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<!DOCTYPE html><!--[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> <!--<![endif]--> <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="https://fonts.googleapis.com/css?family=Forum|Oswald|PT+Sans|Philosopher|Ubuntu+Mono" rel=stylesheet><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=https://likemath.ru/feeds/all.atom.xml type=application/atom+xml rel=alternate title="Блог 529 Atom Feed"><link href=https://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]>
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<![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 октября 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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