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:

• Matthew's Number -- singular -- is a large number, 38,532 digits long, that describes the amount of possible screens in a 160x200 by 16 color graphics mode. It is a TGA (Tandy Graphics Adapter) graphics mode familiar to me at the time I coined this number set. It was used in the original Sierra "Quest" games I played. This number is specifically coined as "Matthew's Number 160x200x16", following a format I decided today for publish.
• Matthew's Numbers -- plural -- are all numbers that describe the amount of possible screens in all popular computer graphics modes. These are an extension of the original Matthew's Number.

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.

Example:

Let's calculate the number of possible screens in Space Quest I on theTandy 1000 SX. Its a 4:3, 160x200 resolution, 16 color bitmap mode. This means it's 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:

• Matthew's Number WIDTHxHEIGHTxCOLORS

Some examples used in this webpage, including a TGA and modern HD modes:

• Matthew's Number 160x200x16
• Matthew's Number 1920x1080x16777216
• etc.

Or use powers of two for colours to help understand the memory bit usage...

• Matthew's Number 160x200x2^4 (2022-MAR-30 update)
• Matthew's Number 1920x1080x2^24 (2022-MAR-30 update)
• etc.

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:

69094740421900874122245780012981740341182858341341952041949311183129599641387359464480582944526900720399521161545666301668372849589288736917366819522290192722543760508714561891675668728638793328179017407468407893826297446760478450476352135903442065307163908254829438569612181010989469026968825121186694219057074101885920710333385554965965522945778235225004245518898796199012494456649878912028458205494182384846480219554112672634892899991809972148545348740138089175551766305303838656572093482156782479157731000325152141716742643462908247585374740359915728652465698388376562701303828897834014295545989797444074030469715602408027243683068091174455731896675268460729816368120941502791419971942224869306352688028579031172507456635914979143559598222400823725147059204546911792545121544821572157157381488105272187569356234472181929064127040645920608739992688036022942495186681955941613362226864670682267813104595756575620993999408279945196120921430469845413643428854578392728373034781486478290047531936618269845128817551884882523493310210159587084034090124837984582297159893842304274775046881925798651884244700467521910425450930386020246091869666892485468408656801840278242412515759723687188578416322966010084564335710629577019447539815193203939645713365705453281565181089448170006784239297313126673083023821720279605099633425167141610948441354956496991436600220288946408473153983803875339274953684358728019808701805979867284395664184811292445458378637265260898722312513471997521457670958764247547552738858788073138995727955235645678914816953583730438958714644171881042500766544167832001446196227612692970885639140230498544276447605810427153265793557321523552267755625067825291324532062897802273569044509180915424429249575946823996216751951160063613359567256076140769389076168938348276612067947492565921603334296927732536841687602888984148500302065611719674357782849285619198899176840053415259414752128361116049909018395075097367258141577743286639086709161603047135057018323588487495912789680362907534981590045996045673270489309553896893040698120900351082008292842832009770387285944215843539006128617608991740013035523533573308546271355677558499628911283475193237020898355688787513801700499540749665023335200977833294137824760519685478859527274549633111383336219507162049009228110187797495541946058130770478569472935350527059690489702639916114261773350907025611354597296350214472471717500817538468356058278526229291410564631743733626393961017195903774567062424211416786519786303088717702206992346794875968463177102060262586997857464053858932163751808202682873251448948963207827497905243232848628208443337969757493638933987796530369596736288093952167472967499397634856386645682163600375409482069969703262454449444217424744275615868378661938224666018363402692898842962445415648302731166012263513618043057456517513985635937586729819087827210367214104621774794013156914155227053727916289853526879193001700116577269350300551342602855784674161410794179139725333237790240229076488456155783146918336898038175326403567866373161823392312990095049107484309632474897537731215596125979904187122024227473796103694645307973208727973863915917706322739289391602033816066325091385148262381705767143721236085700415857130740111035062837155021282836151508378057907244952807324790024579726667813082147695330070308070157618926737900585207166545092730934698620861797352506851023416504927501374393052460443630911081653377656795884747217723832072447419805726776233657489317154409713408722807934113892248688238273035276788805007394223859496042217826778712957154335776228356471594585229802686218999604828166783759636353543919338229523731076459196145569515269895238707130665694296540503984120206867337786243662896561435256376212082479399776581839554384856294516797547794871257888741204978184336837069334406799191851716938876538809448309675329255423487999824475665065993638959425874464233105464960030718087904629116246809534238728496011652567838526219894649823487110455289078384416032124890140835523139854705820235408750836153836654415498983010272772793479035302473855691968712473995088830090100657970409606590091775232097143514128698252035688278088659254541124650948124728666654794475563071678781228594813084295222516939957906404388401838059423550360753386577046032110030703537887194094725639078706533527565392475401855278722710451547859684739499757090862163404686269735866672464545910011546384005832234603272697347096448268295989101630010453019695036690571051667187138047878189739789193372291619711044025290436402904098405796451969858132141715667564714187236534393474206950366271433704053408013817290981410153130347082163138040912933286157378531476226254965940175126221754261223491570623059898863169604391115299796716320844800034694464345892825780541182423395265067600865197106052783939542467940127067334112280800158285504916080230251226961313197416532159438045815610394150975930638329911533765605940030209640067532254808585140135227690849936404398529038420988597910520907794092139794837883957543562820958627674801780385305308407809503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This number is 38,532 digits long.

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.

2022-MAR-30 UPDATE: "376"?

All of my Matthew's Numbers above end in "376." Interesting. Any idea why?

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.

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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.