ttdeci">static void test()</div><div class="ttdoc">Self-test implementations.</div><div class="ttdef"><b>Definition</b> eulers_totient_function.cpp:57</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#aa8dca7b867074164d5f45b0f3851269d">&#9670;&#160;</a></span>test()</h2>
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<p>Self-test implementations. </p>
<dl class="section return"><dt>Returns</dt><dd>void </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 57</span> {</div>
<div class="line"><span class="lineno"> 58</span> assert(<a class="code hl_function" href="../../dd/d47/namespacemath.html#ac37d3ba52eb296597d7a024ba8c4a5a5">math::phiFunction</a>(1) == 1);</div>
<div class="line"><span class="lineno"> 59</span> assert(<a class="code hl_function" href="../../dd/d47/namespacemath.html#ac37d3ba52eb296597d7a024ba8c4a5a5">math::phiFunction</a>(2) == 1);</div>
<div class="line"><span class="lineno"> 60</span> assert(<a class="code hl_function" href="../../dd/d47/namespacemath.html#ac37d3ba52eb296597d7a024ba8c4a5a5">math::phiFunction</a>(10) == 4);</div>
<div class="line"><span class="lineno"> 61</span> assert(<a class="code hl_function" href="../../dd/d47/namespacemath.html#ac37d3ba52eb296597d7a024ba8c4a5a5">math::phiFunction</a>(123456) == 41088);</div>
<div class="line"><span class="lineno"> 62</span> assert(<a class="code hl_function" href="../../dd/d47/namespacemath.html#ac37d3ba52eb296597d7a024ba8c4a5a5">math::phiFunction</a>(808017424794) == 263582333856);</div>
<div class="line"><span class="lineno"> 63</span> assert(<a class="code hl_function" href="../../dd/d47/namespacemath.html#ac37d3ba52eb296597d7a024ba8c4a5a5">math::phiFunction</a>(3141592) == 1570792);</div>
<div class="line"><span class="lineno"> 64</span> assert(<a class="code hl_function" href="../../dd/d47/namespacemath.html#ac37d3ba52eb296597d7a024ba8c4a5a5">math::phiFunction</a>(27182818) == 12545904);</div>
<div class="line"><span class="lineno"> 65</span> </div>
<div class="line"><span class="lineno"> 66</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;All tests have successfully passed!\n&quot;</span>;</div>
<div class="line"><span class="lineno"> 67</span>}</div>
<div class="ttc" id="abasic_ostream_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a></div></div>
<div class="ttc" id="anamespacemath_html_ac37d3ba52eb296597d7a024ba8c4a5a5"><div class="ttname"><a href="../../dd/d47/namespacemath.html#ac37d3ba52eb296597d7a024ba8c4a5a5">math::phiFunction</a></div><div class="ttdeci">uint64_t phiFunction(uint64_t n)</div><div class="ttdoc">Function to calculate Euler's Totient.</div><div class="ttdef"><b>Definition</b> eulers_totient_function.cpp:39</div></div>
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