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<div id="projectbrief">Set of algorithms implemented in C++.</div>
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<div class="title">eulers_totient_function.cpp File Reference</div> </div>
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<div class="contents">
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<p>C++ Program to find <a href="https://en.wikipedia.org/wiki/Euler%27s_totient_function">Euler's Totient</a> function.
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<a href="#details">More...</a></p>
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<div class="textblock"><code>#include <cstdlib></code><br />
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<code>#include <iostream></code><br />
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</div><div class="textblock"><div class="dynheader">
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Include dependency graph for eulers_totient_function.cpp:</div>
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<div class="dyncontent">
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<tr class="heading"><td colspan="2"><h2 class="groupheader"><a name="func-members"></a>
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Functions</h2></td></tr>
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<tr class="memitem:a35e4874a3e1d67eb708dc57944c8aea0"><td class="memItemLeft" align="right" valign="top">uint64_t </td><td class="memItemRight" valign="bottom"><a class="el" href="../../da/d23/eulers__totient__function_8cpp.html#a35e4874a3e1d67eb708dc57944c8aea0">phiFunction</a> (uint64_t n)</td></tr>
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<tr class="separator:a35e4874a3e1d67eb708dc57944c8aea0"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:a0ddf1224851353fc92bfbff6f499fa97"><td class="memItemLeft" align="right" valign="top"><a id="a0ddf1224851353fc92bfbff6f499fa97"></a>
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int </td><td class="memItemRight" valign="bottom"><a class="el" href="../../da/d23/eulers__totient__function_8cpp.html#a0ddf1224851353fc92bfbff6f499fa97">main</a> (int argc, char *argv[])</td></tr>
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<tr class="memdesc:a0ddf1224851353fc92bfbff6f499fa97"><td class="mdescLeft"> </td><td class="mdescRight">Main function. <br /></td></tr>
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</table>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
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<div class="textblock"><p>C++ Program to find <a href="https://en.wikipedia.org/wiki/Euler%27s_totient_function">Euler's Totient</a> function. </p>
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<p>Euler Totient Function is also known as phi function. </p><p class="formulaDsp">
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\[\phi(n) = \phi\left({p_1}^{a_1}\right)\cdot\phi\left({p_2}^{a_2}\right)\ldots\]
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</p>
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<p> where \(p_1\), \(p_2\), \(\ldots\) are prime factors of n. <br />
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3 Euler's properties:</p><ol type="1">
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<li>\(\phi(n) = n-1\)</li>
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<li>\(\phi(n^k) = n^k - n^{k-1}\)</li>
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<li>\(\phi(a,b) = \phi(a)\cdot\phi(b)\) where a and b are relative primes.</li>
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</ol>
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<p>Applying this 3 properties on the first equation. </p><p class="formulaDsp">
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\[\phi(n) = n\cdot\left(1-\frac{1}{p_1}\right)\cdot\left(1-\frac{1}{p_2}\right)\cdots\]
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</p>
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<p> where \(p_1\), \(p_2\)... are prime factors. Hence Implementation in \(O\left(\sqrt{n}\right)\). <br />
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Some known values are:</p><ul>
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<li>\(\phi(100) = 40\)</li>
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<li>\(\phi(1) = 1\)</li>
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<li>\(\phi(17501) = 15120\)</li>
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<li>\(\phi(1420) = 560\) </li>
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</ul>
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</div><h2 class="groupheader">Function Documentation</h2>
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<a id="a35e4874a3e1d67eb708dc57944c8aea0"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#a35e4874a3e1d67eb708dc57944c8aea0">◆ </a></span>phiFunction()</h2>
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<div class="memitem">
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<div class="memproto">
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<td class="memname">uint64_t phiFunction </td>
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<td>(</td>
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<td class="paramtype">uint64_t </td>
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<td class="paramname"><em>n</em></td><td>)</td>
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<td></td>
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</tr>
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</table>
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</div><div class="memdoc">
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<p>Function to caculate Euler's totient phi </p>
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<div class="fragment"><div class="line"><a name="l00032"></a><span class="lineno"> 32</span>  {</div>
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<div class="line"><a name="l00033"></a><span class="lineno"> 33</span>  uint64_t result = n;</div>
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<div class="line"><a name="l00034"></a><span class="lineno"> 34</span>  <span class="keywordflow">for</span> (uint64_t i = 2; i * i <= n; i++) {</div>
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<div class="line"><a name="l00035"></a><span class="lineno"> 35</span>  <span class="keywordflow">if</span> (n % i == 0) {</div>
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<div class="line"><a name="l00036"></a><span class="lineno"> 36</span>  <span class="keywordflow">while</span> (n % i == 0) {</div>
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<div class="line"><a name="l00037"></a><span class="lineno"> 37</span>  n /= i;</div>
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<div class="line"><a name="l00038"></a><span class="lineno"> 38</span>  }</div>
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<div class="line"><a name="l00039"></a><span class="lineno"> 39</span>  result -= result / i;</div>
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<div class="line"><a name="l00040"></a><span class="lineno"> 40</span>  }</div>
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<div class="line"><a name="l00041"></a><span class="lineno"> 41</span>  }</div>
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<div class="line"><a name="l00042"></a><span class="lineno"> 42</span>  <span class="keywordflow">if</span> (n > 1)</div>
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<div class="line"><a name="l00043"></a><span class="lineno"> 43</span>  result -= result / n;</div>
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<div class="line"><a name="l00044"></a><span class="lineno"> 44</span>  <span class="keywordflow">return</span> result;</div>
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<div class="line"><a name="l00045"></a><span class="lineno"> 45</span> }</div>
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