How to calculate 7000! using C++?

1. You need fixed or arbitrary bit width unsigned integers that match result magnitude

   So use some lib for that or implement it yourself (as Array of unsigned integers) this might help:

   • 32bit ALU in C++

   The number of needed bits can be obtained like this:

   int i,n=7000;
   double bits=0;
   for (i=1;i<=n;i++) bits+=log(double(i));
   bits/=log(2.0);
   

   because bit width of result of multiplication is sum of bits of all operands ... The result is:

   expected bits = 79320.8214956583
   

   so use any bigger integer bitwidth than that but beware the more bits you have the slower the computations ...

2. You need to use any fast factorial algorithm

   there are quite a few of them here is mine:

   • Fast Factorial

   otherwise you will computing a long time. If you want to use that do not forget to add all primes up to 7000 to that list or compute them with SoE on the run...

   Other option is to go parallel instead (like with CUDA / GPU) however I am afraid all the fast algorithms will be not paralleli-sable so it will have a benefit only if you have a lot of cores and small n! ...

3. Use fast multiplication

   At least Karatsuba but after you hit the threshold NTT based multiplication is much better for this see (both included there):

   • fast squaring

Putting all together results in this:

time needed: 13341.269 ms
fast_fact(7000) = 88420079569631122478649936968977265151466680638881486140638596171751757547251586495479820703667355925646357791006930644541781709611657521677655318171392137521037946515393843904926361688385348464372984494067447564765087060434482426964215991616495827301379197441739600587455031564842367506811005517790499148488856501426373800630054582580290281811865299798263452179048754687859043163322651123879386396722542235335731933272109449655390256109450343956808991039074772089961015930458651144441967671322357427545823810035264776084511907967315558033621426022857569799375452439094572358740421101878837852733378820485807526409439659040451328513049202261737966791539926664803872369591910016681756479826905288119693875612911515144847168973798635604304887620244692080165205768441538009465474435479884443236739299799764256817785480003184654047333113984566926558839880181532151414838573397388906185632507373858556099854818588444318494306400449873143602066028521825633964734313419493264069907880420141465289868296116800209277705516450639372363938755237638829806981558348778855704648348872144035276138066038740350358423788115675439784361824160812554131712549556999831559192538926065458220945143696678060888546829658648469290150905591710120664206638090792797929725322713987692310172904009823614843086508133219628064583410444222813786108794692012349827695253229423001030635920073856624703998698738893220275881757674368457531113263513313757718248926027136701584305504100577265790463299701028437223456394071685839755677520072055061311107611108613562942926748086005724269818820157889944014612606591216715686221528680902714158163385630948825792114167638569737755926299087946284570525177104171202674816962170489021181151256058824886338859704607653488198721155845074685872412356943644743515923962333058332936320832486286447614054301115587586240359585820966708210622682015369402518757224319108662175305092495220903846299090590197278085499786492117968340374991552773451403839587332365791302640581690246440760781877354608881790897368981774060964487844876160973339308021970317902511623614092832842406010002951207689951929205801201522002591992466770594796212825113133168431817350566911839202731176113171128019800619512943856751812923314186813452272959642189120323042071234572533811420447527647916077476142219372749861208556001845176683128224904350760816912958635522152935135723015038522632948588921272947712948227294117486966751944368579824707772556942430514779012448055008300531370140081644001823441435963485842275961520522115744625195155354032707142386524162085096580614350027062987704946197678149788213665690021962882083789086932579562800924724111899621142782219360239896017887769389975570803046342604164347499894943552950588786356637690585016002603173670476651595949875031338238505604922168587489192629531699983144445903792095015353946926906584328714378619426336456386371380913818487175157505054444234883451134810848869744458981788901959758161262469156184985280793208485584214018214698566204262173432004328310407926043521643564847153106621089474716966925579091848238974982765691135699497550346293520925822794665308401520192623467491072734062127487080935689685860455538270919009008618872638420378849454200143715772459171295703780212328300464879038740447533108552198282456280851138733212754552724704335883837353948466363836046004738500692158378692939059825642958638468127791869004682751693677503718955235202266002906130034589934391420121746280446246112903120132589172839270216504383692083102664117960535026247758910720630186570466248691328933204462185689726579109982687767743957379261982848575114185981120206556950964114728913895002748004224298472670505073943861219449622974128867960950622278068794317059390090758632640573819936639531141015573609149588700162790296881966567738378606327945562001266372983791096024047988184425758479734925063318821456031075821064136954375398808361930148002533798227380147949156998211302496281492814366385655519086971263445070192798237659124916976214973021724178809342149875287157434506597937788046004974005848932938446637783298800876253835739160358731573023397382342570819117947098606779470131222146827493334032019768896936780845931550601875023884185943594148859734803313355346061752081979850194660223467555167504583687398231489219584047112663056355244576712821782457242289981004192284335212297165156135440388737824476337613306371569236799608696085739369276596825077188257703339470341263719225169441522523620838758686009243867645977183181820904801022907131196755455516434685332487553199904517018200585193267550326376564069145559708483868399568735472867058391890398141259646274828309293406513093145516602582987637917148115985621279074222914521958410449977967699835025663284386618246134286350774624587116684587090256713032839347150614242580992185426541123695090997723662698793691763802840823248201829552556886698457790430723383395369980036608189647190378851711772389298003536496624207131302119162091424892854534511056019801008728611137995500221039842737609992410529057749461647454875082851265381802615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result bits = 79321

The computation took ~13.3 sec with my unsophisticated 32 bit code arithmetics on AMD A8-5500 3.2 GHz (using fixed bitwidth 2500*32 bits = 80000 bits )

No numerical fundamental type of C++ can represent such large values.

You can represent arbitrarily large numbers using arrays (constrained by available memory). But in order to perform calculations using arrays that represent numbers, you need to define the operations first. This is called "arbitrary precision arithmetic", and the C++ standard library doesn't provide an implementation of those operations.