Pointless Gigantic List of Numbers - Part 2 (1,000,000 ~ 10^10^1,000,000)

PART 2: THE LITTLE BABIES

1,000,000 ~ 10^10^1,000,000

Here will be numbers that are bigger than a million but less than a millionduplex, including the googol and googolplex themselves. Be prepared for some numbers that aren't so wimpy, but still smallish in the world of googology.

The Third-Block Range

1,000,000 ~ 999,999,999

Entries: 21

One million

1,000,000

One million is a classic benchmark for large numbers, and probably associated with the idea of "really big number" more than any number. It's the first member of the -illion series, and the most widely used by far, with over a billion results on Google. The name is well-known and appears in all kinds of contexts, and it's also often used as hyperbole for a very large number

The word "million" came from an Italian word meaning "great thousand" - therefore it holds a unique place among the origin of number names in English. Here are some examples of how much a million is:

A million cookies would fill up a room or two, and a million seconds is 11.5 days. The Bible (which consists of 66 books) has about a million words. A million people is quite a populous city, although it's considered not a very populous. country or even US state. It would realistically take a year to count to a million if you only use free time for counting, so definitely people have done that. million has been described by Sbiis Saibian as being on the tenuous line between attainable and utterly out of reach. Overall, a million can be described as "truly large" - for a more detailed coverage of how much a million is look here.

A million is the boundary between class 1 and class 2 numbers in Robert Munafo's idea of classes, because it's about the limit of physically perceivable - you can barely put a million dots on a paper and see each individual dot at once. While class 1 numbers are numbers between 6 and a million, class 2 numbers are from a million to 10^1,000,000 (a millionplex). In general, class 0 numbers can be recognized with our number sense alone, class 1 numbers can't be recognized but can by physically perceived, and class 2 numbers cannot be physically perceived, but can be exactly represented in decimal.

With this idea, a billion, a quintillion, Avogadro's number, the number of molecules that make up Earth, a googol, a mililllion, and a googolgong are class 2 numbers.

The SI prefixes for a million are mega- (large) and micro- (small).

Guppybit

1,048,576

This number is 2^20. It's the value of the mebi- binary prefix (see 1024), and therefore under some definitions it's the number of bytes in a megabyte.

Sbiis Saibian gives this number the name guppybit. He defines a guppy as 10^20 and then coins a -bit suffix - x-bit is equal to x in binary, so while a guppy is 1 followed by 20 zeros in decimal, a guppybit is 1 followed by 20 zeros in binary. A guppybit is in fact one of Saibian's smallest googolisms. The next smallest number of his is a pipsqueak, which is another name for ten million.

See also googolplex.

Googolteen

10^100 + 10

Fourteen is 10+4, and fifteen is 10+5. You can extrapolate this to get a suffix -teen which you can apply to any number n, defined as 10+n. Andre Joyce applied this -teen suffix to the googol, creating a humorous number called googolteen. Here is googolteen written out in full:

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,010

Googol plus thirty-three

10^100 + 33

The smallest semiprime (product of two prime numbers) larger than a googol.

Gooprol

10^100 + 267

This is the smallest prime number larger than a googol. Its name was coined in the inactive blog "A googol is a tiny dot" - see also the next few entries and gooprolplex.

Here is gooprol written out in full:

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,267

Booprol

10^100 + 949

The second smallest prime number larger than a googol.

Trooprol

10^100 + 1243

The third smallest prime number larger than a googol.

Quadrooprol

10^100 + 1293

The fourth smallest prime number larger than a googol. For its magnitude it's unusually close to the previous prime.

70!

~ 1.198*10^100

This number is 70 factorial. It's notable for being the smallest larger than a googol, and it's quite close to the googol in size.

57^57

~ 1.216*10^100

The smallest number equal to a whole number raised to its own power which is larger than a googol.

8*10^100+1

This number is equal to 8 times a googol plus 1. It is the number of letters (without the space) in the name for a googolduplex in Conway and Guy's -illion system - for more on that system see here.

Googolbunch / Googolty

10^101 = 10^100 * 10

Forty is 10*4, and fifty is 10*5. We can extrapolate this to get a suffix -ty, which you can apply to any number n to multiply it by ten. Andre Joyce applied this prefix to the googol forming this humorous number along with googolteen.

Sbiis Saibian gives the name "googolbunch" for this number, in analogy to the -speck, -crumb, and -chunk suffixes but with naming multiples of a number instead of fractions of the number. It's equal to exactly ten times a googol.

Googolcrowd

10^105

Once again, this number by Sbiis Saibian is an analogy to the -speck, -crumb, and -chunk suffixes. It's equal to exactly a hundred thousand times a googol, and therefore it can be thought of as a "crowd of googols".

Googolswarm / Eleventyplex

10^110

This number has been named "eleventyplex" by Andre Joyce as an example of one of the many numbers that can be named with the -plex suffix. Sbiis Saibian gives it another name, "googolswarm" - here, "swarm" means multiplying by 10 billion, in contrast with -speck which divides a number by ten billion. This number is a swarm of googols in comparison to the googol, and makes a googolspeck look unimaginably small. This difference is only twenty orders of magnitude, and pretty soon we'll be blasting into the stars where the differences between two numbers on this list have no analogy we can use at all! As rule number 1 of googology says, it only gets worse!!!

This number is also an estimate of how many atoms would be needed to fill up the observable universe - therefore after this point we can no longer use a "sphere of this many atoms" analogy to capture the size of these numbers.

Number of ways a 6x6x6 Rubik’s cube can be arranged

1.57*10^116

This is the number of ways a 6x6x6 Rubik's cube can be arranged, and it's notable for being larger than a googol. We can say that if you ever get your hands on a 6x6x6 Rubik’s cube, you have a real exmple of a number over a googol, right at your fingertips!

To get an idea of how big this value is, gallons of water will no longer help, since this many gallons of water easily dwarfs the observable universe. So we'll need to describe its size some other way, by dividing the universe into this many portions. But even that is a very difficult analogy.

Helium atoms are the smallest out of all atoms, and if this many helium atoms filled up the observable universe, we STILL wouldn't reach this gigantic number. So we need to put this number in terms of something smaller like, say, protons. This many protons in a sphere would be about 115,000 light years, bigger than the Milky Way galaxy.

Rubik's Cube numbers:

2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 7x7x7

Common estimate of all possible chess games

1.47*10^118

This number, equal to 1.47 octotrigintillion or about 1.5 quintillion times a googol, is a common estimate of al possible chess games. It was estimated with very simple methods and guesses, assuming that each player gets 40 turns and that on average there are 30 moves you can make, giving you 30^80 ~ 1.47*10^118.

My upper-bound on the number of possible chess games is much larger, and likely a more reasonable esitmate considering that a game can easily last much longer than 80 turns.

Quadragintillion

10^123

The unofficial 40th -illion, from Latin "quadraginta" meaning 40. The observable universe has a volume of 47 quadragintillion cubic femtometers. Protons and neutrons are about a femtometer wide.

This is also known as a "vigintilliard" in the long scale, the largest power of 1000 to get its own name in the long scale other than the much larger long-scale centillion and centilliard.

Great googol (alternate)

10^150

An alternate definition of Joyce's great googol (see 101000). Logically, this should be the great googol, since a great gross is a gross raised to the 1.5th power, and a googol raised to the 1.5th power is 10150. However, defining it as 101000 is more aesthetically appealing and still makes sense in a way, if you consider how the exponents compare. This is equal to a novemquadragintillion in the -illion system.

Quinquagintillion

10^153

The unofficial 40th -illion, from Latin "quinquaginta" meaning 50. This is approximately the volume of the observable universe in cubic yoctometers (the smallest metric unit of volume).

4^4^4

~ 1.3407*10^153

This is 4^^3, a smallish tetrational number notable for being larger than a googol. This is a better tetrational number but still not quite great, and logarithmically not too far from a googol. It's approximately 13 quinquagintillion and it has exactly 154 digits.

Written out in full it's equal to:

13,407,807,929,942,597,099,574,024,998,205,846,127,479,365,820,592,393,377,723,561,443,721,764,030,073,546,976,

801,874,298,166,903,427,690,031,858,186,486,050,853,753,882,811,946,569,946,433,649,006,084,096

M521 / Thirteenth Mersenne prime

~ 6.865*10^156

This number is the thirteenth Mersenne prime, discovered on January 30, 1952 by Raphael M. Robinson. It is 157 digits long, and it's the smallest one with more than 100 digits. It held the record for largest known prime for a few hours (?) until Robinson discovered another prime number that same day. Five Mersenne primes were discovered in 1952, all by Robinson.

Number of arrangememts of a 7x7x7 Rubik's Cube

~ 1.95*10^160

This is the number of arrangements of a 7x7x7 Rubik's Cube. The number has exactly 161 digits and it's equal to about 19.5 duoquinquagintillion. It's really too big for us to actually comprehend. If you do want an idea of its size, this number is so big that this many cubic yoctometers would fill up ten million observable universes. However, it’s still less than the volume of the observable universe in Planck volumes.

Rubik's Cube numbers:

2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 7x7x7

Sexagintillion

10^183

The unofficial 60th -illion (from Latin "sexaginta" meaning 60). 300 sexagintillion is approximately the number of Planck volumes in the observable universe - see next entry.

Observable universe in Planck volumes

3*10^185

A Planck length is about 1.6*10-35 meters, or 16 geptomicrometers with my extension of SI prefixes, and it's considered the smallest physically meaningful length; a Planck volume is a cubic Planck length. By measuring the volume of the whole observable universe in Planck volumes, we get what is arguably the largest number that actually exists. What's interesting about this figure is that it easily shows us that as big as a googol is, it still has tangible meaning.

In a sense this is the largest number that actually exists in the physical world. Beyond this point it's debatable whether any number actually "exists", though the physical world allows us to generate numbers far larger than this as we will see.

Gargoogol / Fzhundred

10^200

A gargoogol is the square of a googol, formed by applying the gar- prefix (squares the number it's applied to) to a googol. For more on the gar- prefix see gargoogolplex.

A gargoogol is also equal to a fzhundred, using the fz- prefix which takes a number to its own power. It's interesting to note that the fz- prefix achieves the same value as the gar- prefix does for a much larger number.

Septuagintillion

10^213

The unofficial 70th -illion (from Latin "septuaginta" meaning 70). 47 of this would be the volume of the universe in Planck volumes 150 quintillion years after the Big Bang.

Googoc

200^100

~ 1.2676*10^230

This is the best-known example of a number named with the googo- prefix, a prefix that Andre Joyce made from a curious observation about the googol. He noticed that googol can be expressed as 100^50, and googol ends with a letter L, which is 50 in Roman numerals. Therefore, he made a prefix googo-, wheere googo-x is (2x)^x and x is in roman numerals. For example, googoc = (2*100)^100 = 200^100 ~ 1.27*10^230, since C is 100 in Roman numerals. The number has exactly 231 digits and falls between a septuagintillion and an octogintillion.

If I were to only mention one of the googo- numbers, this would be it. Interestingly, it's equal to the product of two other googol-related numbers (googolbit and gargoogol).

Octogintillion

10^243

The unofficial 80th -illion (from Latin "octoginta" meaning 80). 47 of this is the volume of the universe in Planck volumes 1.5 nonillion years after the Big Bang. Some time after that point, the universe may start decomposing, if the theory about proton decay is true.

Quartic hypervolume of the observable universe in Planck hypervolumes (birth-present)

8.01*10^245

Quartic hypervolume is like volume, but it adds time as a 4th dimension. We can get some really big numbers using quartic hypervolume in Planck lengths*Planck lengths*Planck lengths*Planck times, for example of the universe from birth to present. Here’s how I got to this tremendous number:

1.855*1043 Planck times/second * 3.1536*107 seconds/year * 1.37*1010 years of the universe * 3*10185 Planck volumes/universe = 2.40*10246 quartic Planck units

But the universe has expanded since it started, akin to a 4-dimensional hypercone. The volume of a 4-dimensional hypercone is ⅓π*r^4*h, so we need to divide all that by 3 to get this still enormous number.

Nonagintillion

10^273

The unofficial 90th -illion (from Latin "nonaginta" meaning 90). By the time the universe is this many Planck volumes it may have already decayed if the theory of proton decay is correct. But we could still get real-world numbers this big using quatrtic hypervolume of the universe, not to mention the numbers we could get from probability.

2^1000

~ 1.0715*10^301

2^1000 is used in the definition of a transcendental integer, an exotic range of uncomputable huge numbers devised by Harvey Friedman. Currently, no integers have been proven to be transcendental. I honestly have no clue why 2^1000 was chosen - may have to do with binary numerals (see the page for 2), but don't ask me why it was chosen.

Centillion

10^303

A centillion is the largest official -illion, exactly equal to 1 followed by 303 zeros. For a long time it was the last number to get an official name in the English language, before the googol and googolplex came along. It's still the largest -illion that gets an official name. It's unusual that a centillion gets an official name but no -illions between a vigintillion and it does. Nonetheless a centillion is a pretty awesome number.

A centillion is close to too large to represent anything in the universe and much larger than ANYTHING you’ll see in science, but it’s still somewhat in the practical number range. This number is a bit more logarithmically than the number of ways 1000 coins can land (2^1000), and it’s smaller than the probability of certain unspeakably rare occasions as we will see.

Sbiis Saibian has strong personal connections to a centillion - while reading the dictionary, he was fascinated by the names of large numbers, especially the centillion. Therefore a few of his googologisms, like ecetonplex, are based on a centillion.

A way to get the idea of the size of a centillion is to consider q uartic hypervolume, which is like volume but adding time as a 4th dimension. In that case, from the start of the universe to 3.778 septillion years in the future, the quartic hypervolume of the universe is around a centillion quartic Planck units.

Largest number in double floating point

~ 1.7976931*10^308

This is a number with special significance in computing: it's the largest integer that can be stored in the commonly used IEEE754 double-floating-point system, and therefore the largest number storable in double floating point in Java and several other programming lanauages. It's the largest number that can be directly stored in most programming languages at all. This gives it quite some significance in many Javascript applications, such as Google's built in calculator, and indirectly in Robert Munafo's Hypercalc.

The value is defined like this because 53 bits are used to represent the part before the exponent (e.g. 3.5 in 3.5*10^40), 10 to represent the exponent, and one used to say if it's positive or negative, adding up to 64 bits but far more wide ranged than just plus-minus nine quintillion. This number, therefore, is 1.11...(53 1's)...111*10^1111111111 (10 1's's in the exponent) in binary, or just under 21024. It can therefore be expressed in base 10 as ((253-1)/252)*21023.

Current upper-bound to Skewes' problem

1.397*10^316

This is the current upper-bound to the solution of the problem of what the first number where π(x) > li(x), the problem which gave birth to the very large outdated upper-bound known as Skewes' number (see that entry for details). The current lower-bound to the solution is 1014 or 100 trillion.

Quartic hypervolume of the observable universe (birth to proton-decay death)

5.07*10^341

From birth to proton-decay death of the universe (assuming the theory of proton decay), the quartic hypervolume of the universe would be 507 duodecicentillion quartic Planck units. Sbiis Saibian has said that this number is important because it's probably the limit of what we can actually use to measure the universe. Beyond this, numbers can only be physically interpreted in combinatorics and listing possibilities, as well as hypothetical sizes of the whole universe.

Faxul

200! ~ 7.886*10^374

788657867364790503552363213932185062295135977687173263294742533244359449963403342920042840119846

239041772121389196388302576427902426371050619266249528299311134628572707633172373969889439224456

214516642402540332918641312274282948532775242424075739032403212574055795686602260319041703240623

5170085879617892222789623703897374720000000000000000000000000000000000000000000000000 in full.

This number was coined by Lawrence Hollom, creator of hyperfactorial array notation, a notation which extends the factorial into the realm of googology, taking unusually powerful approaches. It's the smallest of Hollom's googolisms, and it's the only one that can be written out in decimal, as shown above.

This number, being the factorial of 200, represents the number of ways 200 books/apples/whatever can be ordered. Its name was chosen arbitrarily, akin to the googol.

Fugafive

5^5^4

~ 7.1821*10^436

This is the fifth member of the fuga- series, a function which achieves staggering hyper-exponential growth. It has exactly 437 digits, and as such it falls between the faxul and the output of pete-3.c.

Output of pete-3.c

9*2^1566

~ 2.3292*10^472

This is the output of the smallest entry in Bignum Bakeoff (the 2001 competition of who could write the 512-character C program with the largest output) that produces a number other than 1. It is a simple program submitted by a guy named Pete that "takes 9 and shifts it left 163 times by 9 and once by 99" (according to David Moews, the man who held Bignum Bakeoff), producing an exponential sized number.

Pete submitted nine different entries (including this one, pete-3) to Bignum Bakeoff, the most of anyone who participated. pete and pete-2 didn't terminate, but pete-4, pete-5, pete-6, and pete-7 produced some very large numbers ranging from w2 to w^w in the fast-growing hierarchy, or from 4-entry arrays to high linear arrays. pete-8 and pete-9 were meant to be refinements, but a bug caused them to only be in the high tetrational range.

Bignum Bakeoff entries:

< (n/a) | pete-3 | pete-9 >

Googolding

10^500

A googolding is a number coined by Sbiis Saibian in analogy to the googolgong (10^100,000), equal to 1 followed by 500 zeros. It's a ridiculously huge number far bigger even the quartic hypervolume of the entire run of the universe, and needs to be compared to entirely different things! If you want an example of its size, the probability that 196 random people in a room all share the same birthday is about 1 in a googolding.

Googocci

402^201 ~ 2.814*10^523

The 201st member of the googo- series - this one is most notable for having an Italian-like name, as stated by Joyce himself.

Tupper's number

~ 4.8585*10^543

This number is the 544-digit number that makes Tupper's self-referential formula work - I call it "Tupper's number". Tupper's self-referential formula is a formula that, when graphed, can visually reproduce the formula itself - however upon further examination it's really more of a magic trick. Here's how it works:

Start with the formula:

1/2 < [mod([y/17]2-17[x]-mod([y],17),2)], where [ ] is the floor function (properly noted as [ ] with the tops removed) and mod(x,y) is the modulo function

Let N = the 544-digit number:

48584506361897134235820959624942020445814005879832445494830930850619347047088099284506447698655243648499972470249151191104116057

39177407856919754326571855442057210445735883681829823754139634338225199452191651284348332905131193199953502413758765239264874613

39490687013056229581321948111368533953556529085002387509285689269455597428154638651073004910672305893358605254409666435126534936

36439571255656959368151843348576052669401612512669514215505395545191537854575257565907405401579290017659679654800644278291314885

48259914721248506352686630476300

When graphing the set of points (x,y) satisfying the inequality, in the area of the graph where 0<x<106 and N<y<N+17, the graph, when turned upside down, produces the formula:

This may seem amazing, but actually this formula is a lot more like the infinite monkey theorem than a self-referential formula. The formula, as it turns out, can graph anything depending on what value of N you pick!

And also, think back to the instructions for the formula. In particular, take note that it requires making use of a specific 544-digit number, not just the formula itself! Because of this, the graph shown above of the formula is not so much in the formula as it is in the definition of the 544-digit number. This means that the formula isn't so much of a self-referential formula as it is a "magic trick" so to speak.

If you want a formula that actually is self-referential in this sense, check this out.

Centillion (long scale)

10^600

This is what a centillion is equal to using the long scale instead of the short scale. In the long scale, which is used in Europe but obsolete in English-speaking countries, x-illion = 10^6x instead of 10^3x+3. x-illiard = 10^6x+3. Therefore in the long scale a centillion would be this value instead of 10^303, with about twice as many digits.

Ducentillion (short scale) / Centilliard (long scale)

10^603

This is the largest official long scale -illion/-illiard, and also the unofficial 200th short scale -illion.

256^256

~ 3.2317*10^616

This number is an example of a very large solution to the four fours puzzle, expressible as (4^4)^(4^4). It isn't quite as big as the gigantic 4^^4, the only solution to transcend a googolplex!

256^256 also shows up in the computation of Steinhaus's mega.

Trecentillion

10^903

The 300th -illion. This number falls just under a googolchime.

9.999...*10^999

Some calculators max out not at 9.999...*10^99, but at 9.999...*10^999, which is just under a googolchime. That is quite a wide range, considering that it's more than can be directly stored in most programming languages (see 1.7976*10308).

The Chime-Gong Range

10^1000 ~ 10^999,999

Entries: 41

Googolchime / Joyce's great googol

10^1000

This number is called googolchime by Sbiis Saibian, with the chime suffix similar in spirit to the -gong suffix. It's equal to one followed by a thousand zeros, similar to how a googol is one followed by 100 zeros. It's the tenth power of a googol and the squared of a googolding, and can be thought of as a "bigger version" of the googol just as the guppy, gogol, and ogol can be thought of as "smaller versions".

Andre Joyce has decided that since a gross is 12^2 and a great gross is 12^3 and a googol is 10^10^2, a great googol should be 10^10^3. Actually, this is wrong: a great googol, with this logic, should be only 10^150! But since 10^1000 can be considered a valid interpretation, we can let it slide. In fact, Joyce has also defined it not only as 10^150 and 10^1000, but also two more broken variants (10^101 and 10^300).

Where do numbers this big crop up in the real world? One place where numbers this big has shown up is in studying the distant future. For example, it's been estimated that somewhere around 10^1500 years in the future may be an age where all matter has turned into iron-56 in the form of iron stars - in that time the stars would be unimaginably far apart from each other and the universe would be a very lonely place.

Quadringentillion

10^1203

The 400th -illion.

M4253 / 19th Mersenne prime

~ 1.908*10^1280

This is the nineteenth Mersenne prime, discovered by 1961 by Alexander Hurwitz with an IBM 7090 computer. It's the first Mersenne prime with over a thousand digits.

Googod

1000^500 = 10^1500

On the entry for 496, I discuss how even perfect numbers are all known to have a specific pattern, but how there are not known to be any odd perfect numbers. If there are any such numbers, the smallest one is at least this big, i.e. it has more than 1500 digits. What is known about odd perfect numbers (if there are any), e.g. at least 9 distinct prime factors and a largest prime factor greater than 100,000,000, is quite strange results, leading many people to believe that there are no odd perfect numbers.

Also, this number can be named googod, the 500th member of the googo- series, with Joyce's googo- naming system.

Quingentillion

10^1503

The 500th -illion.

Sescentillion

10^1803

The 600th -illion.

Septingentillion

10^2103

The 700th -illion.

5^5^5

19110125......08203125 with 2185 digits

This is another rather small tetrational number, expressible as 5^^3 using tetration. Here it is written out in full:

19110125979454775203564045597039645991980810489900943371395127892465205302426158030120593865197398502655864401557944622353592127

88673806972288410146915986602087961896757195701839281660338047611225975533626101001482651123413147768252411493094447176965282756

28519673751439535754247909321920664188301178716912255242107005070906467438287085144995025658619446154318351137984913369177992812

74338404315492368555267835963741021053315460313537253257486369091597786903282664591829838152302869365728736914226481312917437621

36325730321645282979486862576245362218017673224940567642819360078720713837072355305446356153946401185348493792719514594505508232

74922160584891291094518995994868619954314766693801303717616359259447974616422005088507946980448713320513316073913423054019887257

00383298012460501970134673971759090273894939238173157869968458997947810680428224360937839463352654228157043028324423855150823164

90967285712171708123232790481817268327510112746782317410985888683708522000711733492253913322300756147180429007527677793352306200

61828601245525424306100689480544658470482065098266431936096038873625851074707434063628697657670269925864995355797631817390255089

13312232947439303439561613283340728316634982581452268620043077990846881038041873683248009038735962129196336025831207816736737425

33322879296907205490595621406888825991244581842379597863476484315673760923625090371511798941424262270220066286486867868710182980

87280256069310194928083082504419842479679205890881711232719230145558291674679519743054802640464685400273399386079859446596150175

25869658114475685100415686877309037124825353438392853975987494584970500382250124892840018265900562512861876299380444073401423470

62055785305325034918189589707199305662188512963187501743535960282201038211616048545121039313312256332260766436236688296850208839

49614283048473911399166962264994856368523471287329479668088450940589395110465094413790950227654565313301867063352132302846051943

43813998105614006525953007317907727110657834941746426847209561346473277485842382748996687550525043942182321913572230540667153733

74248543645663782045701654593218154053548393614250664498585403307466468541890148134347714650315037954175778622811776585876941680

908203125

We're getting into the somewhat better tetrational numbers, although this number can still be realistically computed, as it still has only about two thousand digits. That means that it can be realistically written by hand; when you write 2 digits per second it would take about 18 minutes. Nonetheless it's still a very big number, far beyond what we can appreciate and somewhere between a septingentillion and an octingentillion in terms of the Latin-based -illions.

777^777

~ 7.19667*10^2245

God is 777777 meters high.

Octingentillion

10^2403

The 800th -illion.

Nongentillion

10^2703

The 900th -illion.

Fzthousand / novemnonagintinongentillion

1000^1000 = 10^3000

An example of a number nameable with the fz- prefix. It's also the -illion right beofre a millillion.

Millillion

10^3003

A millillion is the thousandth -illion, and it's one of the few -illions after the centillion that has a generally agreed-upon name - that's because millillion is the obvious Latin-name continuation from "vigintillion" and "centillion". In fact "millillion" is probably the largest -illion that has an agreed-upon name among number enthusiasts. Even though in Conway and Guy's system it has the slightly different name "millinillion", most other -illion systems either instead opt for "millillion" or use an entirely diferent name.

Milli-untllion

10^3006

This is the 1001st -illion with -illions extended past a millillion - reversing the order of the terms needs to be done for the most logical continuation.

Googom

2000^1000

~ 1.995*10^3301

The thousandth member of the googo- series - this one passes a millillion.

Decyllion

10^4096

This number is known as a decyllion in terms of Donald Knuth's -yllions (see 100,000,000). In one ancient version of the Chinese numeral system this was known as zai (in modern Chinese zai is 10^44), and zai is the largest officially recognized number name in Chinese. For more on Chinese numerals see 10,000.

Googolbell

10^5000

A googolbell is defined by Sbiis Saibian as 1 followed by 5000 zeros. It's a super-huge number that falls somewhere above a millillion, and it's the fiftieth power of a googol and the fifth power of a googolchime. However it's still less than some very huge combinatorical numbers like the famous Hamlet monkey number!

Platillion

10^6000

A platillion is a peculiar illion name found on Jonathan Bowers' -illions page. Bowers says that this name was suggested to him but he has no idea where it came from.

I don't consider "platillion" to be part of Bowers' illion system, since it's the 1999th illion in the short scale and Bowers' system obviously extends upon the short scale, and the 1999th name shouldn't be a milestone name.

10^6000 would be the value of "millillion" (the 1000th illion) using the long scale of -illions instead of the short scale, so perhaps the person who suggested "platillion" to Bowers had the long scale in mind with those names. In any case, this seems like an arbitrary name that is little more than a peculiarity.

M19,937 / 24th Mersenne Prime

~ 4.315*10^6001

This is the 24th Mersenne prime, a number with exactly 6002 digits. It was discovered in 1971 by Briant Tuckerman on an IBM 360 computer.

This number is notable as the period of the Mersenne twister (Wikipedia page), which is by far the most widely used pseudorandom number generator. This means that numbers the Mersenne twister sequence generates repeat with a period of this number. This period is an absurdly huge number (even this many Planck times is about 10^5940 times longer than the age of the universe), and it passes many statistical randomness tests, meaning that the numbers it generates are random for all practical purposes.

Dumillillion

10^6003

The 2000th -illion.

Fugasix

6^6^5

~ 8.0191*10^6050

This is the sixth member of the fuga- series. It has exactly 6051 digits, and it falls somewhere between a googolbell and a googoltoll in terms of Sbiis Saibian's googolisms.

10^8196

This number is equal to one myriad undecyllion in terms of Donald Knuth's -yllion system (see this page). One myriad undecyllion was erroneously given by Andre Joyce (most likely actually Michael Halm) as what a googol is named in terms of Knuth's -yllions. However, this value is much larger than a googol - it's more than raising a googol to the 81st power! In Knuth's -yllions, a googol would actually be called one myriad tryllion quadryllion.

Tremillillion

10^9003

The 3000th -illion. We can continue with "quattuormillillion", "quinmillillion", and so on.

Googolpleiji

3^3^9 ~ 1.505*10^9391

This is a number nameable with the googolple- prefix, a prefix that is similar to the googo- prefix, which was invented by Andre Joyce, but has a far more convoluted history. There are three ways to interpret it:

1. The original verbatim way, where googolple-x = x^(2x)^x where x is written in Roman numerals - that way does not work right.

2. The corrected way, where googolple-x = x^x^x^2 where x is written in Roman numerals - this is the way I normally use googolple-.

3. The -ple- idea, where x-ple-y = y^x where y is written in Roman numerals. This is probably the most logical and the one Joyce used in the revised page, but not as interesting as the second way.

With the second way, which I'm using, we can name this number "googolpleiji" (pronounced /goo-gol-play-jee/), which is just a pronunciation respelling of the unwieldy "googolpleiii".

This is the third member of the googolple- series, and an interesting one in my opinion. It begins 15054164... and ends ...17859227. It can be expressed as 3 to the 19,683rd power, and is larger than googomemem (4000^2000) but less than googomememem (6000^4000), and it's 9392 digits long.

Googoltoll / Great great googol

10^10,000

This number is one followed by ten thousand zeros, a number Sbiis Saibian calls googoltoll, from "toll", the sound bells make at a funeral. It's the tenth power of a googolchime and the 100th power of a googol. It's far far larger than numbers we can actually comprehend, but we can try to imagine it as starting with a sphere of a googol objects, then drawfing the sphere by a factor of itself 99 times. That's really impossible to actually understand, since even a googol objects is well beyond what you can really grasp. Therefore a googoltoll can be seen as a number that is, well, pretty fucking huge ... but tiny in comparison to some FAR MORE MIND-BLOWING numbers!!!

Andre Joyce continues his great googol with the great great googol, great great great googol, etc, allowing us to call this number great great googol.

A googoltoll is also notable for being about as far as we can currently go with testing any single number to be prime - beyond that, special tests need to be used used, which is part of why Mersenne primes are special - they're a lot easier to test for primality than most primes.

M44,947 / 27th Mersenne Prime

~ 8.545*10^13,394

This Mersenne prime was discovered on April 8, 1979 by Harry Lewis Nelson and David Slowinski using a Cray-1 supercomputer. It's the first Mersenne prime to have over 10,000 digits, and therefore the smallest Mersenne prime to be a gigantic prime.

3.5*10^18,726

A lower bound for BB(6) using the busy beaver function (see 4098). This value falls between the 27th Mersenne prime and 2^^5.

2^^5

2003529930................5719156736 with 19,729 digits

This is 2 tetrated to the fifth or 2^2^2^2^2 = 2^65,536. It's a gigantic number with exactly 19,729 digits, and the only non-trivial integer of the form n^^5 that can be written out in full.

Full decimal expansion: http://sites.google.com/site/largenumbers/home/a-1/LNL/265536

Decimillillion / Myrillion

10^30,003

The 10,000th -illion. It's sometimes called myrillion because of the name "myriad" for 10,000, but that name is nonstandard.

Smallest more-than-two-digit repunit prime in base 18

~ 7.06219*10^32,217

This prime number is a string of 25,667 ones in base 18 (equal to 18^25,666 + 18^25,665 + 18^25,664 ... ... + 18^3 + 18^2 + 18 + 1), and it's the smallest string of more than two ones to be prime in base 18 - see the entry for 25,667, the number of digits in this prime in base 18, for why this is special.

My upper-bound on all possible chess games

2.5427*10^33,201

This is my upper-bound on the total number of possible chess games, using the fifty-move rule. I calculated this by first using a maximum of 218 possible moves every turn (218 is the current record most possible moves in any single known chess possition). Then, based on the fifty-move rule and total maximum number of possible captures or pawn moves per player (71), I got 14,296 as an upper-bound of the maximum number of turns a chess game can last (see that entry for a more detailed explanation of how I got that number). Thus we calculate 218^14,296 ~ 10^33,201 as an upper-bound of all possible chess games.

This number is just crazy big, as it has exactly 33,202 digits. This means that it's roughly a millillion times the cube of a googoltoll. Actually it's about 10198 times bigger than that! This is an utterly unbelievable huge number, but small in comparison to numbers we'll examine later.

For more numbers like this see my article on large numbers in probability.

6^6^6 = 6^^3

2659119772................7863878656 with 36,305 digits

This is six tetrated to the third. This number has 36,305 digits, so at this point all the digits placed here would be a bit of a drag to scroll through. Nonetheless it's still small enough to be realistically computed, but don’t expect that to last for much longer.

Full decimal expansion: http://sites.google.com/site/largenumbers/home/a-1/LNL/E6_46656

Hitchhiker's Number

2^276,209, ~ 5.11764*10^83,297

This is a number that appears in The Hitchhiker's Guide to the Galaxy, the same book that made 42 famous. In one chapter, it says that one can survive 30 seconds within the vacuum of space, and the probability of being picked up by a passing spaceship in those thirty seconds is "two to the power of two hundred seventy-six thousand two hundred nine against". The value is much larger than 6^6^6 but smaller than Greg's gazillion.

Sbiis Saibian comments on this number, saying that the improbability seems to be higher than it "should" be - 10^80, the number of atoms in the observable universe, can be considered a very generous upper bound for the probability - even if you make the universe much much bigger, it must be a pretty lonely place for the probability to be this big.

See also 9^9^9, an even larger number that appeared in a work of fiction.

Full decimal expansion: https://sites.google.com/site/largenumbers/home/appendix/a/numbers/E2_276709

Greg's Gazillion

10^86,340

A guy named Greg (source) gave an interesting definition of a "gazillion". He claims that since "gaz" is Latin for "earthly edge", he assumed this to mean it's the Earth's circumference in Greek miles, which is 28,810. Therefore he says that a gazillion should be 1 followed by 28,810 groups of 3 zeros, aka 1 followed by 86,340 zeros. He even bothers to write it out! However, I feel like a "gazillion", if it were a number, should be too big to even write out and have at least a "zillion" zeros. Nonetheless it's an interesting interpretation.

The value is slightly larger than the Hitchhiker's Number. Actually it's just over a millillion times bigger. However on this scale that's considered pretty close.

For another definition of a gazillion see 10^74.

Fugaseven

7^7^6

~ 8.6958*10^99,424

This is the seventh member of the series of numbers formed with the fuga- prefix. It's notable for being quite close to the googolgong and it has nearly 100,000 digits.

Googolgong

10^100,000

A googolgong is 1 followed by 100,000 zeros. It's the thousandth power of a googol, meaning that it's a googol googol googol ... ... ... googol where you say "googol" a thousand times, which is pretty insane. It's a googolism with a rather unsual story.

It all happened when Sbiis Saibian was kid and had recently gotten into large numbers and always talked about them. One day, he mentioned the centillion to his best friend’s father (ironically, he didn’t find out about the googol and googolplex until much later). His best friend’s father then told him that scientists had come up with an enormous number called a “googolgong” which was something like 1 followed by 100,000 zeroes. Saibian was floored and in his mind popped a huge gong that, when struck, would ring for a googolgong years!

The funny thing is that there never even was such a thing as a googolgong; only a googol and a googolplex actually existed and the googolgong was just made up! He was actually incorrectly explaining the much much larger googolplex to Saibian and made that number up out of his mixed-up thoughts. Scientists didn’t even come up with a googolplex; a single mathematician did. I like to imagine that the man thought that scientists came up with a googolgong something like this:

Scientist 1: I finally figured it out!

Scientist 2: Oh goody! What did you find?

Scientist 1: I found out the number of ways this experiment could have gone! I’ve been trying to solve this equation for months!

Scientist 2: Just how big is the number?

Scientist 1: I calculated it as one, followed by one hundred THOUSAND zeroes!

Scientist 2: Oh my goodness! That number is monstrous. We should make a name for this huge number! What are your ideas?

Scientist 1: We should name it after something big! Like ... Google perhaps!

Scientist 2: Wow, great idea! But we should change the spelling so they don’t sue us.

Scientist 1: How about g-o-o-g-o-l?

Scientist 2: That sounds great, but we should end it with something that emphasizes its hugeness!

Scientist 1: I know! We should call it a googolgong!

Scientist 2: This idea is pure gold! We need to publish our studies and allow the world to hear about this huge number we discovered!

Of course, that’s nowhere near the true story of a googolplex.

Nonetheless, Sbiis Saibian has now appropriated a googolgong as his own googolism and uses it as the base of and inspiration for many of his bigger numbers.

Five expofactorial

5!1 = 5^4^3^2^1 = 5^262,144

~ 6.20607*10^183,230

This number is the exponential factorial of 5, compactly expressible as 5 to the power of 262,144. The number has exactly 183,231 digits, so it falls between a googolgong and the 32nd Mersenne prime. This means that it can still be realistically written out, and in fact it's the largest expofactorial that can be written out. It starts 6206069878... and ends ...8212890625.

M756,839 / 32nd Mersenne Prime

~ 1.74*10^277,831

This is the 32nd Mersenne prime, discovered by David Slowinski and Paul Gage on February 17, 1992. The number is the first Mersenne prime with over 100,000 digits, and logarithmically it is fairly close to the famous Hamlet monkey number.

Centimillillion

10^300,003

The 100,000th -illion.

Hamlet monkey number (my version)

~ 1.0381*10^307,383

You've probably heard of the infinite monkey theorem before. It states that a monkey on a typewriter pressing random keys will, after a long bsymbolut finite time, type absolutely every possible string of text at one point or another.

Now imagine a monkey at a typewriter pressing random keys. It would usually type out gibberish but occasionally an intelligible word. This number is the odds against a monkey completely and perfectly typing out the Skhakespearean play Hamlet on its first try. In other words, if we put this many monkeys in a room (we’d need a pretty big room), probably one of them would type Hamlet out on the first try. Many, however, would still type it out, but with a few mistakes.

This number is pretty easy to calculate: my version of this number uses a 47-key typewriter (alphabet, numbers, space, .,?!();:'"/-) and 182,831 characters (including spaces) to get the odds that the monkey would type all the characters right: 47182,831 ~ 10307,383 to one against. There are several variants of this number, some of which only use 35- or 27-key typewriters, and some that use much smaller (likely erroneous) numbers of characters in Hamlet.

Now this number is absurdly huge! Compare it with something like a centilion, itself an unfathomable number. This is about 307,080 orders of magnitude larger than a centillion - the Hamlet monkey number is itself 307,383 orders of magnitude away from one, which seems only a bit bigger than the ratio between it and a centillion! Even multiplying a centillion by itself a thousand times (imagine dwarfing a sphere of a centillion particles by a factor of itself 1000 times) wouldn't quite take you to this number! See my article on large numbers in probability for more on this number, and other numbers like this.

9^(9!)

~ 3.325*10^346,275

One of my old Wikipedia signatures (see 9^9) ended with 99!, which may be interpreted as this big number, though I don't recall ever thinking of it that way. I only interpreted it as the modest 9^9, equal to about 387 million. This number falls between the Hamlet monkey number and the 35th Mersenne prime.

M1,398,269 / 35th Mersenne Prime

~ 8.15*10^420,920

This Mersenne prime was discovered on November 13, 1996 by Joel Armengaud. It's notable because it's the first Mersenne prime to be discovered on GIMPS, the Great Internet Mersenne Prime Search. All Mersenne primes hereafter were discovered by participants in GIMPS.

7^7^7

3759823526................2870132343 with 695,975 digits

7^^3 or 7^7^7. This number has close to a million digits. All of its digits in a text file would take up over a megabyte of space, pretty damn big for a text file. But this number can still be humanly computed, keep that in mind. Googology Wiki has its full decimal expansion.

The Megadigit Range

10^1,000,000 ~ 10^10^99.999

Entries: 61

Millionplex / Milliplexion / Maximusmillion / Googolplux (anonymous)

10^1,000,000

1 followed by a million zeroes is a number that has several different names. I prefer the name "millionplex" which is of course from "million" plus the -plex suffix. Sbiis Saibian suggests an alternate name, "milliplexion", because he never thought millionplex sounded very good. In my opinion "milliplexion" is a cool-sounding name but just not as natural as "millionplex". Two alternate names for this number coined by an unknown author are "googolplux" and "maximusmillion"; the former is also the name of a much larger Joycian googolism.

This number is a borderline case of a hyper-exponential number, a number on the order of a^b^c where c is a reasonably smallish number. The hyper-exponential range (10^1,000,000 - 10^10^1,000,000) corresponds to class 3 numbers, a range from a millionplex to a millionduplex, so this is the boundary between class 2 and class 3 numbers.

Class 3 numbers are numbers that can't easily (or at all, depending on the size) be written out but can be approximated pretty precisely with the leading digits and number of digits - for example, 4^^4 can't be written in full but can have its leading digits and number of digits computed pretty easily. Examples of class 3 numbers are the largest known prime number, trialogue, googolplex, 4^^4, and promaxima.

Fugaeight

8^8^7

~ 10^1,813,917

This is the eighth member of the fuga- series, a number with about 1.8 million digits. It falls between a millionplex and Borges' number.

Borges' number

~ 1.6950*10^1,834,097

This is the number of books in the Library of Babel, sometimes known as Borges' number (named by Robert Munafo). It came from a story of the same name by Jorge Luis Borges, about a universe which is made entirely of a giant library, containing every possible book with 410 pages, 40 lines per page, and 80 characters per line, with 25 different characters (22 letters, space, period, comma), giving 25410*40*80 ~ 101,834,097 books in the Library of Babel. The library at first seems glorious because among those books is information on literally everything - your biography, the biography of anyone you've met, news reports on anything that has happened, a chronology of Earth, your house, etc, written in any language. However, many more of those books would contain false information, ranging from books that botch up a fact or two to books whose information is complete bullshit that seems like someone pulled it out of their ass! Even worse, most of the books would in fact be completely meaningless nonsense to anyone, being a long string of completely random letters. Because of the peculiarities in the Library of Babel, the Library of Babel has been valued as a mathematical thought experiment.

This number is just INSANELY HUGE. It has 1.8 million digits, meaning that you'd need something like a novel to store all its digits! It's a small class 3 number, and it's larger than the number of Planck volumes in the observable universe ... raised to the 995th power. That means you'd need dwarf the observable universe by a factor of its volume in Planck volumes a thousand times to get a universe with Borges' number of Planck volumes! But, as I've said a bunch of times already, this is quite modest to numbers we'll explore soon, not even as big as a googolplex!

See also the Hamlet monkey number and promaxima.

M6,972,563 / 38th Mersenne Prime

~ 4.37*10^2,098,960

This Mersenne prime was discovered on June 1, 1999 by Nayan Hajartwala on GIMPS, the Great Internet Mersenne Prime Search. It's the smallest Mersenne prime to be a megaprime, meaning it's the smallest with over a million digits.

Micrillion / milli-millillion / millinillinillion

10^3,000,003

This number is the millionth -illion. It's a number that has been given multiple names in extensions to the -illions.

It's named "milli-millillion" in Professor Henkle's system for naming one million -illions, and it's the largest -illion in that system. It's a natural-sounding name because milli-milli literally means "thousand thousands", aka "million".

In Conway and Guy's system it's called millinillinillion, which is a name that is certainly less ad hoc than any of the names in Henkle's, but just not as natural of a name as milli-millillion.

In Bowers' system, this number is named micrillion. Why is that?

Notice these numbers and the SI prefixes hidden in them:

decillion is the 10th -illion, deci- divides by 10

centillion is the 100th -illion, centi- divides by 100

millillion is the 1000th -illion, milli- divides by 1000

Therefore it’s natural to continue the pattern call the millionth -illion a micrillion since the prefix micro- divides by a million. Although the name makes it sound like a small number, this system allows for a natural continuation of the -illion names as we will see with later numbers on this list.

Vigintyllion

10^4,194,304

A vigintyllion is the largest -yllion Knuth mentioned in the article where he introduced his -yllions (see this page for more about them). It's one followed by 4,194,304 (222) zeros. It's notable because it's only the 20th -yllion, and yet it's larger than numbers like Henkle's milli-millillion, the largest number named in his system! See my article on them for how Knuth's system is able to work like that.