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The Answer

pursuant to: https://brilliant.org/discussions/thread/how-many-numbers-do-a-given-set-of-primes-can/

I will soon publish the complete equations used to generate how many numbers do a given set of primes generate as a fraction of all natural numbers. But if you decided to take me up on my challenge to try and get the answer for the list containing the first 500 primes I'll post the answer here so that you can check if your answer is the correct one.

If you did manage, congrats, you just did what using brute force only would have taken a supercomputer running at 1 exaflop more than 1000 times the age of the universe to solve!

The 500th prime is 3571

The answer is:

51939918537490815223042767154853154018465570745270456640254362658509603993648445568775630780796265013021776571954845664592945522923026070618511786383419419170277432797488676590029326621437831319788162263658133339378876549060991549922263666468834035926317097559895439943336369084078411195565406861569501158987612106971111698017324914904042330810179460976608237885519747645907573767454892737867605955835420474127881652740047271367680962891271160323851299947664465509396763728161777894213844593830898165225104542305490224609002293834828950049571684374946251898675657442493651263836992280340025700826822939659719402026557253711693650427400139414314006363128300227172062226918388528339747687304257395875290701446279430867441784980374931600583557296767480402614853169012672549828929732151858513723867875433719550807207456587272616600884323600309864137144008509975210815047858394156861796493253923351239329142819179309203285663535350860437064222106745935888271767495170275518594729957930590917207922800284940661956600397373529810586710580144886708314585223655838257407234317637054699545381079782918576863751821772187647948034053960847588432186770133476198220265937267747153388234433498667785997890662341090043348653113760819261045319013587639552726131668321618905645509696533241509663994406858097184505140180637298868545904358732710775871408536172715537728855931880353691201445216560710894365610004262956414917197159252312387646100359038105613845904908753479298728844314517378439149416982869986427478111210565465393664365910490 / 55758984689722289456584343973980218794222360410370486600876521412561125169842671687886659915627578314485846550630950819222944524683009022878904770556027761956176488812292947382221931874253777633437134510302575839253650054921377787870682931197177767075132066757080223218612334159440129580349733289519851712159301522543375931349514587435859733735492107313994296156667043427503419582986673982781662867774668590894039881444170127494445039158987921893949715756676877065894693174740972110652141354362897063044455133797605358546568865726130200450453398931223850324605731252778030970280877206693651731691258544219166428510983029690134652373911775732908004812834312649502260848024341710451795869414983978894963623609913555075593913506181212315124360418666956947337288293197871647777293433519967777021652707770405176662004780202185197896420100824550604276415913952059512150237556052370573695632895271000161855803676535981069482744636004186492414111780359468308325187790134844420749002485723228045872532060620305108426893986989664679237034516531411338218169547038879067681156735284547880657542717850600363029661184202208190403252715716314896654798165882449121636125113013429667194028678491053161776662222026477230231753388170200194252994989570844676016623850478712701514744167948059592159916557939219142408953431605497685148890617736571690321615596678521870072250349196777695604645216560710894365610004262956414917197159252312387646100359038105613845904908753479298728844314517378439149416982869986427478111210565465393664365910490

Which could be reduced to

99230356947816393953078513846327732063037105760593199189351660888415207048418285217520257817696054196832259404064371786753141834296029430888193003527176512322141127580610076662667011686802009804003823737247834826060064504357362836474100996843901358583882526923639467159775678419802345301358349536229634120335114332154409321182554019579535582598316408195286568258916294134150439164786794657392625807236999668842358279102057282127647173970163285336242551299017403246103913875661990665331278460051304605301864382524480893904324439534682653686614589955488229103704469393079847217620133551906115496668885356225151951210932982301626150050617026759197220513248966002089222504358538800536392598639552589487486628147267548579085198741412043403521679747112670395588486298310971946993689705344296300013039577006971872485689523972439275699812837754826560407043282742817820672348919762663670553636690605633505804221924563141991394781133368279010107873917974020274390931706315057720521581894237199000138291402478721255009629852969865800578010370639041617498315023535690291398337873459713116491820897322449861157243559214954832064299103075448651445207789588739294521693210733660219146124502534784092842393280324661452900631532605935503779013 / 106526619786960241461317424761626240575511575099502916718485458224410201683582310664763514326234090874458760137896680051861862887330250919913178749494783821914445566350268447574664933603088583947234973475044257464157873647475313154684305011067729372140617885849024504496653674189366412349429321907058939811528432142287781203543981376443914670767926094891112411361612035315889813578606534389587647725337984720881797874812555560075375221692669037489474642634510693436618045019648107498444165863978173229405718531570869435291191728363793993587041905069327804043272681473740696081678733592897047489537078224139509122905691408369560059053767355204537575956541150337616060119006157205584305067619950508716428047331677360435423562395444408659037231638746947280428871345744684126858966209906417921990703569819029187629924280246597471846203493783083894768505802639326262747802129373308199402928134413623482513277297980196483375623482196992417259194652716225894691936546769405257238979401424526147842816384317791240119501636685619506534000003004478552195258315716080322862534975269540956491820897322449861157243559214954832064299103075448651445207789588739294521693210733660219146124502534784092842393280324661452900631532605935503779013

Note by Daniel Magen
3 months, 1 week ago

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