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<!--[if lt IE 7]> <html class="no-js lt-ie9 lt-ie8 lt-ie7"> <![endif]--><!--[if IE 7]> <html class="no-js lt-ie9 lt-ie8"> <![endif]--><!--[if IE 8]> <html class="no-js lt-ie9"> <![endif]--><!--[if gt IE 8]><!--><html class="no-js"> <head><meta charset="utf-8"><meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"><title>Моё решение задачи 146</title><meta name="description" content><meta name="viewport" content="width=device-width"><link rel="stylesheet" href="../../theme/css/normalize.css"><link href="http://fonts.googleapis.com/css?family=Philosopher&subset=latin,cyrillic" rel="stylesheet" type="text/css"><link href="http://fonts.googleapis.com/css?family=Forum&subset=cyrillic" rel="stylesheet" type="text/css"><link href="//fonts.googleapis.com/css?family=Oswald" rel="stylesheet" type="text/css"><link href="http://fonts.googleapis.com/css?family=Ubuntu+Mono" rel="stylesheet" type="text/css"><link href="http://fonts.googleapis.com/css?family=PT+Sans" rel="stylesheet" type="text/css"><link rel="stylesheet" href="../../theme/css/font-awesome.min.css"><link rel="stylesheet" href="../../theme/css/main.css"><link rel="stylesheet" href="../../theme/css/blog.css"><link rel="stylesheet" href="../../theme/css/github.css"><link href="http://likemath.ru/feeds/all.atom.xml" type="application/atom+xml" rel="alternate" title="Блог 529 Atom Feed"><link href="http://likemath.ru/feeds/all.rss.xml" type="application/rss+xml" rel="alternate" title="Блог 529 RSS Feed"><script src="../../theme/js/vendor/modernizr-2.6.2.min.js"></script></head><body><!--[if lt IE 7]>
<p class="chromeframe">You are using an <strong>outdated</strong> browser. Please <a href="http://browsehappy.com/">upgrade your browser</a> or <a href="http://www.google.com/chromeframe/?redirect=true">activate Google Chrome Frame</a> to improve your experience.</p>
<![endif]--><div id="wrapper"><header id="sidebar" class="side-shadow"><hgroup id="site-header"><a id="site-title" href="../.."><h2><i class="icon-pencil"></i> Блог 529</h2></a><p id="site-desc"> Project Euler и остальное </p></hgroup><nav><ul id="nav-links"><li><a href="../../">Главная</a></li><li><a href="../../pages/projects.html">Мои проекты</a></li><li><a href="../../pages/about.html">Об авторе</a></li><li><a href="../../feeds/feed.atom.xml">Atom feed</a></li></ul></nav><footer id="site-info"><p> Powered by Pelican. </p></footer></header><div id="post-container"><ol id="post-list"><li><article class="post-entry"><header class="entry-header"><time class="post-time" datetime="2016-10-21T17:40:00+03:00" pubdate> Пт 21 Октябрь 2016 </time><a href="../../posts/moio-reshenie-zadachi-146/" rel="bookmark"><h1>Моё решение задачи&nbsp;146</h1></a></header><section class="post-content"><p>Необходимо найти сумму всех натуральных <span class="math">\(n\)</span>, что <span class="math">\(n^2+1\)</span>, <span class="math">\(n^2+3\)</span>, <span class="math">\(n^2+7\)</span>, <span class="math">\(n^2+9\)</span>, <span class="math">\(n^2+13\)</span>, и <span class="math">\(n^2+27\)</span> будут <em>последовательными</em> простыми&nbsp;числами.</p><p>Полное условие можно найти <a href="https://projecteuler.net/problem=146">тут</a></p><p>Хочется отметить, что сложность у задачи 50%, а на текущий момент её решило меньше 4000 человек. Тем не менее, мне она показалось простой. Простейшее решение отработало очень&nbsp;быстро.</p><p>Для начала, можно отметить, что в лоб проверять условие очень долго. Проверять на простоту числа порядка <span class="math">\(10^{15}\)</span> достаточно сложно, поэтому их нужно как-то&nbsp;отсеять.</p><p>Самое простое &#8212; не рассматривать те <span class="math">\(n\)</span>, что хотя бы одно из <span class="math">\(n^2+1\)</span>, <span class="math">\(n^2+3\)</span>, <span class="math">\(n^2+7\)</span>, <span class="math">\(n^2+9\)</span>, <span class="math">\(n^2+13\)</span>, и <span class="math">\(n^2+27\)</span> будет заведомо делиться на какое-то маленькое простое число. Это даёт достаточно хорошие результаты: из 150 миллионов чисел, после отсеивания по простым числам <span class="math">\(&lt; 3000\)</span> (этот параметр я подбирал уже после решения задач: если он слишком маленький, то будет слишком много проверок на простоту, если же слишком большой, то мы делаем слишком много работы, чтобы отсеять несколько чисел), останется меньше <span class="math">\(2000\)</span> чисел. Их уже можно проверить&nbsp;непосредственно. </p><p>Тогда алгоритм может быть&nbsp;таким:</p><ol><li>Находим простые числа меньше <span class="math">\(3000\)</span>.</li><li>Для каждого из них находим допустимые&nbsp;остатки.</li><li>Для каждого из чисел от <span class="math">\(1\)</span> до <span class="math">\(n\)</span> проверяем, что остатки по всем простым&nbsp;хорошие.</li><li>Непосредственно проверяем условие. Важно не забыть проверить <strong>не</strong>простоту оставшихся нечётных чисел из диапазона <span class="math">\(n^2 + 1 \ldots n^2 + 27\)</span> там могут быть (и будут!) другие простые&nbsp;числа.</li></ol><p>Непосредственно сам поиск такой клики можно реализовать тривиально. Ниже мой код на C++11 с использованием библиотек Flint и primesieve. Распараллеливание хоть и просится, но смысла не имеет, т.к. я получил ответ менее, чем за 5&nbsp;секунд.</p><div class="highlight"><pre><span class="code-line"><span></span><span class="cm">/*</span></span>
<span class="code-line"><span class="cm"> * Problem 146 on Project Euler</span></span>
<span class="code-line"><span class="cm"> * Aleksey Lobanov (c) 2016</span></span>
<span class="code-line"><span class="cm"> */</span></span>
<span class="code-line"></span>
<span class="code-line"><span class="cp">#include</span> <span class="cpf">&lt;iostream&gt;</span><span class="cp"></span></span>
<span class="code-line"><span class="cp">#include</span> <span class="cpf">&lt;vector&gt;</span><span class="cp"></span></span>
<span class="code-line"><span class="cp">#include</span> <span class="cpf">&lt;cstdint&gt;</span><span class="cp"></span></span>
<span class="code-line"><span class="cp">#include</span> <span class="cpf">&lt;set&gt;</span><span class="cp"></span></span>
<span class="code-line"><span class="cp">#include</span> <span class="cpf">&lt;iomanip&gt;</span><span class="cp"></span></span>
<span class="code-line"><span class="cp">#include</span> <span class="cpf">&lt;algorithm&gt;</span><span class="cp"></span></span>
<span class="code-line"></span>
<span class="code-line"><span class="cp">#include</span> <span class="cpf">&quot;fmpzxx.h&quot;</span><span class="cp"></span></span>
<span class="code-line"><span class="cp">#include</span> <span class="cpf">&quot;arithxx.h&quot;</span><span class="cp"></span></span>
<span class="code-line"></span>
<span class="code-line"><span class="cp">#include</span> <span class="cpf">&quot;primesieve.hpp&quot;</span><span class="cp"></span></span>
<span class="code-line"></span>
<span class="code-line"><span class="k">using</span> <span class="k">namespace</span> <span class="n">std</span><span class="p">;</span></span>
<span class="code-line"><span class="k">using</span> <span class="k">namespace</span> <span class="n">flint</span><span class="p">;</span></span>
<span class="code-line"></span>
<span class="code-line"><span class="kt">bool</span> <span class="nf">is_prime</span><span class="p">(</span><span class="kt">int64_t</span> <span class="n">num</span><span class="p">)</span> <span class="p">{</span></span>
<span class="code-line"> <span class="n">fmpz_factorxx</span> <span class="n">fact</span><span class="p">;</span></span>
<span class="code-line"> <span class="n">fact</span><span class="p">.</span><span class="n">set_factor</span><span class="p">(</span><span class="n">num</span><span class="p">);</span></span>
<span class="code-line"> <span class="k">if</span> <span class="p">(</span> <span class="n">fact</span><span class="p">.</span><span class="n">size</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">1</span> <span class="p">)</span></span>
<span class="code-line"> <span class="k">return</span> <span class="nb">false</span><span class="p">;</span></span>
<span class="code-line"> <span class="k">if</span> <span class="p">(</span> <span class="n">fact</span><span class="p">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">)</span> </span>
<span class="code-line"> <span class="k">return</span> <span class="nb">false</span><span class="p">;</span></span>
<span class="code-line"> <span class="k">return</span> <span class="nb">true</span><span class="p">;</span></span>
<span class="code-line"><span class="p">}</span></span>
<span class="code-line"></span>
<span class="code-line"><span class="kt">bool</span> <span class="nf">is_possible</span><span class="p">(</span><span class="kt">int64_t</span> <span class="n">num</span><span class="p">,</span> <span class="k">const</span> <span class="n">vector</span><span class="o">&lt;</span><span class="kt">int64_t</span><span class="o">&gt;</span> <span class="o">&amp;</span><span class="n">to_add</span><span class="p">)</span> <span class="p">{</span></span>
<span class="code-line"> <span class="k">for</span> <span class="p">(</span><span class="k">auto</span> <span class="o">&amp;&amp;</span><span class="nl">add</span><span class="p">:</span> <span class="n">to_add</span><span class="p">)</span></span>
<span class="code-line"> <span class="k">if</span> <span class="p">(</span> <span class="o">!</span><span class="n">is_prime</span><span class="p">(</span><span class="n">num</span><span class="o">*</span><span class="n">num</span> <span class="o">+</span> <span class="n">add</span><span class="p">)</span> <span class="p">)</span></span>
<span class="code-line"> <span class="k">return</span> <span class="nb">false</span><span class="p">;</span></span>
<span class="code-line"></span>
<span class="code-line"> <span class="c1">// primes must be consecutive</span></span>
<span class="code-line"> <span class="c1">// so we need check, that other numbers like n^2 + i is not primes</span></span>
<span class="code-line"> <span class="n">vector</span><span class="o">&lt;</span><span class="kt">int64_t</span><span class="o">&gt;</span> <span class="n">other_adds</span><span class="p">;</span></span>
<span class="code-line"> <span class="k">for</span> <span class="p">(</span><span class="kt">size_t</span> <span class="n">i</span> <span class="o">=</span> <span class="n">to_add</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="mi">2</span><span class="p">;</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">to_add</span><span class="p">[</span><span class="n">to_add</span><span class="p">.</span><span class="n">size</span><span class="p">()</span> <span class="o">-</span> <span class="mi">1</span><span class="p">];</span> <span class="n">i</span> <span class="o">+=</span> <span class="mi">2</span><span class="p">)</span> </span>
<span class="code-line"> <span class="k">if</span> <span class="p">(</span> <span class="o">!</span><span class="n">binary_search</span><span class="p">(</span><span class="n">to_add</span><span class="p">.</span><span class="n">begin</span><span class="p">(),</span> <span class="n">to_add</span><span class="p">.</span><span class="n">end</span><span class="p">(),</span> <span class="n">i</span><span class="p">)</span> <span class="p">)</span></span>
<span class="code-line"> <span class="n">other_adds</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">i</span><span class="p">);</span></span>
<span class="code-line"></span>
<span class="code-line"> <span class="k">for</span> <span class="p">(</span><span class="k">auto</span> <span class="o">&amp;&amp;</span><span class="nl">add</span><span class="p">:</span> <span class="n">other_adds</span><span class="p">)</span></span>
<span class="code-line"> <span class="k">if</span> <span class="p">(</span> <span class="n">is_prime</span><span class="p">(</span><span class="n">num</span><span class="o">*</span><span class="n">num</span> <span class="o">+</span> <span class="n">add</span><span class="p">)</span> <span class="p">)</span></span>
<span class="code-line"> <span class="k">return</span> <span class="nb">false</span><span class="p">;</span> </span>
<span class="code-line"></span>
<span class="code-line"> <span class="k">return</span> <span class="nb">true</span><span class="p">;</span></span>
<span class="code-line"><span class="p">}</span></span>
<span class="code-line"></span>
<span class="code-line"><span class="kt">int</span> <span class="nf">main</span><span class="p">()</span> <span class="p">{</span></span>
<span class="code-line"> <span class="k">const</span> <span class="n">vector</span><span class="o">&lt;</span><span class="kt">int64_t</span><span class="o">&gt;</span> <span class="n">to_add</span> <span class="o">=</span> <span class="p">{</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">13</span><span class="p">,</span> <span class="mi">27</span><span class="p">};</span></span>
<span class="code-line"></span>
<span class="code-line"> <span class="k">const</span> <span class="kt">int64_t</span> <span class="n">MAX_N</span> <span class="o">=</span> <span class="mi">1l</span><span class="o">*</span><span class="mi">150</span><span class="o">*</span><span class="mi">1000</span><span class="o">*</span><span class="mi">1000</span><span class="p">;</span></span>
<span class="code-line"> <span class="k">const</span> <span class="kt">int64_t</span> <span class="n">MAX_PRIME</span> <span class="o">=</span> <span class="mi">3000</span><span class="p">;</span></span>
<span class="code-line"></span>
<span class="code-line"> <span class="n">vector</span><span class="o">&lt;</span><span class="kt">int64_t</span><span class="o">&gt;</span> <span class="n">sieve_primes</span><span class="p">;</span></span>
<span class="code-line"> <span class="n">primesieve</span><span class="o">::</span><span class="n">generate_primes</span><span class="p">(</span><span class="n">MAX_PRIME</span><span class="p">,</span> <span class="o">&amp;</span><span class="n">sieve_primes</span><span class="p">);</span></span>
<span class="code-line"></span>
<span class="code-line"> <span class="n">vector</span><span class="o">&lt;</span> <span class="n">vector</span> <span class="o">&lt;</span><span class="kt">int64_t</span><span class="o">&gt;</span> <span class="o">&gt;</span> <span class="n">good_remainders</span><span class="p">;</span> </span>
<span class="code-line"> <span class="k">for</span> <span class="p">(</span><span class="k">auto</span> <span class="o">&amp;&amp;</span><span class="nl">prime</span><span class="p">:</span> <span class="n">sieve_primes</span><span class="p">)</span> <span class="p">{</span></span>
<span class="code-line"> <span class="n">set</span><span class="o">&lt;</span><span class="kt">int64_t</span><span class="o">&gt;</span> <span class="n">remainders</span><span class="p">;</span></span>
<span class="code-line"> <span class="k">for</span><span class="p">(</span><span class="kt">int64_t</span> <span class="n">rem</span> <span class="o">=</span> <span class="mi">0</span><span class="p">;</span> <span class="n">rem</span> <span class="o">&lt;</span> <span class="n">prime</span><span class="p">;</span> <span class="o">++</span><span class="n">rem</span><span class="p">)</span></span>
<span class="code-line"> <span class="n">remainders</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="n">rem</span><span class="p">);</span></span>
<span class="code-line"></span>
<span class="code-line"> <span class="n">set</span><span class="o">&lt;</span><span class="kt">int64_t</span><span class="o">&gt;</span> <span class="n">base_remainders</span><span class="p">;</span></span>
<span class="code-line"> <span class="k">for</span> <span class="p">(</span><span class="k">auto</span> <span class="o">&amp;&amp;</span><span class="nl">base</span><span class="p">:</span> <span class="n">to_add</span><span class="p">)</span></span>
<span class="code-line"> <span class="n">base_remainders</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="n">base</span> <span class="o">%</span> <span class="n">prime</span><span class="p">);</span></span>
<span class="code-line"></span>
<span class="code-line"> <span class="k">for</span> <span class="p">(</span><span class="kt">int64_t</span> <span class="n">rem</span> <span class="o">=</span> <span class="mi">0</span><span class="p">;</span> <span class="n">rem</span> <span class="o">&lt;</span> <span class="n">prime</span><span class="p">;</span> <span class="o">++</span><span class="n">rem</span><span class="p">)</span></span>
<span class="code-line"> <span class="k">for</span> <span class="p">(</span><span class="k">auto</span> <span class="o">&amp;&amp;</span><span class="nl">base_rem</span><span class="p">:</span> <span class="n">base_remainders</span><span class="p">)</span></span>
<span class="code-line"> <span class="k">if</span> <span class="p">(</span> <span class="p">(</span><span class="n">rem</span><span class="o">*</span><span class="n">rem</span> <span class="o">+</span> <span class="n">base_rem</span><span class="p">)</span> <span class="o">%</span> <span class="n">prime</span> <span class="o">==</span> <span class="mi">0</span> <span class="p">)</span></span>
<span class="code-line"> <span class="n">remainders</span><span class="p">.</span><span class="n">erase</span><span class="p">(</span><span class="n">rem</span><span class="p">);</span></span>
<span class="code-line"></span>
<span class="code-line"> <span class="n">good_remainders</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int64_t</span><span class="o">&gt;</span><span class="p">(</span><span class="n">remainders</span><span class="p">.</span><span class="n">begin</span><span class="p">(),</span> <span class="n">remainders</span><span class="p">.</span><span class="n">end</span><span class="p">()));</span></span>
<span class="code-line"> <span class="p">}</span></span>
<span class="code-line"></span>
<span class="code-line"> <span class="kt">size_t</span> <span class="n">cnt</span> <span class="o">=</span> <span class="mi">0</span><span class="p">;</span></span>
<span class="code-line"> <span class="kt">size_t</span> <span class="n">sum</span> <span class="o">=</span> <span class="mi">0</span><span class="p">;</span></span>
<span class="code-line"></span>
<span class="code-line"> <span class="cm">/* WARNING</span></span>
<span class="code-line"><span class="cm"> * for small n can be that</span></span>
<span class="code-line"><span class="cm"> * n^2 + [1 or 3 or .. ] is prime in sieve primes</span></span>
<span class="code-line"><span class="cm"> * but there is only one n &lt; 315410 is 10,</span></span>
<span class="code-line"><span class="cm"> * so we need add 10 to n</span></span>
<span class="code-line"><span class="cm"> */</span></span>
<span class="code-line"> <span class="n">sum</span> <span class="o">+=</span> <span class="mi">10</span><span class="p">;</span></span>
<span class="code-line"></span>
<span class="code-line"> <span class="k">for</span> <span class="p">(</span><span class="kt">int64_t</span> <span class="n">i</span> <span class="o">=</span> <span class="mi">1</span><span class="p">;</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">MAX_N</span><span class="p">;</span> <span class="o">++</span><span class="n">i</span><span class="p">)</span> <span class="p">{</span></span>
<span class="code-line"> <span class="kt">bool</span> <span class="n">is_good</span> <span class="o">=</span> <span class="nb">true</span><span class="p">;</span></span>
<span class="code-line"> <span class="k">for</span> <span class="p">(</span><span class="kt">size_t</span> <span class="n">prime_ind</span> <span class="o">=</span> <span class="mi">0</span><span class="p">;</span> <span class="n">prime_ind</span> <span class="o">&lt;</span> <span class="n">sieve_primes</span><span class="p">.</span><span class="n">size</span><span class="p">();</span> <span class="o">++</span><span class="n">prime_ind</span><span class="p">)</span> <span class="p">{</span></span>
<span class="code-line"> <span class="k">if</span> <span class="p">(</span> <span class="o">!</span><span class="n">binary_search</span><span class="p">(</span><span class="n">good_remainders</span><span class="p">[</span><span class="n">prime_ind</span><span class="p">].</span><span class="n">begin</span><span class="p">(),</span> <span class="n">good_remainders</span><span class="p">[</span><span class="n">prime_ind</span><span class="p">].</span><span class="n">end</span><span class="p">(),</span> <span class="n">i</span> <span class="o">%</span> <span class="n">sieve_primes</span><span class="p">[</span><span class="n">prime_ind</span><span class="p">])</span> <span class="p">)</span> <span class="p">{</span></span>
<span class="code-line"> <span class="n">is_good</span> <span class="o">=</span> <span class="nb">false</span><span class="p">;</span></span>
<span class="code-line"> <span class="k">break</span><span class="p">;</span></span>
<span class="code-line"> <span class="p">}</span></span>
<span class="code-line"> <span class="p">}</span></span>
<span class="code-line"> <span class="k">if</span> <span class="p">(</span> <span class="n">is_good</span> <span class="p">)</span> <span class="p">{</span></span>
<span class="code-line"> <span class="n">cnt</span><span class="o">++</span><span class="p">;</span></span>
<span class="code-line"> <span class="k">if</span> <span class="p">(</span> <span class="n">is_possible</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">to_add</span><span class="p">)</span> <span class="p">)</span> <span class="p">{</span></span>
<span class="code-line"> <span class="n">sum</span> <span class="o">+=</span> <span class="n">i</span><span class="p">;</span></span>
<span class="code-line"> <span class="p">}</span> </span>
<span class="code-line"> <span class="p">}</span></span>
<span class="code-line"> <span class="p">}</span></span>
<span class="code-line"> <span class="n">cout</span> <span class="o">&lt;&lt;</span> <span class="s">&quot;count = &quot;</span> <span class="o">&lt;&lt;</span> <span class="n">cnt</span> <span class="o">&lt;&lt;</span> <span class="n">endl</span><span class="p">;</span></span>
<span class="code-line"> <span class="n">cout</span> <span class="o">&lt;&lt;</span> <span class="s">&quot;Result is: &quot;</span> <span class="o">&lt;&lt;</span> <span class="n">sum</span> <span class="o">&lt;&lt;</span> <span class="n">endl</span><span class="p">;</span></span>
<span class="code-line"> <span class="k">return</span> <span class="mi">0</span><span class="p">;</span></span>
<span class="code-line"><span class="p">}</span></span>
</pre></div><p>Ответ: <strong>676333270</strong></p><script type="text/javascript">if (!document.getElementById('mathjaxscript_pelican_#%@#$@#')) {
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