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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 134</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=2016-10-30T17:40:00+03:00 pubdate> Вс 30 октября 2016 </time> <a href=../../posts/moio-reshenie-zadachi-134/ rel=bookmark><h1>Моё решение задачи&nbsp;134</h1></a> </header> <section class=post-content> <p>Назовём <em>порождающим</em> для двух последовательных простых <span class=math>\(p_1 &lt; p_2\)</span> наименьшее натуральное число, что оно закачивается на <span class=math>\(p_1\)</span> и при этом делится на <span class=math>\(p_2\)</span>. Необходимо найти сумму порождающих для всех <span class=math>\(p_1 \in \left[ 5; 10^6&nbsp;\right]\)</span></p> <p>Например, если <span class=math>\(p_1 = 19\)</span>, то следующее простое <span class=math>\(p_2 = 23\)</span>. Тогда порождающим будет число <span class=math>\(1219\)</span>, при этом <span class=math>\(1219 \: \vdots \: 23\)</span>.</p> <p>Полное условие можно найти <a href="https://projecteuler.net/problem=134">тут</a></p> <p>Несмотря на то, что сложность задачи 45%, для её решения достаточно выписать&nbsp;условие.</p> <p>Пусть <span class=math>\(p_1\)</span> содержит в себе <span class=math>\(k\)</span> цифр, т.е. <span class=math>\(n = r \cdot 10^k + p_1\)</span>, где <span class=math>\(r\)</span> &#8212; какое-то натуральное число с отрезка <span class=math>\(\left[ 1; p_2-1&nbsp;\right]\)</span></p> <p>Давайте посчитаем остатки по модулю <span class=math>\(p_2\)</span>: <span class=math>\(n \equiv r \cdot 10^k + p_1 \equiv 0\)</span>. Отсюда получим явную формулу для <span class=math>\(r\)</span>: </p> <div class=math>$$ r \equiv -p_1 \cdot 10^{-k} \equiv -p_1 \cdot 10^{p_2 -1-k} $$</div> <p>Комментарии:</p> <ol> <li>Так как <span class=math>\(a^p \equiv a \mod p\)</span>, то верно что <span class=math>\(a^{-k} \equiv a^{p -1-k} \mod&nbsp;p\)</span></li> <li>Это всё бессмысленно, если не знать про <a href=https://ru.wikipedia.org/wiki/Алгоритмы_быстрого_возведения_в_степень>алгоритм быстрого возведения в степень</a>, который делает асимптотическую сложность возведения в степень&nbsp;логарифмической.</li> </ol> <p>У нас есть явная формула для порождающего, и мы знаем как её быстро посчитать. Ниже приведён код на Python с использованием <a href=http://www.sympy.org/ru/ >sympy</a>.</p> <div class=highlight><pre><span class=code-line><span></span><span class=kn>from</span> <span class=nn>sympy</span> <span class=kn>import</span> <span class=n>primerange</span> <span class=c1># для получения простых чисел</span></span>
<span class=code-line></span>
<span class=code-line><span class=c1># быстрое возведение в степень по модулю</span></span>
<span class=code-line><span class=k>def</span> <span class=nf>fast_pow</span><span class=p>(</span><span class=n>x</span><span class=p>,</span> <span class=n>y</span><span class=p>,</span> <span class=n>modulo</span><span class=p>):</span></span>
<span class=code-line> <span class=k>if</span> <span class=n>y</span> <span class=o>==</span> <span class=mi>0</span><span class=p>:</span></span>
<span class=code-line> <span class=k>return</span> <span class=mi>1</span></span>
<span class=code-line> <span class=n>p</span> <span class=o>=</span> <span class=n>fast_pow</span><span class=p>(</span><span class=n>x</span><span class=p>,</span> <span class=n>y</span> <span class=o>//</span> <span class=mi>2</span><span class=p>,</span> <span class=n>modulo</span><span class=p>)</span></span>
<span class=code-line> <span class=n>p</span> <span class=o>=</span> <span class=p>(</span><span class=n>p</span> <span class=o>*</span> <span class=n>p</span><span class=p>)</span> <span class=o>%</span> <span class=n>modulo</span></span>
<span class=code-line> <span class=k>if</span> <span class=n>y</span> <span class=o>%</span> <span class=mi>2</span><span class=p>:</span></span>
<span class=code-line> <span class=n>p</span> <span class=o>=</span> <span class=p>(</span><span class=n>p</span> <span class=o>*</span> <span class=n>x</span><span class=p>)</span> <span class=o>%</span> <span class=n>modulo</span></span>
<span class=code-line> <span class=k>return</span> <span class=n>p</span></span>
<span class=code-line></span>
<span class=code-line><span class=c1># нам нужно первое простое, которое больше 10^6 -- 10^6+3</span></span>
<span class=code-line><span class=n>primes</span> <span class=o>=</span> <span class=nb>list</span><span class=p>(</span><span class=n>primerange</span><span class=p>(</span><span class=mi>5</span><span class=p>,</span><span class=mi>10</span><span class=o>**</span><span class=mi>6</span><span class=o>+</span><span class=mi>4</span><span class=p>))</span> </span>
<span class=code-line></span>
<span class=code-line><span class=n>sm</span> <span class=o>=</span> <span class=mi>0</span></span>
<span class=code-line></span>
<span class=code-line><span class=k>for</span> <span class=n>i</span> <span class=ow>in</span> <span class=nb>range</span><span class=p>(</span><span class=nb>len</span><span class=p>(</span><span class=n>primes</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=n>digs</span> <span class=o>=</span> <span class=nb>len</span><span class=p>(</span><span class=nb>str</span><span class=p>(</span><span class=n>primes</span><span class=p>[</span><span class=n>i</span><span class=p>]))</span> <span class=c1># количество цифр</span></span>
<span class=code-line> <span class=n>r</span> <span class=o>=</span> <span class=p>(</span><span class=n>primes</span><span class=p>[</span><span class=n>i</span><span class=o>+</span><span class=mi>1</span><span class=p>]</span><span class=o>**</span><span class=mi>2</span> <span class=o>-</span> <span class=n>primes</span><span class=p>[</span><span class=n>i</span><span class=p>]</span> <span class=o>*</span> <span class=n>fast_pow</span><span class=p>(</span><span class=mi>10</span><span class=p>,</span> <span class=n>primes</span><span class=p>[</span><span class=n>i</span><span class=o>+</span><span class=mi>1</span><span class=p>]</span> <span class=o>-</span> <span class=mi>1</span> <span class=o>-</span> <span class=n>digs</span><span class=p>,</span> <span class=n>primes</span><span class=p>[</span><span class=n>i</span><span class=o>+</span><span class=mi>1</span><span class=p>]))</span> <span class=o>%</span> <span class=n>primes</span><span class=p>[</span><span class=n>i</span><span class=o>+</span><span class=mi>1</span><span class=p>]</span></span>
<span class=code-line> <span class=n>sm</span> <span class=o>+=</span> <span class=n>r</span> <span class=o>*</span> <span class=mi>10</span><span class=o>**</span><span class=n>digs</span> <span class=o>+</span> <span class=n>primes</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>print</span><span class=p>(</span><span class=s1>&#39;Result is {}&#39;</span><span class=o>.</span><span class=n>format</span><span class=p>(</span><span class=n>sm</span><span class=p>))</span></span>
</pre></div> <p>Ответ: <strong>18613426663617118</strong></p> <script type=text/javascript>if (!document.getElementById('mathjaxscript_pelican_#%@#$@#')) {
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