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Primes Ali Salman Soutcho Toure Prof. Geoffrey Exoo Prof. Jeff Kinne.

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Presentation on theme: "Primes Ali Salman Soutcho Toure Prof. Geoffrey Exoo Prof. Jeff Kinne."— Presentation transcript:

1 Primes Ali Salman Soutcho Toure Prof. Geoffrey Exoo Prof. Jeff Kinne

2 15 computers, 4 cores each. Each core can do around 1billion operations/s. 60 cores running 24/7. SETI@home run 225,534 cores. Computer Power

3 Goals ● Goal 1 ● Find large primes composed of 2digits. ● Like « 1616161661666111 » or « 15515151515511 » ● Goal 2 ● Find large primes. ● Goal 3 ● Find large Sophie Germain primes.

4 Primes Example : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 … How can we check if a number is prime ? Trial Division Lucas Theorem Is 8,876,044,532,898,802,067 (19 digit) a prime ?

5 Trial Division Checks if n can be divided by any number that’s greater than 1 and less than n. Example, 47 47 can only be divided by {1, 47} Example, 49 49 Can be divided by {1, 7, 49} 8,876,044,532,898,802,067 { 1500450271, 5915587277 }

6 Checks if n is prime by taking the factors of n-1 and doing a sequence of tests on them. Lucas theorem Example N = 47 N-1 = 46 Factors of 46 {2, 23}

7 Results 1. Finding primes that are made out of 1’s and 5’s. n = 5,551,151,151,151,111,151 (19 Digits).

8 n = 511151111511111515115115555151155155511515 151515155151151511111151155151115151111151 5151115555151111 ( 100 digit ) factors for (n-1): 1: 2 2: 5 3: 29 4: 755861 5: 42189106789 6: 4363477260190361 7:1266707453760963996064804203139157071722 98481447007567082016637811 I was able to get 30-40-70 digit primes.

9 Lucas Theorem Example N = 47 N-1 = 46 Factors of 46 {2, 23} 2. Finding the biggest prime What if we took random primes and multiplied them together ? (2*5)+1 = 11 (prime) (2*7)+1 = 15 (not a prime)

10 797926194455194518549810721230049327406848190328391199958202457357511916456694155899293937842761528374298 985584493441749216763236184074182970267655984633446863679852193105291980133799870978525763176703829941106 019589517450330987016308323171647617229977275444203922465476347466932183267004353230283667361009967820895 115687128404997236160821466089223455323527777542578283678833421147952445440635174416371082884804638989836 158362712251307688193169545651017643090427720358332152485383120574210422424303183831988852810468841529727 363136566743991072346307185536159631244368776505221220214082571654023991597195198937849706403245226594877 091354053317715699480049878169458635328152639268612003335671405606368575685556554731781561926853568867469 495790387704715421800327679006160376484641878402388731084119469046614140187588241325832087983068294414877 624615589727796243037651843379589186884630078269493891576764571038895991227218584787737245162073597389075 038134301416905749195313602399696947545226806613206176800904131870416173857114115432683931843878904570340 545931243365274557581237073808858482524612696897083251161177307804558879169757417573344174605908322717857 099365674673744500805451710251989452832889902282941312670802509021928730168440678361035100700538243067738 500458225434489696866970796348168724036005602240006861577510737669324566576996537890781440681507503853433 029348382518425276818262408128998638636445292613141990546308117761283541329195762316523799668500171541177 948938637903928351282007248089111380335502792357346715146734634592086300584736895228698088952096390241754 808223976106654394611381911987421352844112857304653522270455963659677890905408038742570059132511068736231 742778348673574570482444548373137383995676338641837720089270775429942129161975171411953135637317015564050 311389926943669665757498193660117817107080752403598474591366860361573342009813190794699699348726979666141 549464620759764204332432965800208230392039216250365141189647148542790125220969548820212630525276099624526 651605963430035287042839436342845086898735850476375081551042555589167713847096536332548362456509882496715 101869711314865186129855527282024964286312538338495149009864908220268830487891278228844064820313703048225 638865281369759860051546527323006552114094880006275271848377552736814145383490917687490236082681691058250 483503449645831549279679330173665149597636070770691420028230597896800341538953442535556425280043729027892 635552390222092828656924037272009747195841939439615605608118553339396068909844471698043861560164857861054 965642420952857479894844400943022291031109757435892800029353509532716913589088968665082952471571068652237 107995797814970908758343216950485910572614538990406492670659023653027396094222870486145612791587457782263 216647960098044021763265177984670796741889262319111766332564565669112400990758437044387184394604425698717 900599030016745738719866550916500925976548214467962018902014222451329049697749175418904785727213789430339 364332165373945754148150430270721400247262131150141804231847684606128764937336682378560479017738419627665 832459153152821135520412977866100046638361139875991534621061398444626993034733203308669161405365444838978 525403437970382768981912953381158832265646762458185419978890745472098843299446173696156472567442679628687 194633043463355840388159235581719923826545407203145511514908310756735565784563572950356457376620082231657 096080685215791514348153241240316890115585319858165879472565309312354046397300827146941944682243776947765 080619343076478304436188186813983159154725851257924698664768282776511095237414440547089091768643741535586 931442705299716430453067796470031294292054281186741829197746212635130849995356590209367070216558741103403 857817086153826043634889670684377143549732427461538770945417561841445245513521006624576174139269981594526 997511350901986353571612352331515125511722799279928287567739323296792376315177233693990020279946656027372 072563266244050623437631603731495204330891139106642149665399319263711304745988700771816327374326285572824 199818397373163614996329666635945448903443559038115504369707112350868293713362246390647036446173938798690 744982994225551887684982309367243127776151838030773214596466652192541844822516610963004879449347392072547 70686859087566632772654825123495254909556106520478218015146899 ( around 4,200 digits ).

11 Currently working on Sophie Germain prime p and 2p+1 are both prime. p = 2981625476911166341900898692545033822173452735465 0172738418743608467276189780160915299673113327070 48954384883904760278503 (121 digits) 2p+1 = 5963250953822332683801797385090067644346905470930 0345476837487216934552379560321830599346226654140 97908769767809520557006 p = 5 2p+1 = 11 5 and 11 are primes. Therefore, 5 is a Sophie Germain prime.

12 Future Goal Reaching the lists of the largest prime Or Sophie Germain prime. Records 5000th largest known prime (as of now) is 287,407 digits. 20th largest Sophie Germain prime (as of now) is 29,628 digits.

13 The End Students : Ali Salman Soutcho Toure Po-Ching Liu Troy Schotter Karthik Tottempudi Professors : Jeff Kinne Geoffrey Exoo


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