Matthew's Numbers

Thursday, May 14, 2020
By: Matthew Doucette

These are a set of large numbers I conceived and named after myself as a child. Do not pay too much attention to the nomenclature. I was only young.

 

Definitions:

The relevancy is the appreciation these are finite numbers. No matter what computer graphics mode your devices are using, there are only a finite number of possible screens they can display. These numbers highlight how large "large numbers" can be and how finite numbers can be so large as to effectively be infinite in practical use.

 

History:

When I was a child, I played and loved Sierra Quest games, including King's Quest, Space Quest, and Police Quest series. Among many graphics modes used by Sierra, one of the modes used was the TGA (Tandy Graphics Adapter) 160x200 resolution with 16 colors mode, on Tandy systems. See the example below from Space Quest I.


Space Quest I - Full Playthrough

Playing these games lead me to the idea there were "only" a finite amount of possible screens in TGA. I think it helped that there were only 16 colors and the graphics were pixel art. Each new location in the Sierra games was one of these possible screens. Every game ever made for TGA uses one of these screens, including games that are not yet made.

I was curious just how many screens were possible. It was something I was unable to calculate until much later in life.

 

Formula:

N(w,h,c) = c^(w*h)

Where:
N() = number of possible screens
w = width (in pixels)
h = height (in pixels)
c = number of colors

Note: This formula assumes a bitmap mode with individual pixel access, not a character based mode.

Let's calculate the number of possible screens in Space Quest I on the Tandy 1000 SX, Matthew's Number 160x200x16:

N(160,200,16)

16^(160*200)

16^32000

16*16*16*...*16 performed 32,000 times!

A 38,532 digit long number.

 

See the full number below.

 

Nomenclature:

The name of each Matthew's Number can be constructed by the resolution and colors, as follows:

Here are some examples used in this webpage, including a TGA  and modern HD modes:

Why are they called "Matthew's Numbers"? I was a child, following other popular numbers such as Graham's number. I still do not know if it is proper for me to name a set of numbers my own name.

 

Matthew's Number 160x200x16:

The original Matthew's Number, which I coined as Mathew's Number -- singular -- and it is the amount of possible screens in the 160x200 16 color TGA mode:

690947404219008741222457800129817403411828583413419520419493111831295996413873594644805829445269007203995211615456663016683728495892887369173668195222901927225437605087145618916756687286387933281790174074684078938262974467604784504763521359034420653071639082548294385696121810109894690269688251211866942190570741018859207103333855549659655229457782352250042455188987961990124944566498789120284582054941823848464802195541126726348928999918099721485453487401380891755517663053038386565720934821567824791577310003251521417167426434629082475853747403599157286524656983883765627013038288978340142955459897974440740304697156024080272436830680911744557318966752684607298163681209415027914199719422248693063526880285790311725074566359149791435595982224008237251470592045469117925451215448215721571573814881052721875693562344721819290641270406459206087399926880360229424951866819559416133622268646706822678131045957565756209939994082799451961209214304698454136434288545783927283730347814864782900475319366182698451288175518848825234933102101595870840340901248379845822971598938423042747750468819257986518842447004675219104254509303860202460918696668924854684086568018402782424125157597236871885784163229660100845643357106295770194475398151932039396457133657054532815651810894481700067842392973131266730830238217202796050996334251671416109484413549564969914366002202889464084731539838038753392749536843587280198087018059798672843956641848112924454583786372652608987223125134719975214576709587642475475527388587880731389957279552356456789148169535837304389587146441718810425007665441678320014461962276126929708856391402304985442764476058104271532657935573215235522677556250678252913245320628978022735690445091809154244292495759468239962167519511600636133595672560761407693890761689383482766120679474925659216033342969277325368416876028889841485003020656117196743577828492856191988991768400534152594147521283611160499090183950750973672581415777432866390867091616030471350570183235884874959127896803629075349815900459960456732704893095538968930406981209003510820082928428320097703872859442158435390061286176089917400130355235335733085462713556775584996289112834751932370208983556887875138017004995407496650233352009778332941378247605196854788595272745496331113833362195071620490092281101877974955419460581307704785694729353505270596904897026399161142617733509070256113545972963502144724717175008175384683560582785262292914105646317437336263939610171959037745670624242114167865197863030887177022069923467948759684631771020602625869978574640538589321637518082026828732514489489632078274979052432328486282084433379697574936389339877965303695967362880939521674729674993976348563866456821636003754094820699697032624544494442174247442756158683786619382246660183634026928988429624454156483027311660122635136180430574565175139856359375867298190878272103672141046217747940131569141552270537279162898535268791930017001165772693503005513426028557846741614107941791397253332377902402290764884561557831469183368980381753264035678663731618233923129900950491074843096324748975377312155961259799041871220242274737961036946453079732087279738639159177063227392893916020338160663250913851482623817057671437212360857004158571307401110350628371550212828361515083780579072449528073247900245797266678130821476953300703080701576189267379005852071665450927309346986208617973525068510234165049275013743930524604436309110816533776567958847472177238320724474198057267762336574893171544097134087228079341138922486882382730352767888050073942238594960422178267787129571543357762283564715945852298026862189996048281667837596363535439193382295237310764591961455695152698952387071306656942965405039841202068673377862436628965614352563762120824793997765818395543848562945167975477948712578887412049781843368370693344067991918517169388765388094483096753292554234879998244756650659936389594258744642331054649600307180879046291162468095342387284960116525678385262198946498234871104552890783844160321248901408355231398547058202354087508361538366544154989830102727727934790353024738556919687124739950888300901006579704096065900917752320971435141286982520356882780886592545411246509481247286666547944755630716787812285948130842952225169399579064043884018380594235503607533865770460321100307035378871940947256390787065335275653924754018552787227104515478596847394997570908621634046862697358666724645459100115463840058322346032726973470964482682959891016300104530196950366905710516671871380478781897397891933722916197110440252904364029040984057964519698581321417156675647141872365343934742069503662714337040534080138172909814101531303470821631380409129332861573785314762262549659401751262217542612234915706230598988631696043911152997967163208448000346944643458928257805411824233952650676008651971060527839395424679401270673341122808001582855049160802302512269613131974165321594380458156103941509759306383299115337656059400302096400675322548085851401352276908499364043985290384209885979105209077940921397948378839575435628209586276748017803853053084078095035701193751303490476900945543301072273036248498868235496078961912361150876907711721056049941235064155096677750326098096825513465942341167329206615303786488120805272094974418971358475914826388209332235445598750621100352523912092754891860665235047438566333199874419846158689508690465675902467393588834857833633466936055976587874279671878811288723196812617530691416684017651134552268950332320153112359368138593638190508692059587808621119629769534425810765046965346818330831362557677156663295422837108986445594663057487503858771959001201061549298809801983584361568853530752787617271482779301780934447996241317215385917812899509668203469344336723894450454680983814420739535418535669354042700173618689035637242994368970089186428335642166968546819000960254581416999376030670628256469903990941177286219737127237745522465798625570235176170629396176605490148269193813534772500544853626745680175884979515182211994902013568753887945745851928459361108208000972074638616397317386772161909582884750560600818143994578727560517643852485511496624087576223219765500737683888839102364875085013845422717921166217897034315405566579288421123455395854119593242774532919159630341215680362005603564137064095929467595282817688561636459073121902770077472657024036992324528218625035952382226040249329424254777261037455013606201012275445073920509078459375168852057065064868458443675492133542215894078186675590882799419598011941419982321140152268549794083459796722496839018358373311832766343501048120782182440586785615425294973032319163625725345802916568320821939300692796848931576728784631081198090988763848385622037147663212517818700407681411961786028942170604441013042655677483626025459316707655052529096971949708274988205921882662533088860428333144938860720849934888487177925088393863493759046770486703300837118260358480350844569785187440440025879011125442789984902063847843687091001217912581450496019253928446854862712338571385394998638513748402518353984849371690036890346694050324984787179184589338803661752957828318743328026217680525648165968367556981184587574346952490168967528730992920887407767501599664050512631537131684887447280317285490777432212403661548095036729956536489038485402055184343803913005214311597776845817450094655652584806115069166466222011900168816522440890980851612492276789942317373194877152137823553160001739945987390776541451604675430458015182853686860091347024084776507843176946646305507104992230566497517656809869114245519940953336315716429747254254661656958084644555445622039643730709339481731370545941372219594146925206198555488471284669149992825901089938351076792849607444393914704728210389248510935192114285757242630079163141042945526535049579334700353935390961547179711187143437907092694200907712551023795543308342123785968393321220863362666182880138959741934411233932348751678883360443080073545211654912892018149149318125697837623564238853771240292743598917294126539366622838124542974436400700789824801284693349955969705122395919150430294705949445794646406874751813696901445427354038337933653388078123384386830421115653882875216719416910445003172528341069396405755842056306737017080050094010140098588853600730461815495912665374143762138902395826914806872530816452744395097713136883136860865742900038243456291454652684896388943809504764498858670885004971731107609332588681342878546518085370899786373259647378237508591877823937467058234784125545511562144726992277604040873594250824442107911553176519603809311881624152269891540456863336020177232653827982257326560217075263752439625098416046720098250014555291536282790972912477587912962821190243100811323971477711698470337829756432080563651296680191220712847666840108789794676490004407822604104491219157295785483708636706478110596142611050716143867475804271620186006117583888812750915899047505026872243936899517531314864102097724644882361300182677395372626752825538290335862557891048788189893553453596180249661586658681460649402643202668322547981681447290347671323669220800276725787598619772237121965757473809644627066589165643863958078337972481985749093761206393245858999909812038238467680204992596518991881479626742544944413092153972551761783438174593537575018388305003317138860925032771716443307397478978334036368982083288777166372190331675435448499698055381073621459086548691684347713003247651495927538649681414920472719789718767116739424257808824252484320650948911487509525896888492034389469280412364744417094885209228615833623141446201149493627978587983057643687314556703692578405792257529237022600766349922303423361400358577065715699025199428395325923387126943366454603486282945073746027262658273155905914458900804817230078834763997099593487268341398132754214724991748712107716403724073526042288905757428788987663179609306661915057709797525872380240722357840536785564025712466179216103317497985220672973365714746448169193891038050952338945431807642484087999552019760759730852831623000038092088852231416481433159869866775920665180281011518477010682516021482556583165512139377998533566765956752719639659473692096697386381177847006062442914598466808842779641384234433647611924804951987004217153745869035560281331197813432804379445175113127715899124433287408270161404921568883979165522552030710619307909897123322269724413967685087409714761686264388873448242806725205430834842482203018553041651096485486149924867145079979755442253341452949419744654830800172807600266713916598084717886013548852338972418815310363085800503555111937722440455954787041063987171040704755036918206791047768509558172191618896329355875960450234423934969591283053978773321905068474625863512023182670717560099693751551752230960351052699895831531026668053379509922513487094347840230255046587695236314322922665278589896294649791633392408267158515638559227177287893993526499324965380825323229553578833935714214217632418273884254471431888491601375480919229723767738172279090798427416815243118793183847436438037291534184090535817593018459422003935208201319473459729876588051643899824044422306451342709922524345221942347600753989614968765308403765447030888019069913317911561093656239815792271556526884407931029966060571028684385085118578608593332309266926328542245810309490177789720082850381345255608096322939080216124144995289484950848704514680984891392652423018648081498472648966569448227673908996969044619336401581743522739061195403829488740076363226829703008620764672358871584830809629291745012813892511346865453905482760013555420325653349482853065425828327027079157399501046388692808163539849759060036630580079830208155300911659233787175610658000620517204930559483078208627601781282868563648849498983685792680987728265263503334924154475549151808535710404489867955228214082532989879722935763984137838528806422763385776141248196878060293414979684329079658256270440861583866741268168171830188805510581713954014924385339867415169311219379660284151147692658115849508188009997956036440822610902436260432330255694425270456679564016391529277735958406710153956379589041303085876858400975987634193947805262244876510149614035503561921018755354508408673152675231657100211354127810876451725923773590618670993450086519645625774176654730364700183636973658311545018833968066241941469906973145072561082829660674766019892134396015255281517620326126458644537766540664802592761769732684139104848321570623248782109917124832647991146246591250773421837016692667024774550398740417035538860106929051186350073035651876616321449816887424240422034787661759462546656456181889289711415319407063265355704189514840102286085272354482176538209591944091373966323153078768041330354042348840335009910451946898126714561134013334814823005918384951867885633444908198810198857402229395665469159109303870502547008598406364674100217566344748479618420908402078151484738745146395960517371396858920961519153469461388258882354568249877391895946060928679069915925140629571782990861810985981949686567262437822591559250484562340523908678338228243264176070503267402345090155395853039436353782128646208936230374591590151579918215302726955510690930330901159580909873898691131173041596543437773395207358761781264371178943159596110053349074635509044577935017004683523499102803866305394375401771952739322552300519470132317059015846956789028122230984967293786907716800297427984960063014751897223107289850599862626200133483961903576387111308009696904922136255977373904704924652869958991244728771718420139868959804882581427159574065477131281447068690221861195893271943457546993322415595194553368930231650130054293611139088649631182642268354260128545479735594050862414824872978191289787498201726239894802768994674454228793515393350392870523721159399875202661731486549703064293867050464968971425290516148437630755679920605769004580853821930898973250620777943995832857252790908903246756119148818241998972775779411799961954524827926604537520882056228839047512463987857480888667768634549041631131716731655521852929240666843761640487959262667052839739304775472815458716315605694491588040215219148288908449890593622406236648076843521047884374203734113775094482719748154196647642809322272742664570352631206849227503559846159836466389189536178328398645487245088929463238287510822364402314170259804890989207848428898924947405855304641113007447879671504681257875790588060895550843807785468295747719021790336564154337371019432251891133147002477960838048242822255027361302086543680127081414182521416664096878881959740588626994394196199138643635936098232156672476938642808354135965084451202719877420028245746746513016889649466475789617594133627493992201035585857886545249727298464602294559612830823035611075250509344137071447396770293425118777117802899766998401382835028680549407245947576760934514961822207362162022429664179087177320177220684596956176214132538609323652052987338091270822458182365421395828681255169913114093148097397601079696465526380992424949702009781189386074662887577247522574347242430330884999407250061557374702765055325741609393776140673753516907547002559945404872179289655091955488856442596132143740590814126196446985384010612811645495010882572881253129329858367319821323132966904400294436602701327858971025507490170934889866469825017009852168520814426179588422225968666492423385963091603300245951962544152981629773807635644862757982867291683961423270117534263868258996216399148336139804865292322214788790698572059906361594279480803997355850407067769373732065281595997465939033530447278654719790643195801909499820526943516357779625074898698010284999615902004240978125460962584720786679186085280282968782485871903223095108433538907669504193797984248958417954778079961364422910023116065257786506881836811568870998232134965111859912346885762518200388530477021809175475540870601772615966908933899186247025171289067708817707880379486252537655804993040432424747963124892560652737489350929921684309132261270387640100740570415716004175010895748674352606886829692821188001949627304647424181224014220561160106616335873262869328956362304197256824197870739748595987634966376311101689867064221957453819509295947882131241855812484712338001158539321158356582693240780977523098358068085108068823020591688242359355083467602397592112226367179696878642779459401029489465791743193759865418751046014265594696816918158672332533875481011141483824523718911141932927832644997580189297060413608539919321999312344533247623677370912484770778297169946864860325293931577424146271605645206760728365689822628138924020118277465595652574756419683928408189563062730348593305969497623196065806408652258618994787128312731400933491773223776372834741959754979140567745112249805362357722097696906747552091516100743458768181658067553966089206009708487059559506768256009840233848341882783972602936713219766493412658476006452327359878435551276289350853754587774919506393006471247921853410075173987321673567555682202222299299184301438491999977019805247264675368139741687423284795643733667178311770413797962763174519262334205681087547497355584326039046238513578821794822031337184284664839297109269157632712371006387376643741808347522486905937464539514150968641308840454489205588520852634907152832381313863786085983968396502781457986067582170983085634149947478481333316346305754960625790927502804100007518218867680864516694507128060470281730540875443465966148992490890498945660461437775616461276432800879568495954943621592568406979528023634676199899798232083744925168142807726063739548545993404700802867424027456512751680282740832443592892531769874501731531986109837577705947961596377425357245562142649799220088507952763552423340160556854945979869148946382826337803049303797483297947996189993962574219237044759921041744040998003632021636888022796662096130382126053627062542760544936253147906022919801161006759389186237469924929792777800944747825300241955377229771239124514609535947289011712617659656927793523809215413147225465429625978291139712711381307107997378647584669852357536504402780362840140010528100651894751529405425082238787072113906780373169402270444734488576758028460573911653978498324826769282936611271048062048872477315095465005287076174833504844327751192825968105826744499893605731823520470499632104330068776922127831788851747423094682489333565759537261934700577045338314256108486275981162092159518879367744240476329188260157542517105523285098537243175508402503100169202539562476438194313229074617931505295976169304575083960173529746845732438320703512215610247942994385379152166605118254373769649542441030672825846343866834322545395476763957912348873635513020302069985944271659060181924854702476126244071968365362072546207420360841932829957997640296693197977626703495111218298884008041063885894160854320651889547333045500098595884567258564053722333312359759886358696407006444020888719092240713531845556638338016825798156047043173875068130893748749046953857994844959599879971243226108299029778879825768423786739233239255954326339207558279984612437082126722719301174491791351733625291453757254370207483894436057900952097113521358116489256213979259826367185146818426295639600358326266877319880763159337598588432615079467084538759400368384399449513510897666419302566496127496083122714941468280789680111032143154904623586325461646917600365162977267008777899315009519709056304967936331923109702801604231160720793650791794661390914716263199290304158629659761856499305562889220653881877999656194441540348898802695607433491221822370216805708551774840968414917636074655061803162190888307815332627642717164408090852641001650649778419757307891841771448048756940327037390840573226634941285713113016505625886372516175312931287722612173315966073758006590531545916785036263684966189052309691056880069961894283591985470457532895735259259515618340666947893042137087351488651628508553036965290870857697337716641253362748845423932109242644146363432835627253680272981244681235999940982743002096148420987038414060929569649733252660599294385243733664279589515286365426668169430278683468072165384924293679842386095423339170731680760587288204568869336389689007492641837767920965625610189990228038716756754854230166082916244358388408739604654646232659798220126901936610629768536038310170410013032670297598286160823506710315945150762317237529306100299692909840384300497341583394448163443025964663014414434300416457838878964003557700787851909819880094394827676334836732741593446006212833013477484153918606927264510659735394595204604208585481037143580535564508559870833451354731507096424280631153377999132848065901386936086151071566201880601717637137105018478648184338722947685103830700168785827579745830805164748987956015273990381977364121085466360764044287432474792735758599843326435859930782555523856275395768559623471597056822132364152695784082644345668480054674406497589458871109303581685583217080210401631817467070580131162272408581307333395632261551915362867749358970624072401161238432553424669012586633940528903643078950233449794276171476187864523917785046287569664733028417158779458830991319807265818156516879845238462066458371321712171747854497601959730373422058745739879394918344447554632115099422243349258998452661552103971155703060356430848863258222463658307821615780553322527794320229580569436056514120553592771680595543125832719989563197204673961561334149250851595332596657885221887543232685237759104906996164325638558447187705127244012014865856721166950010052010014148933932857515060680988670269904389113860094516587057714323544038657380406232985222482466744123873027812150302974249089928707682025032438932533614790180564784577963381428462753163976502205524665008057309468738898335887278333416570601646192193961219214627640380405111731426551640634910684226026863002208087217283903327267118852894467250845679762725339407505536987404571496084919632243380484696715218323316688416608076801275432120196975977330157689271194718343954117204649225980137934899474805634554812263680048980165087109400698333349005841870725817012002210827087892479476566294947483956996656625369661139030129104936418674151374494439662749903917399495050846561532288332699500013511036829876822463204820433170930283044947634406940907546957458696707079724814551607338566511790218389044885640416486307099652777508851290956374682810576014405459212608607112325443594996823432023081528157389502050251626493011100027145497259305001275310213034405723791720462641124229768674464354775669703599852084040909424333759412773337613535076615286688017240926446943600124363266304512833576155846713782563524904337003201075465824002111234848888447593390442091841152814437481266039528696082044968589851895197418729954903408508425880871078787907614828990998500804683962503711905748584723399989475978207320845617735589272920107728641596156935374247297552652303441999412677658590582206317960255608848171531955944674311993881791178427476350635870733494644130597708970962739228344403173444597668190024237303368727301510498664278989832876042715012933086423876698461164939192732789609627153218053797630658054869164290263834117285590126987213555074256555995035502702155768562278784769939667590070431564728603955306023769788033896548997295612672900327450282314845237962109891216929339583814646266592728342874921947935479910765869827958379473521126324328078542346183344436539523509599842664155606967200407542520457980286226497620976825057963015981681081008490672936841230966040816842520755549015947047938919068890273116705998186213535710425983037512119303820652590387776548968462927334889266571186779686394677763023913576193894249536078067385429034784145584769260741063486721483895971545012586330444852334661635212271443284032642131779165614398550989458250053009930707395109899644873110562620548638453801691770928179596612160363783013500341598521269301868646577543554460927258323455744064355470543790996390265507509418667083996517959806193680404919811930276951820169006298083826659574811992559347181829765236322376378929078014159329578176439077797413171467852070396485237944342688261452097313687460856447373412653531803365469844234575283464523779901251349013231948369949878686823065900907373599773861678817218898590597087625392892618240597079440043832436673523540953212956451050766198646688075779289756276630859557463737891724143497800309671669541993911973218599825064741193027222907004547406589104160788128276129730813625892093601809187302506632988936827256671506633628680799951872747487077762493028664851230023724245943683022354660807353993828949399969341533801975476970448849263227385170643397710967830990392453403095877849308390114888158430080773999597500968700502963091531571832562023712383209761021408764402079639931352568353871792099933418280274894314883081065720310061499470945442884680462265434967891449198865951775851674390216616685122643584746371509984702117717454172007663269506378527753558005608707464736935202504129864062201033920996848846256777640828381383225524115809104357109969636137861620360817454222656101274598165947563982488500270156981480392864913442626877507564059150589586633595036833744351143732624227885426785584072003578173756569685037502476145613077392294900219648767356788453452322461889311264712747008067841329445844048710650708693976843089381693211551150925161810504497531336994401136203916794692782187847491158778483379439888138261296254891537428557613137364210232499578510825035027942215443711773004020841424429660637080977128906641627651836115704541001089396454964334139405336688093699737861437030990811316161050149080124769742179551557204898163151015350162070375701428513682629907270487505488329434616797254278111046359471858729692090237628106830165789774906601559415346371690653864177603568103689708522719680418569209040583832749602648869738886106933736273243198466151377046491439531074351573897582960360961588776527236952752588737461086376392024946203374659043694101467422705136565905685536283537877771028808613369733544622976791894193417348588997680538179684186886049591901392672603279366330290161134483222477850206648896048934520022746317726348715578496456059524415523961795166808074709545822212421296045099463550207706903138044995298323354096006123066535054478397428929794056861217320407918241699721540475904958501801209586723431692102265743843807682970163694233931872574311011237424129698639799901012965657462880486807585809567806203101545429667827953923151417163913942215046170775979316895751713208485599380450476906927874617189319805194660818284306375478378168529936158665967715892731945905455647153536628454135412999757933845706593305321205079638439198559513127821616074526580250959868442042848837703516955161812496349401766770031256541433682085435681460460960806395572751403759970608911739085828514144729307166275835650981767881790452770771556640228249774724937653521858796908504438177447549383548398357923566849741972953329408059272188809751945351677636809785540309856844562091163075182690505511494778035355616047348472049416176147552586764419533156706678040634990402962090229180869129301473377918032215595039578235247485801517575747452373310421390297474690057289600643979201086166180748928178634048608398071659233731914493935443352142520111874453604768392331122094295628844029802420753186494599323805291479139021158170997542190635648551236003724525091331161344236403090271126662249140576144689997097548488146598574479292183134407500729814026569403979145718664054587093530642971095183776407216499279221030093048520336389206052820856509981988257329362979182140234276459846942108390905622129163637822521304195380428279874249390559847342575487427218587213188965358346867761550283999505596296002350426749855425966819704648002773906114991589737585997105830020050865848563479038459089423357931960949716478038998681781959255534340844576518667652517647418602954324482829902995871400625524664075381507907392389134160491414086608112508573751557414540952721413413620171882255706105137786813260560914593954073871436843876571733007000785596397183424942047168980313430198314856488752576727828241150927273603461561210291665691964764237587540448379563159205095564437232785669845353067313375763683748661692385073859056752912460163834555974173083946775967565254922967659877338866704658706383659205254781729461550440710915631036139754104331659094621851436733068188601623232076838450125788053355780031330701530396841964450862841371634836207235824521294547443896488438149448748902202090558806792141623716988038678646529053201245760517176778526492866295581770770748741968382037733589225245246844272603865863278995025162585011021191906922334577124741802468262683001773937806422331469538636136448654704727520817973588864265938622361541645729220729786988343123395422641306215344283020336444949899769867360781042723880006638224080539549844866435578699368633460920875110212984468936928669328210459965313328662809492251282042243812089899304939348059518477859660341069536097094848069637262246832141021203996312705923768179576748104858892151567153415365980782593184262660522912472349383229409599561719474263613461518157861176315927146354615791702720117407175960368436008638333537646306227579366771888674762766293038294210573826718634620563413841144220339297323170307511587197182663459929300210643055191355006289351617339523452116379440374267906294105650412140208441323204096070331845890304317709439687048443880179167504360240700362046871836306423157020480956218109241406285912708825436037395456502023398540871492873377865279062306568080949036441164857609982816800595353507460430156530733982366688141566666943425185918640511337619207246501282603879235111383459917755717382283759889318633659708189869440724477663601447413773948272574599584387549079183445779755722338902398781771458014946337758012640953047134036477972867148710683688460651125951482083625117825738825344390575145772972555019081378347001896753104513111669765738064864559337842988484939060924825453081164630680312121639907439479634030378638992803190518325452579124205508700672875871534955762131545345739428708016890957828009874786176427858468153794116723310343767056511367172291343504578155637613016024009968015912422211203355306302566528259724607982172317140994267353334967849901870205989723826598231411034930111455438595422863490415436187219121815408939499116589170148624502374957804831807503255912562943717272913410544445665830847795248440707913298432086610816988214323295443449335645341131067930083952214622685530466726671322474312711578797449786566486737739295734370879702430753308029898167360067158256981995458223286802004361557901993767723524287584058846973477289059382597786098225152456137038087026937367111346080019493968751012291386153125418741691345783881379453913074604825448893399284428523933456666852919366733979616111119840186832604033527307631166807232907975287140825867724201750492824014011076352720227487404112018692836571705091089692886119357411848708054807951285344784248285255700140421584757568755451549722338252280675401090668117056118781305330990332365396457953051979391997150769440896695513230059939671646803870827650800193368303315258025616546434896520553609397087604740749682996346791282648026057295300046017637848791316267098442400486244497821884847583816535446186536459794509233024092094373688193310563739293957018685461101964190267420267779766687460310672936441046008996866274961174246566697291057100788244256701831892400797711528255147312081701944688655606604178177619772675448422089807783530536397169999761323656632873905786368791780908084439944201655587903582892932111930500255902770557953560800588528085769507864684472772992775727096290551054308023560962204517228097296463194890436115978022303143193063836966938352029919469989402615138012063479381868551220319906406976450089395478443071375857432706058539541582446608973991790044585528157911840552932023895600786941582801315368293675491165242674440895810704672739718884335277602084014199728817790160721729036025291738462359530262361798358231299932405855338216730877480530018843197520259089310068501434391165996028359184771475408841115323145726412946393944204038302296176940159643940500772830877503913784375945362172922416182327713615354053033616501212584278298964583558779181420384958171033893351789900790152470923025550501530598186180243457178380760373659257716041649530393863171426051491261660631382307904497313461247294836649757919776615357805551585879282896532676330024775516047819923112680164480757234603831671302525684680576925036238463154391882688626212590618473913620440932539890251601214937109730396335855372399038574307446032290317315764327743591934270228929603078590215329582775925378558142853138452399492656260475412073110313571785247016261055398907555085680146520935062345013853607101795899984992256020484416142648028496963333211850086701663524403707538755317380188220636577184545643913851071641054712155100812882110132133884187046114941182307920087039208179159172114674074383263091812689231498705444513503498108137773625846876104618420797046399977141495544686527380000683453793427188122373804514800086373383851080502568043974245375264340779424331740342739270023650087575711206008591676703250800385013210290210538041958666723452364311609283705850140425878367718221711493508957288348607637513978253339346215269490158706758043774014127973318954154568172487004633538445182625048375943971821318771724150131569029620124192911399188743079280602925124117501148798373064291080978510277632047106641923337252866243494410575613204307237461819494687045471418768323100469665287581056439366103662880818848580270824108486812508573893704562901362580925880336156141070564203140076818049141780652280248844315436883889505643388763721387304663709950342496807297776518114306043654264412519243974080557015026390304303136099262158099498298083918560173024288746692239911217902033187705604886481657603630414169044712090930572351936942502354732473063543065487684386848475966703320639489653123573568280156302239397404918720686509965775069473508285370140469532392660545259266895002535350002151844293141635405528944596032447897524184358726789636991727357429461362817470797618895282196689387435291532107528245524298969387638242986408156120947787870249794902713040580244489954415196343011274495240416292210212593155553485242322893205230086474451997783163824981481260474845402200875039052672668408779567390074755118961908361466508135723282040020362602760652165478515290270285838452059049673068120434040130618376171048026320405581426931151655968070313202711661321555669210630229677296885185107875157085077414902407325622305318578976587844164477949715532008015482157500015474661893690854874518813422043427839087030790022109746371444248400024407489074941498765308415527615314929749825975882079785323731076781316574863126612879188848196511268763724148950417531281299538351334781251100870047565338431371323217018665089472851331996723641959584882883963033198394476203689465831210437273009483706729645938097122611317530757992400638274250888225266875779719612802806211826848463695991213515977414439573383245641786326517692000438404130896707556860582231527747951968265672367549240834601419086536898904773000701984718483972955906928054001925965375405414513953988945082456680310182217604595098479096969854993988628310242086570388631668244026133138954003303670958401782822400402346199846209347176799021328706882021468360914484747788001565093149385921456634085417680405348778350006253890330714372829393311479001329789390758649742407695457749190228348009829833696000714723661506610703506845363582077984551163095214326869057148394198347537813658679556465446949827062288992562415070381787803474237049838498777508291502423166615248402294674592087769943944716496265619640250742773358839792621365755301032297924952214461615781736460029273446154255844498058619879967523023532475514269892285398088425455940956936623919268523111436566446404555877605133804941312609813302569873609381715320766478947755900124009757838514045053488314107682830956667357767919627378499391786236134285037599114507287564824888822484771340907194667541869509087897535807736942384706477697891286055234002683468896665499254360355107366602868208172732628932962543375429423614509499568181566879579402197014119545883670609492682581803235543694063230455855581758838172052041401100231100045117997829153346402043379082999282341052314941566438090685725838081262792862688822814883152808994752873548939070520008571618223839210106761214576002833434146682080471120568972120076865517932622118265184035487431554080535008803094645791691154453296845484340073530860638931364149020029213840731911556809654900518005659305877055868795804827971120777777705573948467236344818727084573679205996955580955107781830205894069788863038746545624824422292825036704292122033872281606722795386312429267124417099787830647764386293997538171199283162215363647799615343304742090906298380084124230078177824925253766221878399319419268573321972599540890549746559383516008409810750143580211823910248785287404164585143341763410065081238774213753418243509680074596550584126478275344382917408605860086920243706729700429141755296945832643346020897411276238380826874735372343237223424403650386658265554814679348086487377269484881267594560406806580565542663969956686047824084839388902539537099501993270053003853887836039691912782648282206634366765510848295685163143531404013268748250676164561735202043684931440229597646004797863551624277517898061813169184442793438949399300054947525057789094593184226267970478964276394714527381828387542112330044264350528181871228806626997172751667495123967635292693378340774578132235914632591681134307961924125201118690539042478462316469126578096098296572087627262961568594863671444841559019146175073424860242328690603641587090509574620409397258033342243288086034202590695762910680969023781019225862043019770020618966593096702148995394319140893906522366963283780034899571762758507054571812087831300376049407311490660960485441865395109423225129200006110969931778577137517924048225319855332303375446559993354283017226637869832241435197927977193020875464610297229510740748909370614253120835033375043956511636886996825913887216179491713085261803193719002795025769557035464920170870502529927853882501312137800308451730543645403579294818902072928102622367033879469792318127965062310323290865280601050288524744297562659404929193640365712159850841949217537422935690539638399133880401297618018553815538416726633131883411007535732088464349058786233752590320124719445970286464798405623207278546880620315099787324658275014025814725364074238078102818282148501398271198491442760751154755431933001350793450538853816808526763987220002929006264726854178285426158935164113057774827973162554328835970928676252178650898981369923508875425164286450539228099547434198712860078984328535689957445525641205194965963971455119388892523225946612571279872918170916830803523176981598195240980895639347861656548230904043587623789000308909763756641320171592200705500451053617615711645466921507670291540057992982984596384184596056823796885057911133405590262505967608278395479209831489366540239593876853147503349989376

This number is 38,532 digits long.

 

Downloads:

There are many Matthew's Numbers -- plural -- one for each computer graphics mode. I will show you some:

*For a human to write out, assuming writing one digit per second, full-time work, 8 hours per day, 5 days per week, no vacations or breaks.

 

Large Numbers in Print:

To help grasp how big these numbers are, Numberphile printed out the Mersenne prime 2^74,207,281 - 1 that has 22,338,618 digits:


22 million digit number printed out.

 

2020-MAY-22 Update - New Largest Matthew's Number:

This is the largest number I have ever created.

Calculations for Matthew's Number 3840x2160x16777216 ended. It took 3.5 days on my gaming laptop. It is the largest Matthew's Number I have calculated. It has 59,924,717 digits.

For comparison, the largest known prime number is 2^82,589,933 − 1 and has 24,862,048 digits. Matthew's Number 3840x2160x16777216 is 2.4 times longer. That said, calculating Matthew's Numbers is simple compared to testing for primality.

 

Also See:

See our other math theory work.

 

That is all.

 

 

About the Author: I am Matthew Doucette of Xona Games, an award-winning indie game studio that I founded with my twin brother. We make intensified arcade-style retro games. Our business, our games, our technology, and we as competitive gamers have won prestigious awards and received worldwide press. Our business has won $180,000 in contests. Our games have ranked from #1 in Canada to #1 in Japan, have become #1 best sellers in multiple countries, have won game contests, and have held 3 of the top 5 rated spots in Japan of all Xbox LIVE indie games. Our game engines have been awarded for technical excellence. And we, the developers, have placed #1 in competitive gaming competitions -- relating to the games we make. Read about our story, our awards, our games, and view our blog.