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‎main/algebra/factorization.html

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<ul class="metadata page-metadata" data-bi-name="page info" lang="en-us" dir="ltr">
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Last update:
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<span class="git-revision-date-localized-plugin git-revision-date-localized-plugin-date" title="April 16, 2024 21:08:03">April 16, 2024</span>&emsp;
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If <span class="arithmatex">$p$</span> is smaller than <span class="arithmatex">$\sqrt{n}$</span>, the repetition will likely start in <span class="arithmatex">$O(\sqrt[4]{n})$</span>.</p>
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<p>Here is a visualization of such a sequence <span class="arithmatex">$\{x_i \bmod p\}$</span> with <span class="arithmatex">$n = 2206637$</span>, <span class="arithmatex">$p = 317$</span>, <span class="arithmatex">$x_0 = 2$</span> and <span class="arithmatex">$f(x) = x^2 + 1$</span>.
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From the form of the sequence you can see very clearly why the algorithm is called Pollard's <span class="arithmatex">$\rho$</span> algorithm.</p>
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<center>![Pollard's rho visualization](pollard_rho.png)</center>
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<div style="text-align: center;">
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<img src="pollard_rho.png" alt="Pollard's rho visualization">
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</div>
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<p>Yet, there is still an open question.
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How can we exploit the properties of the sequence <span class="arithmatex">$\{x_i \bmod p\}$</span> to our advantage without even knowing the number <span class="arithmatex">$p$</span> itself?</p>
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<ul class="metadata page-metadata" data-bi-name="page info" lang="en-us" dir="ltr">
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<span class="contributors-text">Contributors:</span>
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‎main/algebra/sieve-of-eratosthenes.html

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The next unmarked number is 5, which is the next prime number, and we mark all proper multiples of it.
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And we continue this procedure until we have processed all numbers in the row.</p>
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<p>In the following image you can see a visualization of the algorithm for computing all prime numbers in the range <span class="arithmatex">$[1; 16]$</span>. It can be seen, that quite often we mark numbers as composite multiple times.</p>
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<center>![Sieve of Eratosthenes](sieve_eratosthenes.png)</center>
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<div style="text-align: center;">
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<img src="sieve_eratosthenes.png" alt="Sieve of Eratosthenes">
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<p>The idea behind is this:
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A number is prime, if none of the smaller prime numbers divides it.
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<span class="contributors-text">Contributors:</span>
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<ul class="contributors" data-bi-name="contributors"><li><a href="https://github.com/jakobkogler" title="jakobkogler" data-bi-name="contributorprofile" target="_blank">jakobkogler</a> (42.7%)</li><li><a href="https://github.com/ellen-interpret" title="ellen-interpret" data-bi-name="contributorprofile" target="_blank">ellen-interpret</a> (23.36%)</li><li><a href="https://github.com/hieplpvip" title="hieplpvip" data-bi-name="contributorprofile" target="_blank">hieplpvip</a> (16.79%)</li><li><a href="https://github.com/Morass" title="Morass" data-bi-name="contributorprofile" target="_blank">Morass</a> (5.47%)</li><li><a href="https://github.com/tcNickolas" title="tcNickolas" data-bi-name="contributorprofile" target="_blank">tcNickolas</a> (3.65%)</li><li><a href="https://github.com/adamant-pwn" title="adamant-pwn" data-bi-name="contributorprofile" target="_blank">adamant-pwn</a> (2.19%)</li><li><a href="https://github.com/boxlesscat" title="boxlesscat" data-bi-name="contributorprofile" target="_blank">boxlesscat</a> (2.19%)</li><li><a href="https://github.com/roll-no-1" title="roll-no-1" data-bi-name="contributorprofile" target="_blank">roll-no-1</a> (1.09%)</li><li><a href="https://github.com/TrietMinh799" title="TrietMinh799" data-bi-name="contributorprofile" target="_blank">TrietMinh799</a> (0.73%)</li><li><a href="https://github.com/jxu" title="jxu" data-bi-name="contributorprofile" target="_blank">jxu</a> (0.73%)</li><li><a href="https://github.com/zdr256" title="zdr256" data-bi-name="contributorprofile" target="_blank">zdr256</a> (0.36%)</li><li><a href="https://github.com/infiniteasp8" title="infiniteasp8" data-bi-name="contributorprofile" target="_blank">infiniteasp8</a> (0.36%)</li><li><a href="https://github.com/joaquingx" title="joaquingx" data-bi-name="contributorprofile" target="_blank">joaquingx</a> (0.36%)</li></ul>
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‎main/data_structures/fenwick.html

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<span class="git-revision-date-localized-plugin git-revision-date-localized-plugin-date" title="July 31, 2024 14:14:13">July 31, 2024</span>&emsp;
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<p>where <span class="arithmatex">$|$</span> is the bitwise OR operator.</p>
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<p>The following image shows a possible interpretation of the Fenwick tree as tree.
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The nodes of the tree show the ranges they cover.</p>
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<h2 id="implementation">Implementation<a class="headerlink" href="#implementation" title="Permanent link">&para;</a></h2>
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<h3 id="finding-sum-in-one-dimensional-array">Finding sum in one-dimensional array<a class="headerlink" href="#finding-sum-in-one-dimensional-array" title="Permanent link">&para;</a></h3>
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<span class="contributors-text">Contributors:</span>
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