/mycila-math/trunk/todo/webpages/Prime Numbers, Factorization and Euler Function .html

http://mycila.googlecode.com/ · HTML · 784 lines · 663 code · 94 blank · 27 comment · 0 complexity · ad0846f6a24837f3fa3540e2d8d13a1a MD5 · raw file 

    

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  265. <li><a href="http://www.topcoder.com/wiki/display/tc/TopCoder+UML+Tool">UML Tool</a></li>
    266. 

  266. <li class="exp"><a href="javascript:void(0)" onclick="Javascript:toggleMenu(this.parentNode,'m_des_stats')">Statistics</a>
    267. 

  267. <ul id="m_des_stats">
    268. 

  268. <li><a href="http://www.topcoder.com/tc?module=CompList&amp;ph=112">Past Contests</a></li>
    269. 

  269. <li><a href="http://www.topcoder.com/tc?module=ColorChange&amp;ph=112">Recent Color Changes</a></li>
    270. 

  270. <li><a href="http://www.topcoder.com/wiki/display/tc/Component+Design+Data+Feeds">Data Feeds</a></li>
    271. 

  271. </ul></li>
    272. 

  272. <li><a href="http://www.topcoder.com/tc?module=ReviewBoard&amp;pt=1">Meet the Review Board</a></li>
    273. 

  273. <li class="exp"><a href="javascript:void(0)" onclick="Javascript:toggleMenu(this.parentNode,'m_des_support')">Support / FAQs</a>
    274. 

  274. <ul id="m_des_support">
    275. 

  275. <li><a href="http://www.topcoder.com/wiki/display/tc/How+to+Compete+in+Component+Design+Competitions">How to Compete</a></li>
    276. 

  276. <li><a href="http://www.topcoder.com/wiki/display/tc/Component+Design+Documentation">Documentation</a></li>
    277. 

  277. <li><a href="http://www.topcoder.com/wiki/display/tc/Component+Design+Ratings">Ratings</a></li>
    278. 

  278. <li><a href="http://www.topcoder.com/wiki/display/tc/Component+Design+Reliability+Ratings">Reliability Ratings</a></li>
    279. 

  279. </ul></li>
    280. 

  280. </ul></li>
    281. 

  281. <li class="exp"><a href="javascript:void(0)" onclick="Javascript:toggleMenu(this.parentNode,'m_dev_competitions')">Software Development</a>
    282. 

  282. <ul id="m_dev_competitions">
    283. 

  283. <li><a href="http://www.topcoder.com/tc?module=ViewActiveContests&amp;ph=113">Active Contests</a></li>
    284. 

  284. <li><a href="http://www.topcoder.com/tc?module=ContestStatus&amp;pt=2">Contest Status</a></li>
    285. 

  285. <li><a href="http://www.topcoder.com/tc?module=ViewReviewProjects&amp;pt=2">Review Opportunities</a></li>
    286. 

  286. <li><a href="http://software.topcoder.com/review">Submit &amp; Review</a></li>
    287. 

  287. <li><a href="http://www.topcoder.com/wiki/display/tc/TopCoder+UML+Tool">UML Tool</a></li>
    288. 

  288. <li class="exp"><a href="javascript:void(0)" onclick="Javascript:toggleMenu(this.parentNode,'m_dev_stats')">Statistics</a>
    289. 

  289. <ul id="m_dev_stats">
    290. 

  290. <li><a href="http://www.topcoder.com/tc?module=CompList&amp;ph=113">Past Contests</a></li>
    291. 

  291. <li><a href="http://www.topcoder.com/tc?module=ColorChange&amp;ph=113">Recent Color Changes</a></li>
    292. 

  292. <li><a href="http://www.topcoder.com/wiki/display/tc/Component+Development+Data+Feeds">Data Feeds</a></li>
    293. 

  293. </ul></li>
    294. 

  294. <li><a href="http://www.topcoder.com/tc?module=ReviewBoard&amp;pt=2">Meet the Review Board</a></li>
    295. 

  295. <li class="exp"><a href="javascript:void(0)" onclick="Javascript:toggleMenu(this.parentNode,'m_dev_support')">Support / FAQs</a>
    296. 

  296. <ul id="m_dev_support">
    297. 

  297. <li><a href="http://www.topcoder.com/wiki/display/tc/How+to+Compete+in+Component+Development+Competitions">How to Compete</a></li>
    298. 

  298. <li><a href="http://www.topcoder.com/wiki/display/tc/Component+Development+Documentation">Documentation</a></li>
    299. 

  299. <li><a href="http://www.topcoder.com/wiki/display/tc/Component+Development+Ratings">Ratings</a></li>
    300. 

  300. <li><a href="http://www.topcoder.com/wiki/display/tc/Component+Development+Reliability+Ratings">Reliability Ratings</a></li>
    301. 

  301. </ul></li>
    302. 

  302. </ul></li>
    303. 

  303. <li class="exp"><a href="javascript:void(0)" onclick="Javascript:toggleMenu(this.parentNode,'m_assembly_competitions')">Software Assembly</a>
    304. 

  304. <ul id="m_assembly_competitions">
    305. 

  305. <li><a href="http://www.topcoder.com/wiki/display/tc/Assembly">Overview</a></li>
    306. 

  306. <li><a href="http://www.topcoder.com/tc?module=ViewAssemblyActiveContests">Active Contests</a></li>
    307. 

  307. <li><a href="http://www.topcoder.com/tc?module=ContestStatus&amp;pt=14">Contest Status</a></li>
    308. 

  308. <li><a href="http://www.topcoder.com/tc?module=ViewReviewProjects&amp;pt=14">Review Opportunities</a></li>
    309. 

  309. <li><a href="http://software.topcoder.com/review">Submit &amp; Review</a></li>
    310. 

  310. <li><a href="http://www.topcoder.com/tc?module=ReviewBoard&amp;pt=14">Meet the Review Board</a></li>
    311. 

  311. <li class="exp"><a href="javascript:void(0)" onclick="Javascript:toggleMenu(this.parentNode,'m_assembly_stats')">Statistics</a>
    312. 

  312. <ul id="m_assembly_stats">
    313. 

  313. <li><a href="http://www.topcoder.com/tc?module=CompList&amp;ph=125">Past Contests</a></li>
    314. 

  314. </ul></li>
    315. 

  315. <li class="exp"><a href="javascript:void(0)" onclick="Javascript:toggleMenu(this.parentNode,'m_assembly_support')">Support / FAQs</a>
    316. 

  316. <ul id="m_assembly_support">
    317. 

  317. <li><a href="http://www.topcoder.com/wiki/display/tc/How+to+Compete+in+Assembly+Competitions">How to Compete</a></li>
    318. 

  318. <li><a href="http://www.topcoder.com/wiki/display/tc/Assembly+Competition+Tutorial">Tutorial</a></li>
    319. 

  319. <li><a href="http://www.topcoder.com/wiki/display/tc/Assembly+Competition+Documentation">Documentation</a></li>
    320. 

  320. </ul></li>
    321. 

  321. </ul></li>
    322. 

  322. <li class="exp"><a href="javascript:void(0)" onclick="Javascript:toggleMenu(this.parentNode,'m_testing_competitions')">Application Testing</a>
    323. 

  323. <ul id="m_testing_competitions">
    324. 

  324. <li><a href="http://www.topcoder.com/wiki/display/tc/Testing">Overview</a></li>
    325. 

  325. <li><a href="http://www.topcoder.com/tc?module=ActiveContests&amp;pt=13">Active Contests</a></li>
    326. 

  326. <li><a href="http://www.topcoder.com/tc?module=ContestStatus&amp;pt=13">Contest Status</a></li>
    327. 

  327. <li><a href="http://www.topcoder.com/tc?module=ViewReviewProjects&amp;pt=13">Review Opportunities</a></li>
    328. 

  328. <li><a href="http://www.topcoder.com/tc?module=ReviewBoard&amp;pt=13">Meet the Review Board</a></li>
    329. 

  329. <li class="exp"><a href="javascript:void(0)" onclick="Javascript:toggleMenu(this.parentNode,'m_testing_stats')">Statistics</a>
    330. 

  330. <ul id="m_testing_stats">
    331. 

  331. <li><a href="http://www.topcoder.com/tc?module=CompList&amp;ph=124">Past Contests</a></li>
    332. 

  332. </ul></li>
    333. 

  333. <li class="exp"><a href="javascript:void(0)" onclick="Javascript:toggleMenu(this.parentNode,'m_testing_support')">Support / FAQs</a>
    334. 

  334. <ul id="m_testing_support">
    335. 

  335. <li><a href="http://www.topcoder.com/wiki/display/tc/How+to+Compete+in+Testing+Competitions">How to Compete</a></li>
    336. 

  336. <li><a href="http://www.topcoder.com/wiki/display/tc/Testing+Competition+Tutorial">Tutorial</a></li>
    337. 

  337. <li><a href="http://www.topcoder.com/wiki/display/tc/Testing+Competition+Documentation">Documentation</a></li>
    338. 

  338. </ul></li>
    339. 

  339. </ul></li>
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  340. <li class="exp"><a href="javascript:void(0)" onclick="Javascript:toggleMenu(this.parentNode,'m_ui_prototype_competitions')">UI Prototype</a>
    341. 

  341. <ul id="m_ui_prototype_competitions">
    342. 

  342. <li><a href="http://www.topcoder.com/wiki/display/tc/UI+Prototype+Competitions">Overview</a></li>
    343. 

  343. <li><a href="http://www.topcoder.com/tc?module=ActiveContests&amp;pt=19">Active Contests</a></li>
    344. 

  344. <li><a href="http://www.topcoder.com/tc?module=ContestStatus&amp;pt=19">Contest Status</a></li>
    345. 

  345. <li><a href="http://www.topcoder.com/tc?module=ViewReviewProjects&amp;pt=19">Review Opportunities</a></li>
    346. 

  346. <li><a href="http://www.topcoder.com/tc?module=ReviewBoard&amp;pt=19">Meet the Review Board</a></li>
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  347. <li class="exp"><a href="javascript:void(0)" onclick="Javascript:toggleMenu(this.parentNode,'m_ui_prototype_stats')">Statistics</a>
    348. 

  348. <ul id="m_ui_prototype_stats">
    349. 

  349. <li><a href="http://www.topcoder.com/tc?module=CompList&amp;pt=19">Past Contests</a></li>
    350. 

  350. </ul></li>
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  351. <li class="exp"><a href="javascript:void(0)" onclick="Javascript:toggleMenu(this.parentNode,'m_ui_prototype_support')">Support / FAQs</a>
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  352. <ul id="m_ui_prototype_support">
    353. 

  353. <li><a href="http://www.topcoder.com/wiki/display/tc/How+to+Compete+in+IU+Prototype+Competitions">How to Compete</a></li>
    354. 

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    355. 

  355. </ul></li>
    356. 

  356. <li class="exp"><a href="javascript:void(0)" onclick="Javascript:toggleMenu(this.parentNode,'m_ria_build_competitions')">RIA Build</a>
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    359. 

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  365. <li><a href="http://www.topcoder.com/tc?module=CompList&amp;pt=24">Past Contests</a></li>
    366. 

  366. </ul></li>
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  368. <ul id="m_ria_build_support">
    369. 

  369. <li><a href="http://www.topcoder.com/wiki/display/tc/How+to+Compete+in+RIA+Builds">How to Compete</a></li>
    370. 

  370. </ul></li>
    371. 

  371. </ul></li>
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  372. <li class="exp"><a href="javascript:void(0)" onclick="Javascript:toggleMenu(this.parentNode,'m_bugraces_competitions')">Bug Races</a>
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  374. <li><a href="http://www.topcoder.com/wiki/display/tc/Bug+Races">Overview</a></li>
    375. 

  375. <li><a href="http://www.topcoder.com/wiki/display/tc/Active+Bug+Races">Active Contests</a></li>
    376. 

  376. <li><a href="http://www.topcoder.com/wiki/display/tc/Bug+Races+Reviews">Review Opportunities</a></li>
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  377. <li class="exp"><a href="javascript:void(0)" onclick="Javascript:toggleMenu(this.parentNode,'m_bugraces_support')">Support / FAQs</a>
    378. 

  378. <ul id="m_bugraces_support">
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  379. <li><a href="http://www.topcoder.com/wiki/display/tc/How+to+Compete+in+Bug+Races">How to Compete</a></li>
    380. 

  380. </ul></li>
    381. 

  381. </ul></li>
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  382. <li class="exp"><a href="javascript:void(0)" onclick="Javascript:toggleMenu(this.parentNode,'m_hs_competitions')">High School</a>
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  383. <ul id="m_hs_competitions">
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  385. <li><a href="javascript:arena();">Launch Arena</a></li>
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  386. <li class="exp"><a href="javascript:void(0)" onclick="Javascript:toggleMenu(this.parentNode,'m_hs_stats')">Statistics</a>
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  387. <ul id="m_hs_stats">
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  388. <li><a href="http://www.topcoder.com/tc?module=HSRoundOverview">Match Overview</a></li>
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  389. <li><a href="http://www.topcoder.com/tc?module=HSRoundStatsTeam">Match Results (Team)</a></li>
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  391. <li><a href="http://www.topcoder.com/tc?module=Static&amp;d1=hs&amp;d2=match_editorials&amp;d3=archive">Match Editorials</a></li>
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  393. <li><a href="http://www.topcoder.com/tc?module=Static&amp;d1=hs&amp;d2=support&amp;d3=dataFeed">Data Feeds</a></li>
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  395. <li><a href="http://www.topcoder.com/tc?module=Static&amp;d1=hs&amp;d2=sponsorship">Sponsorship</a></li>
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  398. <li><a href="http://www.topcoder.com/tc?module=Static&amp;d1=hs&amp;d2=support&amp;d3=index">How to Compete</a></li>
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  399. <li><a href="http://www.topcoder.com/tc?module=Static&amp;d1=hs&amp;d2=support&amp;d3=faqIndex">FAQs</a></li>
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  405. <li class="exp"><a href="javascript:void(0)" onclick="Javascript:toggleMenu(this.parentNode,'m_long_contests')">Marathon Matches</a>
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  407. <li><a href="http://www.topcoder.com/longcontest/?module=ViewActiveContests">Active Contests</a></li>
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  414. <li><a href="http://www.topcoder.com/longcontest/stats/?module=MatchWinners">Match Winners</a></li>
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  419. <ul id="m_long_support">
    420. 

  420. <li><a href="http://www.topcoder.com/longcontest/?module=Static&amp;d1=instructions">How to Compete</a></li>
    421. 

  421. <li><a href="http://www.topcoder.com/longcontest/?module=Static&amp;d1=support&amp;d2=ratings">Rating System</a></li>
    422. 

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    513. 

  513.             <span class="bigTitle">Prime Numbers, Factorization and Euler Function</span> <br>
    514. 

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    515. 

  515.             <div class="authorPhoto"> <img src="Prime%20Numbers,%20Factorization%20and%20Euler%20Function%20_files/medv_big.jpg" alt="Author"> </div>
    516. 

  516.             <div class="authorText"> By&nbsp;
    517. 

  517.                 <a href="http://www.topcoder.com/tc?module=MemberProfile&amp;cr=269207&amp;tab=alg" class="coderTextBlue">medv</a>
    518. 

  518.                 <br>
    519. 

  519.                 <em>TopCoder Member</em> </div>
    520. 

  520.             <br clear="all">
    521. 

  521.             <p><i>In addition to being a TopCoder member,
    522. 

  522.                 <a href="http://www.topcoder.com/tc?module=MemberProfile&amp;cr=269207&amp;tab=alg" class="coderTextBlue">medv</a>
    523. 

  523.                 is a lecturer in Kiev National University's cybernetics faculty.</i></p>
    524. 

  524.             <p>Prime
    525. 

  525. numbers and their properties were extensively studied by the ancient
    526. 

  526. Greek mathematicians. Thousands of years later, we commonly use the
    527. 

  527. different properties of integers that they discovered to solve
    528. 

  528. problems. In this article we&#8217;ll review some definitions, well-known
    529. 

  529. theorems, and number properties, and look at some problems associated
    530. 

  530. with them.</p>
    531. 

  531.             <p><strong>A prime number</strong> is a positive integer, which is divisible on 1 and itself. The other integers, greater than 1, are <strong>composite</strong>. <strong>Coprime</strong> integers are a set of integers that have no common divisor other than 1 or -1. </p>
    532. 

  532.             <p><strong> The fundamental theorem of arithmetic:</strong><br>
    533. 

  533. Any positive integer can be divided in primes in essentially only one
    534. 

  534. way. The phrase 'essentially one way' means that we do not consider the
    535. 

  535. order of the factors important. </p>
    536. 

  536.             <p>One
    537. 

  537. is neither a prime nor composite number. One is not composite because
    538. 

  538. it doesn&#8217;t have two distinct divisors. If one is prime, then number 6,
    539. 

  539. for example, has two different representations as a product of prime
    540. 

  540. numbers: 6 = 2 * 3 and 6 = 1 * 2 * 3. This would contradict the
    541. 

  541. fundamental theorem of arithmetic. </p>
    542. 

  542.             <p><strong>Euclid&#8217;s theorem:</strong><br>
    543. 

  543.                 There is no largest prime number. </p>
    544. 

  544.             <p><strong> </strong>To prove this, let's consider only <em>n</em> prime numbers: <em>p</em><sub>1</sub>, <em>p</em><sub>2</sub>, &#8230;, <em>p<sub>n</sub></em>. But no prime <em>p<sub>i</sub></em> divides the number </p>
    545. 

  545.             <p>N = <em>p</em><sub>1</sub> * <em>p</em><sub>2</sub> * &#8230; * <em>p<sub>n</sub></em> + 1, </p>
    546. 

  546.             <p>so N cannot be composite. This contradicts the fact that the set of primes is finite. </p>
    547. 

  547.             <p><strong>Exercise 1.</strong> Sequence <em>a<sub>n</sub></em> is defined recursively: </p>
    548. 

  548.             <p><img src="Prime%20Numbers,%20Factorization%20and%20Euler%20Function%20_files/prime-eq1.jpg" alt=""></p>
    549. 

  549.             <p>Prove that <em>a</em><sub><em>i</em></sub> and <em>a</em><sub><em>j</em></sub>, <em>i</em> <font face="Symbol">š</font> <em>j</em> are relatively prime. </p>
    550. 

  550.             <p><strong> </strong>Hint: Prove that <em>a</em><sub><em>n</em>+1</sub> = <em>a</em><sub>1</sub><em>a</em><sub>2</sub>&#8230;<em>a</em><sub><em>n</em></sub> + 1 and use Euclid&#8217;s theorem. </p>
    551. 

  551.             <p><strong>Exercise 2.</strong> Ferma numbers F<sub><em>n</em></sub> (<em>n</em> &#8805; 0) are positive integers of the form </p>
    552. 

  552.             <p><img src="Prime%20Numbers,%20Factorization%20and%20Euler%20Function%20_files/prime-eq2.jpg" alt=""></p>
    553. 

  553.             <p>Prove that F<em><sub>i</sub></em> and F<em><sub>j</sub></em>, <em>i</em> &#8800; <em>j</em> are relatively prime. </p>
    554. 

  554.             <p> Hint: Prove that F<sub><em>n</em> +1</sub> = F<sub>0</sub>F<sub>1</sub>F<sub>2</sub>&#8230;F<sub><em>n</em></sub> + 2 and use Euclid&#8217;s theorem.</p>
    555. 

  555.             <p><strong>Dirichlet&#8217;s theorem about arithmetic progressions:</strong><br>
    556. 

  556.                 For any two positive coprime integers <em>a</em> and <em>b</em> there are infinitely many primes of the form <em>a</em> + <em>n*b</em>, where <em>n</em> &gt; 0. </p>
    557. 

  557.             <p><strong> Trial division: <br>
    558. 

  558.                 </strong>
    559. 

  559. Trial division is the simplest of all factorization techniques. It
    560. 

  560. represents a brute-force method, in which we are trying to divide <em>n</em> by every number <em>i</em> not greater than the square root of<em> n</em>. (Why don't we need to test values larger than the square root of <em>n</em>?) The procedure <em>factor</em> prints the factorization of number <em>n</em>. The factors will be printed in a line, separated with one space. The number <em>n</em> can contain no more than one factor, greater than <em>n</em>. </p>
    561. 

  561.             <pre>     void factor(int n) 
    562. 

  562.      { 
    563. 

  563.        int i; 
    564. 

  564.        for(i=2;i&lt;=(int)sqrt(n);i++) 
    565. 

  565.        { 
    566. 

  566.          while(n % i == 0) 
    567. 

  567.          { 
    568. 

  568.            printf("%d ",i); 
    569. 

  569.            n /= i; 
    570. 

  570.          } 
    571. 

  571.        } 
    572. 

  572.        if (n &gt; 1) printf("%d",n); 
    573. 

  573.        printf("\n"); 
    574. 

  574.      } 
    575. 

  575. </pre>
    576. 

  576.             <p> Consider a problem that asks you to find the factorization of integer <em>g</em>(-2<sup>31</sup> &lt; <em>g</em> &lt;2<sup>31</sup>) in the form </p>
    577. 

  577.             <p><em>g</em> = <em>f</em><sub>1</sub> x <em>f</em><sub>2</sub> x &#8230; x <em>f</em><sub><em>n</em></sub> or <em>g</em> = -1 x <em>f</em><sub>1</sub> x <em>f</em><sub>2</sub> x &#8230; x <em>f</em><sub>n</sub></p>
    578. 

  578.             <p> where <em>f</em><sub><em>i</em></sub> is a prime greater than 1 and <em>f</em><sub><em>i</em></sub> &#8804; <em>f</em><sub><em>j</em></sub> for <em>i</em> &lt; <em>j</em>. </p>
    579. 

  579.             <p> For example, for g = -192 the answer is -192 = -1 x 2 x 2 x 2 x 2 x 2 x 2 x 3. </p>
    580. 

  580.             <p> To solve the problem, it is enough to use trial division as shown in function <em>factor</em>. </p>
    581. 

  581.             <p><strong>Sieve of Eratosthenes:</strong><br>
    582. 

  582. The most efficient way to find all small primes was proposed by the
    583. 

  583. Greek mathematician Eratosthenes. His idea was to make a list of
    584. 

  584. positive integers not greater than <em>n</em> and sequentially strike out the multiples of primes less than or equal to the square root of <em>n</em>. After this procedure only primes are left in the list. </p>
    585. 

  585.             <p>The procedure of finding prime numbers <em>gen_primes </em>will use an array primes[MAX] as a list of integers. The elements of this array will be filled so that </p>
    586. 

  586.             <p><img src="Prime%20Numbers,%20Factorization%20and%20Euler%20Function%20_files/prime-eq3.jpg" alt=""></p>
    587. 

  587.             <p>At the beginning we mark all numbers as prime. Then for each prime number <em>i</em> (<em>i</em> &#8805; 2), not greater than &#8730;MAX, we mark all numbers <em>i</em>*<em>i</em>, <em>i</em>*(<em>i</em> + 1), &#8230; as composite. </p>
    588. 

  588.             <pre>     void gen_primes() 
    589. 

  589.      { 
    590. 

  590.        int i,j; 
    591. 

  591.        for(i=0;i&lt;MAX;i++) primes[i] = 1; 
    592. 

  592.        for(i=2;i&lt;=(int)sqrt(MAX);i++) 
    593. 

  593.          if (primes[i]) 
    594. 

  594.            for(j=i;j*i&lt;MAX;j++) primes[i*j] = 0; 
    595. 

  595.      } 
    596. 

  596. </pre>
    597. 

  597.             <p> For example, if MAX = 16, then after calling <em>gen_primes</em>, the array &#8216;primes&#8217; will contain next values: </p>
    598. 

  598.             <table class="stat" style="width: 100%;" cellpadding="0" cellspacing="0">
    599. 

  599.                 <tbody><tr nowrap="nowrap">
    600. 

  600.                     <td class="valueC"><em>i</em></td>
    601. 

  601.                     <td class="valueC"> 0 </td>
    602. 

  602.                     <td class="valueC"> 1 </td>
    603. 

  603.                     <td class="valueC"> 2 </td>
    604. 

  604.                     <td class="valueC"> 3 </td>
    605. 

  605.                     <td class="valueC"> 4 </td>
    606. 

  606.                     <td class="valueC"> 5 </td>
    607. 

  607.                     <td class="valueC"> 6 </td>
    608. 

  608.                     <td class="valueC"> 7 </td>
    609. 

  609.                     <td class="valueC"> 8 </td>
    610. 

  610.                     <td class="valueC"> 9 </td>
    611. 

  611.                     <td class="valueC"> 10 </td>
    612. 

  612.                     <td class="valueC"> 11 </td>
    613. 

  613.                     <td class="valueC"> 12 </td>
    614. 

  614.                     <td class="valueC"> 13 </td>
    615. 

  615.                     <td class="valueC"> 14 </td>
    616. 

  616.                     <td class="valueC"> 15 </td>
    617. 

  617.                 </tr>
    618. 

  618.                 <tr>
    619. 

  619.                     <td class="valueC"> primes[<em>i</em>] </td>
    620. 

  620.                     <td class="valueC"> 1 </td>
    621. 

  621.                     <td class="valueC"> 1 </td>
    622. 

  622.                     <td class="valueC"> 1 </td>
    623. 

  623.                     <td class="valueC"> 1 </td>
    624. 

  624.                     <td class="valueC"> 0 </td>
    625. 

  625.                     <td class="valueC"> 1 </td>
    626. 

  626.                     <td class="valueC"> 0 </td>
    627. 

  627.                     <td class="valueC"> 1 </td>
    628. 

  628.                     <td class="valueC"> 0 </td>
    629. 

  629.                     <td class="valueC"> 0 </td>
    630. 

  630.                     <td class="valueC"> 0 </td>
    631. 

  631.                     <td class="valueC"> 1 </td>
    632. 

  632.                     <td class="valueC"> 0 </td>
    633. 

  633.                     <td class="valueC"> 1 </td>
    634. 

  634.                     <td class="valueC"> 0 </td>
    635. 

  635.                     <td class="valueC"> 0 </td>
    636. 

  636.                 </tr>
    637. 

  637.             </tbody></table>
    638. 

  638.             <p><strong>Goldbach's Conjecture:</strong><br>
    639. 

  639.                 For any integer <em>n</em> (<em>n</em> &#8805; 4) there exist two prime numbers <em>p</em><sub>1</sub> and <em>p</em><sub>2</sub> such that <em>p</em><sub>1</sub> + <em>p</em><sub>2</sub> = <em>n</em>. In a problem we might need to find the number of essentially different pairs (<em>p</em><sub>1</sub>, <em>p</em><sub>2</sub>), satisfying the condition in the conjecture for a given even number<em> n</em> (4 &#8804; <em>n</em> &#8804; 2 15). (The word &#8216;essentially&#8217; means that for each pair (<em>p</em><sub>1</sub>, <em>p</em><sub>2</sub>) we have <em>p</em><sub>1 </sub>&#8804;<em> </em><em>p</em><sub>2</sub>.) </p>
    640. 

  640.             <p>For example, for <em>n</em> = 10 we have two such pairs: 10 = 5 + 5 and 10 = 3 + 7. </p>
    641. 

  641.             <p><strong><em> </em></strong>To solve this,<strong></strong>as <em>n</em> &#8804; 2<sup>15</sup> = 32768, we&#8217;ll fill an array primes[32768] using function <em>gen_primes</em>. We are interested in primes, not greater than 32768. </p>
    642. 

  642.             <p>The function <em>FindSol</em>(<em>n</em>) finds the number of different pairs (<em>p</em><sub>1</sub>, <em>p</em><sub>2</sub>), for which <em>n</em> = <em>p</em><sub>1</sub> + <em>p</em><sub>2</sub>. As <em>p</em><sub>1</sub> &#8804; <em>p</em><sub>2</sub>, we have <em>p</em><sub>1</sub> &#8804; <em>n</em>/2. So to solve the problem we need to find the number of pairs (<em>i</em>, <em>n</em> &#8211; <em>i</em>), such that <em>i</em> and <em>n</em> &#8211; <em>i</em> are prime numbers and 2 &#8804; <em>i</em> &#8804; <em>n</em>/2. </p>
    643. 

  643.             <pre>     int FindSol(int n) 
    644. 

  644.      { 
    645. 

  645.        int i,res=0; 
    646. 

  646.        for(i=2;i&lt;=n/2;i++) 
    647. 

  647.          if (primes[i] &amp;&amp; primes[n-i]) res++; 
    648. 

  648.        return res; 
    649. 

  649.      } 
    650. 

  650. </pre>
    651. 

  651.             <p><strong> Euler&#8217;s totient function<br>
    652. 

  652.                 </strong> The number of positive integers, not greater than <em>n</em>, and relatively prime with <em>n</em>, equals to Euler&#8217;s totient function &#966; (<em>n</em>). In symbols we can state that </p>
    653. 

  653.             <p> &#966; (<em>n</em>) ={<em>a</em> <font face="Symbol">Î</font> N: 1 &#8804; <em>a</em> &#8804; <em>n</em>, gcd(<em>a</em>, <em>n</em>) = 1} </p>
    654. 

  654.             <p>This function has the following properties: </p>
    655. 

  655.             <ol>
    656. 

  656.                 <li> If <em>p</em> is prime, then &#966; (<em>p</em>) = <em>p</em> &#8211; 1 and &#966; (<i>p<sup>a</sup></i>) = <em>p</em><em> a</em> * (1 &#8211; 1/<em>p</em>) for any <em>a</em>. </li>
    657. 

  657.                 <li> If <em>m</em> and <em>n</em> are coprime, then &#966; (<em>m</em> * <em>n</em>) = &#966; (<em>m</em>) * &#966; (<em>n</em>). </li>
    658. 

  658.                 <li> If <em>n</em> = <img src="Prime%20Numbers,%20Factorization%20and%20Euler%20Function%20_files/prime-eq4.jpg" alt="">, then Euler function can be found using formula: </li>
    659. 

  659.             </ol>
    660. 

  660.             <p>&#966; (<em>n</em>) = <em>n</em> * (1 &#8211; 1/<em>p</em> 1) * (1 &#8211; 1/<em>p</em> 2) * ... * (1 &#8211; 1/<em>p k</em>) </p>
    661. 

  661.             <p>The function fi(<em>n</em>) finds the value of &#966;(<em>n</em>): </p>
    662. 

  662.             <pre>     int fi(int n) 
    663. 

  663.      { 
    664. 

  664.        int result = n; 
    665. 

  665.        for(int i=2;i*i &lt;= n;i++) 
    666. 

  666.        { 
    667. 

  667.          if (n % i == 0) result -= result / i; 
    668. 

  668.          while (n % i == 0) n /= i; 
    669. 

  669.        } 
    670. 

  670.        if (n &gt; 1) result -= result / n; 
    671. 

  671.        return result; 
    672. 

  672.      } 
    673. 

  673. </pre>
    674. 

  674.             <p>For example, to find &#966;(616) we need to factorize the argument: 616 = 2<sup>3</sup> * 7 * 11. Then, using the formula, we&#8217;ll get: </p>
    675. 

  675.             <p>&#966;(616) = 616 * (1 &#8211; 1/2) * (1 &#8211; 1/7) * (1 &#8211; 1/11) = 616 * 1/2 * 6/7 * 10/11 = 240. </p>
    676. 

  676.             <p>Say you've got a problem that, for a given integer <em>n</em> (0 &lt; <em>n</em> &#8804; 10<sup>9</sup>), asks you to find the number of positive integers less than <em>n</em> and relatively prime to <em>n</em>. For example, for <em>n</em> = 12 we have 4 such numbers: 1, 5, 7 and 11. </p>
    677. 

  677.             <p>The solution: The number of positive integers less than <em>n</em> and relatively prime to <em>n</em> equals to &#966;(<em>n</em>). In this problem, then, we need do nothing more than to evaluate Euler&#8217;s totient function. </p>
    678. 

  678.             <p>Or consider a scenario where you are asked to calculate a function Answer(<em>x</em>, <em>y</em>), with <em>x</em> and <em>y</em> both integers in the range [1, <em>n</em>], 1 &#8804;<em> n</em> &#8804; 50000. If you know Answer(<em>x</em>, <em>y</em>), then you can easily derive Answer(<em>k</em>*<em>x</em>, <em>k</em>*<em>y</em>) for any integer <em>k</em>. In this situation you want to know how many values of Answer(<em>x</em>, <em>y</em>) you need to precalculate. The function Answer is not symmetric.<strong></strong></p>
    679. 

  679.             <p>For example, if <em>n</em>
    680. 

  680. = 4, you need to precalculate 11 values: Answer(1, 1), Answer(1, 2),
    681. 

  681. Answer(2, 1), Answer(1, 3), Answer(2, 3), Answer(3, 2), Answer(3, 1),
    682. 

  682. Answer(1, 4), Answer(3, 4), Answer(4, 3) and Answer(4, 1). </p>
    683. 

  683.             <p>The solution here is to let res(<em>i</em>) be the minimum number of Answer(<em>x</em>, <em>y</em>) to precalculate, where <em>x</em>, <em>y</em><font face="Symbol"> Î</font>{1, &#8230;, <em>i</em>}. It is obvious that res(1) = 1, because if <em>n</em> = 1, it is enough to know Answer(1, 1). Let we know res(<em>i</em>). So for <em>n</em> = <em>i</em> + 1 we need to find Answer(1, <em>i</em> + 1), Answer(2, <em>i</em> + 1), &#8230; , Answer(<em>i</em> + 1, <em>i</em> + 1), Answer(<em>i</em> + 1, 1), Answer(<em>i</em> + 1, 2), &#8230; , Answer(<em>i</em> + 1, <em>i</em>). </p>
    684. 

  684.             <p>The values Answer(<em>j</em>, <em>i</em> + 1) and Answer(<em>i</em> + 1, <em>j</em>), <em>j</em> <font face="Symbol">Î</font>{1, &#8230;, <em>i</em> + 1}, can be found from known values if GCD(<em>j</em>, <em>i</em> + 1) &gt; 1, i.e. if the numbers <em>j</em> and <em>i</em> + 1 are not common primes. So we must know all the values Answer(<em>j</em>, <em>i</em> + 1) and Answer(<em>i</em> + 1, <em>j</em>) for which <em>j</em> and <em>i</em> + 1 are coprime. The number of such values equals to 2 * &#966; (<em>i</em> + 1), where &#966; is an Euler&#8217;s totient function. So we have a recursion to solve a problem: </p>
    685. 

  685.             <blockquote>res(1) = 1,<br>
    686. 

  686.                 res(<em>i</em> + 1) = res(<em>i</em>) + 2 * j (<em>i</em> + 1), <em>i</em> &gt; 1 </blockquote>
    687. 

  687.             <p><strong>Euler&#8217;s totient theorem:</strong><br>
    688. 

  688.                 If <em>n</em> is a positive integer and <em>a</em> is coprime to <em>n</em>, then <em>a</em> <sup>&#966; (<em>n</em>)</sup> <font face="Symbol">?</font> 1 (mod <em>n</em>). </p>
    689. 

  689.             <p><strong> Fermat&#8217;s little theorem:</strong><br>
    690. 

  690.                 If <em>p</em> is a prime number, then for any integer <em>a</em> that is coprime to <em>n</em>, we have </p>
    691. 

  691.             <p><em> a <sup>p</sup></em> &#8801; <em>a</em> (mod <em>p</em>) </p>
    692. 

  692.             <p> This theorem can also be stated as: If <em>p</em> is a prime number and <em>a</em> is coprime to <em>p</em>, then </p>
    693. 

  693.             <p><em>a</em> <sup><i>p</i> -1</sup> &#8801; 1 (mod <em>p</em>) </p>
    694. 

  694.             <p>Fermat&#8217;s little theorem is a special case of Euler&#8217;s totient theorem when <em>n</em> is prime. </p>
    695. 

  695.             <p><strong>The number of divisors:<br>
    696. 

  696.                 </strong> If <em>n</em> = <img src="Prime%20Numbers,%20Factorization%20and%20Euler%20Function%20_files/prime-eq4.jpg" alt="">, then the number of its positive divisors equals to </p>
    697. 

  697.             <p>(<em>a</em><sub>1</sub> + 1) * (<em>a</em><sub>2</sub> + 1) * &#8230; * (<em>a</em><sub><em>k</em></sub> + 1) </p>
    698. 

  698.             <p> For a proof, let A<em> i</em> be the set of divisors <img src="Prime%20Numbers,%20Factorization%20and%20Euler%20Function%20_files/prime-eq5.jpg" alt="">. Any divisor of number <em>n</em> can be represented as a product <em>x</em><sub>1</sub> * <em>x</em><sub>2</sub> * &#8230; * <em>x</em> <sub><em>k</em></sub> , where <em>x</em><sub><em>i</em></sub> <font face="Symbol">Î</font> A<sub><em>i</em></sub>. As |A<sub><em>i</em></sub>| = <em>a</em><sub><em>i</em></sub> + 1, we have </p>
    699. 

  699.             <p>(<em>a</em><sub>1</sub> + 1) * (<em>a</em><sub>2</sub> + 1) * &#8230; * (<em>a</em><sub><em>k</em></sub> + 1) </p>
    700. 

  700.             <p>possibilities to get different products <em>x</em><sub>1</sub> * <em>x</em><sub>2</sub> * &#8230; * <em>x</em><sub><em>k</em></sub>.</p>
    701. 

  701.             <p>For
    702. 

  702. example, to find the number of divisors for 36, we need to factorize it
    703. 

  703. first: 36 = 2? * 3?. Using the formula above, we&#8217;ll get the divisors
    704. 

  704. amount for 36. It equals to (2 + 1) * (2 + 1) = 3 * 3 = 9. There are 9
    705. 

  705. divisors for 36: 1, 2, 3, 4, 6, 9, 12, 18 and 36. </p>
    706. 

  706.             <p>Here's another problem to think about: For a given positive integer <em>n</em> (0 &lt; <em>n</em> &lt; 2<sup>31</sup>) we need to find the number of such <em>m</em> that 1 &#8804; <em>m</em> &#8804; <em>n</em>, GCD(<em>m</em>, <em>n</em>) &#8800; 1 and GCD(<em>m</em>, <em>n</em>) &#8800;<em> m</em>. For example, for <em>n</em> = 6 we have only one such number <em>m</em> = 4. </p>
    707. 

  707.             <p>The solution is to subtract from <em>n</em> the amount of numbers, coprime with it (its amount equals to &#966;(<em>n</em>)) and the amount of its divisors. But the number 1 simultaneously is coprime with <em>n</em> and is a divisor of <em>n</em>. So to obtain the difference we must add 1. If <em>n</em> = <img src="Prime%20Numbers,%20Factorization%20and%20Euler%20Function%20_files/prime-eq4.jpg" alt=""> is a factorization of <em>n</em>, the number <em>n</em> has (<em>a</em><sub>1</sub> + 1) * (<em>a</em><sub>2</sub> + 1) * &#8230; * (<em>a</em><sub><em>k</em></sub> + 1) divisors. So the answer to the problem for a given <em>n</em> equals to </p>
    708. 

  708.             <p><em>n</em> &#8211; &#966;(<em>n</em>) &#8211; (<em>a</em><sub>1</sub> + 1) * (<em>a</em><sub>2</sub> + 1) * &#8230; * (<em>a</em><sub><em>k</em></sub> + 1) + 1 </p>
    709. 

  709.             <p><strong> Practice Room:</strong><br>
    710. 

  710.                 Want to put some of these theories into practice? Try out these problems, from the <a href="http://www.topcoder.com/tc?module=ProblemArchive">TopCoder Archive</a>:</p>
    711. 

  711.             <ul>
    712. 

  712.                 <li><a href="http://www.topcoder.com/stat?c=problem_statement&amp;pm=2986&amp;rd=5862"> Refactoring</a> (SRM 216) </li>
    713. 

  713.                 <li><a href="http://www.topcoder.com/stat?c=problem_statement&amp;pm=3458&amp;rd=5869"> PrimeAnagrams</a> (SRM 223) </li>
    714. 

  714.                 <li><a href="http://www.topcoder.com/stat?c=problem_statement&amp;pm=4481&amp;rd=6538"> DivisibilityCriteria</a> (SRM 239) </li>
    715. 

  715.                 <li><a href="http://www.topcoder.com/stat?c=problem_statement&amp;pm=4475&amp;rd=8012"> PrimePolynom</a> (SRM 259) </li>
    716. 

  716.                 <li><a href="http://www.topcoder.com/stat?c=problem_statement&amp;pm=6186&amp;rd=9823"> DivisorInc</a> (SRM 302) </li>
    717. 

  717.                 <li><a href="http://www.topcoder.com/stat?c=problem_statement&amp;pm=6063&amp;rd=9824"> PrimePalindromic</a> (SRM 303) </li>
    718. 

  718.                 <li><a href="http://www.topcoder.com/stat?c=problem_statement&amp;pm=6195&amp;rd=9825"> RugSizes</a> (SRM 304) </li>
    719. 

  719.                 <li><a href="http://www.topcoder.com/stat?c=problem_statement&amp;pm=6408&amp;rd=9826"> PowerCollector</a> (SRM 305) </li>
    720. 

  720.                 <li><a href="http://www.topcoder.com/stat?c=problem_statement&amp;pm=6083&amp;rd=9987"> PreprimeNumbers</a> (SRM 307) </li>
    721. 

  721.                 <li><a href="http://www.topcoder.com/stat?c=problem_statement&amp;pm=2017"> EngineersPrimes</a> (SRM 181) </li>
    722. 

  722.                 <li><a href="http://www.topcoder.com/stat?c=problem_statement&amp;pm=2342"> SquareFree</a> (SRM 190) </li>
    723. 

  723.             </ul>
    724. 

  724.             <p>&nbsp; </p>
    725. 

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