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### UFNL

Sbiis Saibian's
ULTIMATE
FINITE NUMBERS LIST

Welcome to my Ultimate Finite Numbers List (UFNL). If your here for the first time check out the introduction link which describes the purpose and formatting of this page.

ULTIMATE FINITE NUMBERS LIST
PART I

Numbers expressible within 100 generations of primitive recursive functions or less...

Total Entries: 1352

Before the Beginning

Googologically Negative Numbers Epoch
(-,-10100]
Entries: 27

[Indescribable]

Minus Sam's Number

On February of 2014 a user by the name of SammySpore added "Sam's Number" to the googology wiki without any citation. The page simply read:

"Sam's number is so gigantically huge it cannot be described. It boggles the mind. Actually, it would boggle a megafugafzgargoogolplex minds.

If you want a small glimpse of how big it is, here. Sam's Number is enormously larger than Rayo's Number. It can fill a greagol multiverses. Actually it can fill so much more than that, it is undescribable."

-- SammySpore

Although the text is ambiguous, it has popularly been interpretted to mean that Sam's Number is so big that it is impossible for us to actually describe it in practice no matter what methods we employ, rather than the more modest interpretation that it was simply too large for Sammy to describe. For the former interpretation to make sense we need to assume that the number of possible descriptions in practice is strictly finite. Assuming this to be the case Sam's Number would have to be larger than any number yet described by googologists, or indeed larger than any number that could be described by anyone ever, by definition. If this is to be interpretted this way that Sam's Number is indeed a big "number". But the problem is that an infinite number of numbers would fit that description if that were so. No problem, you say, just let Sam's Number be the smallest such number. Here we have a contradiction however. If Sam's number is defined as the smallest indescribable number, then guess what, ... we have just described it. What's more we can now describe bigger numbers like Sam's Number+1 which should be in theory indescribable. For the purposes of googology this isn't considered good enough to count. However there is nothing wrong with speculating about such number that are literally too big to be described. In fact, we only need to make two presuppositions to make such numbers and inevitability. Firstly assume that whatever we can do in practice is strictly bounded by some finite number. Second assume the information in a number can only be reduced up to a certain point. Believe it or not, the more contentious point is actually the second, but if we agree that we can't condense information indefinitely, then there will strictly be a finite limit on what we can describe in practice.

Since we can't define a unique number with this property of "indescribability", itself not properly defined as that would require a description that would in principle give us a way to describe the indescribable, we can simply assert that there must be such numbers and they lie beyond the boundary of the describable. For the purposes of this entry a negative Sam Number can be defined as any number so much less than 0 that there is no description which can bound it from below except negative infinity. Negative Sam Numbers act very much like negative infinity in that, no matter what negative number we think of, if we are thinking of it, it's not less enough! Weird.

It's debatable whether this entry should even be included on this list, but it acts as a nice way to delineate a boundary that isn't strictly infinite. Whatever other numbers we come up with for the list, they will be describable and therefore exist within the boundaries of negative and positive sam numbers. This entry is more a range than an actual entry, and can just be thought of as the space in the interval (-inf,-describable number). Another neat way to think of it is that its simply the realm of numbers we've yet to describe. So every time we describe a new number we empty the sam boundaries, but since their infinite they never get completely exhausted. In that way we can meaningfully talk about Sam Numbers, as they can act as the current limit of googology.

????????

minus Utter Oblivion

(See Utter Oblivion for a description)

????????

minus Oblivion

(see Oblivion for a description)

-FOOT10(10100)

minus BIG FOOT

The negative of what is the largest number currently recognized by the googology community.

This is the first entry that we can have some confidence is well defined enough to actually be a number. Unlike Sam's Number, it doesn't claim to be indescribable, and unlike Oblivion and Utter Oblivion we don't need to make presuppositions about the nature of information. Instead we define a definite language for describing numbers (and much else besides) and then simply diagonalize over it.

-Rayo(10100)

minus Rayo's Number

The negative of what held the record for "largest number in googology" until October of 2014 when BIG FOOT officially took the title.

-D5(99)

The negative of what's considered to be the largest computable number in googology.

{{L100,10}10,10&L,10}10,10

minus meameamealokkapoowa oompa

(See meameamealokkapoowa oompa)

{L100,10}10,10

minus meameamealokkapoowa

(See meameamealokkapoowa)

-{10100&10&10}

minus golapulus

(See golapulus)

-E100{#,#,1,2}100

minus blasphemorgulus

The negative of one of my largest and most popular googolism's.

-{{10,10,100}&10}

minus humongulus

(See humongulus)

-{10^^^100&10}

minus kungulus

No one but a googologist would ever think up such a number! We can think of this as a VERY VERY VERY "Large" Negative number, though normally it would be called a "very very very small number". As I've argued before however, small should refer to numbers between 0 and 1. We can then break up the negative numbers into "large negative numbers" (numbers between -infinity and -1), and "small negative numbers" (numbers between -1 and 0).

-{3&3&3}

minus triakulus

Here is the negative version. Negative triakulus is inconceivably less than the next entry :)

-{10^^100&10}

minus goppatoth

( See goppatoth )

-E100#^^#100

minus tethrathoth

The tethrathoth is one of the larger numbers in my system. So here is negative tethrathoth. When googologist's invent large numbers they also make it possible to define a whole family of related numbers. For every large number a googologist defines, a reciprocal can be defined, a negative, and a negative reciprocal. So googologist's really get four terms for the price of one! Although I only consider real numbers as relevant to googology because they can be "ordered", one can also use large numbers to create large imaginary numbers, the sums large numbers and large imaginary numbers, or even add reciprocals and negatives into the mix. The number of possible derivative terms quickly multiplies as we include even more unorthodox things such as quaternion and octonion units. For our purposes however, none of this stuff has much baring on googology since i, the imaginary unit, can not be put anywhere along the real axis. In all cases we are just moving away from zero and the only thing that changes is the direction we are moving away from it. In that case the positive direction is just the simplest case and therefore the most efficient.

-{10,10(100)2}

minus gongulus

In an article called "Why Does God Exist?" written by Jonathan Bowers, famed googologist and inventor of array notation, he makes mention of "minus gongulus" in passing to make a point that every number has "trueness". Technically this makes a minus gongulus the least real number explicitly mentioned on Jonathan Bowers' entire site! That's got to count for something!

-E100#^#100

minus godgahlah

(See godgahlah)

-{10,100(1)2}

minus goobol

-{10,10(1)2}

minus iteral

-G(64)

minus Graham's Number

'Cuz if I don't someone else will (See Graham's Number).

-E100##100

minus gugold

-{10,10,100}

minus boogol

-E100#100

minus grangol

-10^^100

minus giggol

-10^^10

minus dekalogue

-10^10^100

minus googolplex

Because if I don't mention it, someone else will. In googology if there is a googolism, there is probably a negative version of it somewhere. The more notorious the number, the more likely that's to be so.

-10100

minus googol
Ditto.

Ordinary Negative Number Epoch
(-10100,0)
Entries: 31

-19,500,000,000,000

Current US National Debt
(As of 2016)

The current US National Debt is estimated at 19.5 trillion. Here is an example of a fairly large negative number with practical significance. Of course such numbers are no where near googological in size! Could you imagine having a googologically large debt? Yikes!!!

-1,000,000

minus million

Imagine having that as a debt; Tell me that isn't a "real number" then!

-459.67

minus four hundred fifty-nine point six seven

This is absolute zero as read in the fahrenheit temperature scale. This is the theoretically lowest possible reading in fahrenheit. This represents no molecular activity whatsoever, which is actually impossible due to quantum effects. What is possible is to approach abritrarily close to this temperature from above. Why is this temperature not simply 0? Because the fahrenheit scale was defined with a different 0 in mind. The Rankine scale corrects this by simply adding +459.67 to fahrenheit, giving a proper 0 point. (See 459.67). This is again an example of a fairly large negative number that comes up in real life.

-273.15

minus two hundred seventy-three point one five

This is absolute zero as read in the celsius scale. This is the lowest possible temperature as read in celsius. The kelvin scale corrects for this by simply adding +273.15 to celsius. (See 273.15).

-40

minus forty

This is a number in which celsius and fahrenheit are equal to each other. That is -40F = -40C. This is a unique point. Fahrenheit defines its freezing point at 32F and boiling point at 212F for a 180 degree difference. Celsius defines its freezing point at 0C and boiling point at 100C for a 100 degree difference. This means that every degree celsius is actually worth 1.8 degrees of fahrenheit. If we multiply by this and correct by adding 32 we can go from celsius to fahrenheit:

F = 1.8C+32

Now assume there is a place at which F=C. We can then solve the following equation:

F = 1.8F+32

-0.8F = 32

F = -32/0.8 = -40

It's interesting how this number arises incidentally and just happens to be negative.

-6.36221590585...

loglog1.000001

double logarithm of one point zero zero zero zero zero one

As the argument of the double logarithm approaches 1 from above, the output approaches negative infinity . Even when we use 1.000001 as the argument however, it doesn't result in a very large negative number. In fact the number of zeroes after the decimal point is roughly the negative number that will result. Thus this is an inefficient method for generating very large negative numbers.

Curiously, if the argument of the double logarithm is larger than about 1.25 then the result is a small negative number, and if it's less than 1.25 but greater than 1 then the result is a large negative number. (See -1)

-1.38307639985...

loglog1.1

double logarithm of one point one

-1.07918124605...

log(1/12)

logarithm of one twelfth

-1.04139268516...

log(1/11)

logarithm of one eleventh

-1

negative one

Negative one is kind of special among the negative numbers. If I was only going to mention a single negative number, this one would be it. It is the square of the imaginary unit: i^2= -1. It also pops up in this very strange equation:

e^(i*pi) = -1

This equation can be used to develop a system of complex exponentiation! For googology, it's purpose is simply to define the predecessor of any integer. It is used explicitly in the definition of the predecessor function:

P(n) = n - 1

Beyond that negative numbers don't really have much use in googology. After all, we aren't interested in making numbers smaller, but larger! However one of the catches to this is that you need to take a step back now and then when defining googological functions or else the function does not terminate. Every googological function must have a base case, and every googological function must make use of the predecessor function so that the evaluation of any expression is eventually forced back to the base case. Hence "minus one" is being used implicitly all the time in googology, even though we usually never think of it as a number in it's own right. Yet without those implicit "-1"s, googology wouldn't even function the way it does. So I'd say some credit is due to negative one.

Just a small note of passing interest: -1 = loglog1.25892541179... = log0.1

-0.954242509439

log(1/9)

logarithm of one ninth

Since 1/10 < 1/9, it follows from the fact that the logarithm is a strictly increasing function that log(1/10) < log(1/9). Thus log(1/9) must be greater than log0.1 which is -1. The absolute value of this number is log9.

Although ordering negative numbers seems confusing at first, just remember that in this case a "larger negative" is less than a "smaller negative". In other words the order is reversed. In the case of negative numbers, the number closer to zero is always greater. This is in contrast to positive numbers where the number further from zero is always greater.

log(1/8)
-0.903089986992...
logarithm of one eighth

log(1/7)
-0.845098040014...
logarithm of one seventh

log(1/6)
-0.778151250384...
logarithm of one sixth

loglog1.5
-0.754262201319...
double logarithm of one point five

log0.2
-0.698970004336...

logarithm of one fifth

log0.25
-0.602059991328...

logarithm of one quarter

loglog2
-0.521390227654...
double logarithm of two

This number has some importance in googology believe it or not. When attempting to compute 2^^6, we find that 2^^6 = 2^2^65,536. Naturally we want to convert this into base 10 form. Roughly speaking we could change the 2's into 10's but that isn't very accurate, especially for a number this small (tetrationally speaking). So instead we use logarithms:

2^2^65,536 = (10^log2)^(10^log2)^65,536 = (10^log2)^10^(65,536log2) =

10^(log2*10^(65,536log2)

log2 is approximately 0.301, so we can simplify 65,536log2 to about 19,728. Since in log2*10^19728 the log2 won't have much effect on 10^19278 it is sometimes ignored. However it has a small reducing factor, that can be accounted for by 10^(19,728+loglog2). Since loglog2 is negative, it means that it slightly reduces the top most exponent to about 19,727.7804056. Interestingly loglog2 is just enough to decrease the top most exponents integer part from 19,728 to 19,727, so it's effect is not completely negligible, especially considering it's a second exponent. In fact:

(10^10^19,727.7804)^3.32 ~ 10^10^19,728.3017

In otherwords, the corrected reduced estimate has to be cubed to get the rough estimate. Considering how large 10^10^19,727.7804 is, you have to imagine that shrinking to an unimaginably small dot amongst exactly that many dots, then imagine that as an unimaginably small dot amongst that many dots to get close to the rough estimate. So you can imagine, there is a big difference from factoring it in, from an ordinary perspective. Although we routinely ignore huge differences like this in googology (numbers are often so far apart that such differences are insignificant), such accuracy for smaller numbers is sometimes necessary to settle a close call. So the number loglog2, although negative, does in fact serve a practical purpose in googology.

log(1/3)
-0.47712125472...

logarithm of one third

loglog3
-0.321371236131...

double logarithm of three

log0.5
-0.301029995664...

logarithm of a half

This number is also the additive inverse of log2. It turns out that |logx| = |log(1/x)|.

loglog4
-0.22036023199...

double logarithm of four

log(2/3)
-0.176091259056...

logarithm of two thirds

Two thirds is the simplest non-unit fraction. It's logarithm is -0.176091259056...

loglog5
-0.155541461208...

double logarithm of five

loglog6
-0.108935980359...

double logarithm of six

loglog7
-0.073092905527...

double logarithm of seven

loglog8
-0.044268972935...

double logarithm of eight

loglog9
-0.020341240467...

double logarithm of nine

loglog9.9
-0.001899759965

double logarithm of nine point nine

loglog9.999999
-0.000000018861...

double logarithm of nine point nine nine nine nine nine nine

As the argument of the double logarithm approaches 10 from below the result gets arbitrarily close to zero from below. This would be an example of a very small negative number. If this was your bank account balance it would be a debt so small that it would be virtually indistinguishable from breaking even. If the debt were to compounded at 7% interest annually, it would take roughly 195 years for you to owe the bank a whole penny!

-1/E100#^^#100

negative reciprocal of a tethrathoth

This is the smallest negative number on this list. It is one of the four possible flavors of a "tethrathoth" using negatives and reciprocals. Since it is both negative and reciprocal it is probably the strangest out of the four possible combinations.

Singularity

0

zero

"Zero" can be thought of as the smallest quantity possible. After all you can't have less than nothing, or can you? Sometimes "negatives" are thought of being "smaller" than zero, but this seems to defy logic since you can't have something smaller than something which is infinitely small to begin with! It is better to think of negatives as "less than zero" rather than "smaller than zero".

Negative numbers do not relate so much to the concept of quantity as to "position". An axis can have a "central point" at zero, with negatives to one side and positives to another. Quantity however can only travel in one direction from zero, namely, towards positive infinity.

The exclusion of negatives from this list much easier to justify than the exclusion of zero (although I've decided to include them none the less). This is because the negatives really serve almost no purpose in the large numbers field. In order for algorithms to terminate it is necessary to have a minimum value for every argument. This means we have to choose a least number allowable as an argument. Common choices for the minimum value are 1 and 0. If however we allow any integer value, including negatives, we kind of drop the bottom out so to speak and the result is either a function which does not terminate for all values, some values, or requires at least 3 rules (a base case, a less-than-base case, and a more-than-base case). None of these options provides any advantages over simply deciding on a minimum integer value for the function. This is why zero has some importance in the large number field, because it serves as a beginning value. Some of the uses of zero in googology are as the minimum value of an argument in the Ackermann function, and the minimum order of a separator in array notation. It is also the minimum arity of an array. In cantor's system of ordinals, it is the smallest possible ordinal.

Zero crops up when attempting to extend the hyper-operators to all integers. By definition a^^1=a. Since logaa^^n = a^^(n-1), for n>2, we can define this as a law for all integer values. Thus we obtain that a^^0 = logaa^^1 = logaa=1. Thus a^^0=1. Next let a^^(-1) = logaa^^0 = loga1 = 0. Thus we find that any positive integer>1, a, that a^^(-1)=0.

Zero
is equal to the double logarithm of 10: loglog10=0. It is also equal to log1.

Zero sometimes leads to undefines as in 1/0 and log0. For this reason I have occasionally been wary of zero. However we rarely have such problems in googology since the functions we work with are usually integral and not continuous.

In some respects zero serves as a natural starting point for large numbers. We can think of it as the starting line, and any positive real becomes part of the race coarse. The end goal can be thought to be at infinity, although it is probably better to think of this as a race with no end goal! We can begin this race, but we can never finish it! Better get going then ...

Googologically Small Epoch
(0,10-100)
Entries: 8

10-E100{#,#,1,2}100

blasphemorgulminex

This micronym is *slightly* smaller than the blasphemorgulminutia using my own special suffix -minutia. This uses Conway's -minex prefix where (n)-minex = 10-n. This number is 0.0000...00001 where there are a blasphemorgulus minus one zeroes between the decimal point and one. This number is incomprehensibly small in a way analogous to how the blasphemorgulus is incomprehensibly large.

1/E100{#,#,1,2}100

blasphemorgulminutia

This is an example of a micronym, a special name for an extremely small number. micronyms are rarer than macronyms in googology but there are a few examples. The most famous example is Conway's googolminex.

1/E100#^^#100

tethrathoth-minutia

1/G64

reciprocal of Graham's Number

Let's begin our journey very slowly. We will have plenty of time to accelerate towards the infinite! Our first non-zero entry is a number so incredibly small that you'd have to multiply it by Graham's Number (seen later on this list) to get 1. By necessity this number must begin as 0.0000000000000000000000000000000000000000000000000000000000000000............ but we have no way of knowing exactly what the first non-zero digit is, or where exactly it would occur! It's that small! As far as the race towards infinity, it's as if we haven't even left the starting line yet, but in fact we have.

2/G64

two divided by Graham's Number

This number is just as far from our last entry as our last entry was from 0. If we want to get somewhere however we're going to have to pick up the pace because we'd have to have a Graham and one entries just to reach one!

1/(10^10^100)

googolminex

Conway and Guy have suggested the name "googolminex" for the reciprocol of a googolplex (seen later). It's an example of an extremely small number with an actual name! One of the many consequences of being able to define very large numbers, is that we can also define very small ones. We simply have to take the reciprocal of some large number, and we get it's inverse: a number that is just as small as the original number was large! You can imagine this number as 0.0000000000000000000000000000000000000..................................000000001 where there a googol-1 zeroes after the decimal point. This number is tremendous when compared to the reciprocal of Graham's Number, and yet it is still mind-bogglingly "googol-scopic". If we were to continue with the multiples of a googolminex, such as two googolminexthree googolminex, etc. We would never even have a hope of reaching 1, let alone actually large numbers. So once again we must pick up the pace...

10^-110

googol-minutia-speck

This is the smallest of my googolism's explicitly listed in the ExE Numbers list. It is googolism #32.

10-100

0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001

googol-minutia

Here is an extremely small number so large that I can actually write out it's decimal expansion in full. This is the reciprocal of a googol (seen later). It can be most compactly defined as 10^-100. This number is smaller than some of the very smallest numbers in physics. 10^-100 meters would be a distance so small that it couldn't even theoretically be measured, no matter how powerful our particle accelerators got. The reason for this is because quantum effects would distort space and time so much that no meaningful measurements could be made! This number, though vastly larger than the previous entries, is still uselessly small. Even if we were to continue with the multiples of this number we still wouldn't have any hope of reaching 1! Let's now explore some really small numbers in science...

Ordinary Small Numbers Epoch
(10^-100,1)
Entries: 29

10-43

Planck Time (in seconds)

This is the smallest time scale that can theoretically be measured. You can think of this as the length of time for a single "frame" of the universe. Another way of looking at it is that the universe has a "frame rate" of about 10^43. See 1.2x10^-17 for comparison.

10-35

0.00000000000000000000000000000000001

Planck Length
(in meters)

This is the smallest distance that can theoretically be measured. This distance is so small that even the diameter of an electron would be about 10^17 Planck Lengths! In string theory this is said to be the size of a typical "string particle". It would also be roughly the size of the curled up dimensions. You can think about it this way. A string is as small relative to an electron as an electron is to us, making a string a "particles particle".

10-18

0.000000000000000001

electron diameter
(in meters)

This is the theoretical diameter of an electron. In truth however the meaningfulness of this is doubtful. Physicists have long since ceased thinking of sub-atomic particles as little billiard balls, and prefer to think of them as mathematical points with an associated "field" surrounding them. The idea that an electron has a size however can be given some meaning based on the proximity another particle has to be in order to interact with it. This distance of interaction should be around this value. This distance is also known as an "attometer".

1.2x10-17

0.000000000000000012

Smallest Measured Time
(in seconds)

According to a wikipedia article[1], this is the smallest amount of time scientists have ever actually measured. Interestingly this is equivalent to about 10^26 Planck Times. As you can see we are still a long way from being able to measure changes by the Planck Time!

1/100

0.01

one hundredth

One hundredth of something is considered to be a pretty small precentage of anything. This number represents the same concept as 1%. However, this number is still huge compare some of the really small numbers in science. (check out the previous entries)

1/12
0.08333333333333333333333333333...
one twelfth

1/11
0.090909090909090909090909090909...
one eleventh

1/10
0.1
one tenth

1/9
0.1111111111111111111111...
one ninth

1/8
0.125
one eighth

1/7
0.142857142857...
one seventh

1/6
0.1666666666666666666666666666...
one sixth

1/5
0.2
one fifth

1/4
0.25
one quarter

log2

0.301029995663981195213738894724...

logarithm of two

Here is a small number with an important connection to googology. The common logarithm of two, or log2 is a number such that 10^log2 = 2. The upshot of this is that it allows us to change the base of a power tower of base 2 to base 10. For example, we can estimate 2^65,536 to a high degree of accuracy, obtaining the correct order of magnitude, simply by using this number. Observe:

2^65,536 = (10^log2)^65,536 = 10^(65,536log2) ~ 10^(65,536*0.3) ~ 10^19,661

This is pretty accurate despite the very rough rounding. The number 2^65,536 actually has exactly 19,729 digits.

The common logarithms of the primes are also useful because we can compute the logarithms of other positive integers by decomposing them into their prime factors and then using the laws of logarithms to figure out the value. As an example, we can compute the logarithm of 8 easily if we know the logarithm of 2:

log8 = log(2^3) = 3log2 ~ 3*0.301 = 0.903

Logarithms are indispensable to the study of tetrational class numbers and allow us to make estimates and bounds on numbers like Skewe's Number, or Ballium's Number. A lot can be learned about numbers of this size, without an impractical amount of computation. Thus even small numbers play an important role in googology.

1/3
0.333333333333333333333333333333333...

one third

This is the "decimal expression" for the fractional value, 1/3. This value is of some importance in my early exploration of mathematics and large numbers. This number has the somewhat irksome property that it can not be expressed as a "finite decimal expression". That is, it is not expressible in the form A/B where B is a whole power of 10. The decimal expression for 1/3 can be thought of as an infinite number of 3s following the decimal point. The discovery that the long division of the fraction would not terminate was something of a revelation to me. I was suspicious of infinity to begin with, but up until that point I figured infinity was something "out there". Now I saw that infinity could also be invoked even with the very close regions between 0 and 1.

There is also a curious feature of the decimal expansion of 1/3. It follows from the definition of 1/3 that 3(1/3) = 1. What happens if we multiply the decimal expansion by 3:

3(0.333333...) = 0.999999...

It turns out that 0.999999... is the same as 1. This may seem incredible. However consider what happens when we had 0.333333... to 0.999999...:

0.333333...+0.999999... = 1.333333...

We get 1+1/3. This might seem like a contradiction since 0.3+0.9=1.2, 0.33+0.99=1.32, 0.333+0.999=1.332, etc. However since neither decimal expression is finite, the "2" never "shows up" and the result is an infinite sequence of 3s. This just illustrates the rather counter-intuitive properties of the infinite.

log3
0.477121254719662437295027903255115...

logarithm of three

Normally written as log3, it is the unique real number such that 10^log3 = 3. With this number, and log2, we can approximate many common logarithms without a calculator. For example:

log6 = log(2*3) = log2+log3 ~ 0.301+0.477 = 0.778

log9 = log(3*3) = log3+log3 = 2log3 ~ 2(0.477) = 0.954

log27 = log(3^3) = 3log3 ~ 3(0.477) = 1.431

We can compute any common logarithm in this manner as long as the integer decomposes into 2s and 3s. We would not be able to compute log5, log7, log10, log11, etc. with only log2 and log3.

0.5

one half

0.5 is the decimal expression of the unit fraction, 1/2. This fraction is the simplest possible that can not be simplified as a whole number. It lies exactly half way between zero and one. It is important enough that it even gets the special name "one half".

0.5772156649...

lim(n->inf)sigma(1/i,i,1,n)-ln(n)

Euler-Mascheroni Constant
"gamma"

This seemingly innocuous number between 0 and 1 is a mysterious constant that emerges as the error between the harmonic series, an a relatively "good" approximation of it, ln(n). (See later for some of the harmonic sums, also on this list). The harmonic series is 1/1+1/2+1/3+1/4+... It is divergent, but it is a very very slow divergence. In fact it can be bounded logarithmically. Since it's such a slow growing function, one thing that can be done to get large numbers from this series is to ask what the smallest partial sum that exceeds a certain number is. This can often be very large. For example, to exceed 10 takes at least 22,027 terms. We can get even crazier things with larger numbers. It takes approximately 10434,294 terms to reach a mere 1,000,000. To get these values we need some method of approximation. recall that ln(x) = int(1/t,t,1,x). Using riemann sums we can show that the harmonic series 1/1+1/2+...1/n > ln(n). We can use a similar trick to show it's less than ln(n)+1. Let H(n) be the nth partial sum of the harmonic series, a natural question is, if its true that ln(n) < H(n) < ln(n)+1 , then how does H(n) behave within this interval. It turns out that H(n) approaches ln(n)+gamma, as n goes to infinity. But another way lim(n->inf)(H(n)-ln(n)) = 0.5772156649... This number occurs in physics, often in unexpected places. For our purposes however it is mainly interesting as a way to get accurate bounds on H(n) for very large n. So we can estimate that H(22,027) ~ ln(22,027)+0.5772156649 ~ 10.5772399175. So we know this is at least enough terms to exceed 10. Like wise H(10434,294) ~ 434,294*ln(10)+0.5772156649 ~ 999,999.46 so this is a bit of an underestimate.

log4
0.602059991328...
logarithm of four

2/3
0.6666666666666666666...
two thirds

This is the simplest non-unit fraction. It was important enough that there was a special symbol for it in eygptian mathematics.

log5
0.698970004336...
logarithm of five

log6
0.778151250384...
logarithm of six

log7
0.84598040014...
logarithm of seven

log8
0.903089986992...
logarithm of eight

log9
0.954242509439...
logarithm of nine

log9.9
0.995635194598...
logarithm of nine point nine

log9.999999
0.999999956571...
logarithm of nine point nine nine nine nine nine nine

As the argument of the logarithm approaches 10 from below, the output approaches the value of 1.

0.9999999999999999999999999999999..................999999999976974149....

w/googol-1 9s

10^(-10^(-10^100))

googolduminex

A series can be created with the googol and plex suffix by repeatedly appending it creating the sequence:

googol, googolplex, googolduplex, googoltriplex, ...

This sequence is strictly increasing and grows tetrationally. What happens if we do the same thing with the minex? We get a series of googolism's : googol , googolminex , googolduminex, googoltriminex, etc. This is not the same as having the reciprocals of googolplex, googolduplex, googoltriplex, etc. That would be googolminex, googolpleximinex, googoldupleximinex, etc. (See googologically small numbers epoch). Rather we get a behavior that remains strictly bounded in the interval (0,1). To understand what googolduminex, remember that it is googolmineximinex. And that just as:

googolminex = 10-googol

we have:

googolduminex = 10-googolminex

googolminex is a number only slightly larger than zero, so minus googolminex is a number slightly less than 0. Recall that 10^0 = 1. So 10^(-0.000000.....0001) will be a number just a little smaller than 1. Since 10^x approaches 1+xln10 as x approaches 0, we can conclude that googolduminex is close to 1-(ln10)*googolminex = 1 - ln10/googolplex. ln10 = 2.30258509299... so this amounts to 0.99999....9999976974149... where there are a googol minus one 9s, far more than could be written out. The digits after this will match perfectly with 10-ln10 up to approximately a googol more digits, and then diverge. What's interesting is we got this number, just ever so less than 1 (yet not equal to 1 unlike 0.999999...) by simply applying a common googological suffix twice.
(See googoltriminex for more insanity).

NEITHER LARGE NOR SMALL

1

one

This is a number SO AMAZINGLY LARGE ... THAT it's not small! In fact, it's the smallest ... (Read More)

Next up, the Large Numbers ...

SBIIS SAIBIAN'S ULTIMATE LARGE NUMBER LIST
(ULNL)
BEGINS HERE

LARGE NUMBERS
(1,∞)

Small Superuniary Epoch
(1,2)
Entries:24

10^10^(-E100{#,#,1,2}100)

blasphemorgulminexiplex

This is an honest to goodness googolism formed by combining a "blasphemorgulus", a googolism created by me, and applying conway's minex suffix followed by the plex suffix.
It's a googolism that is large...er than 1 ... barely :/

Perhaps it looks like this number should be huge! Well it IS larger than 1 at least, but of course you mean googologically large. To understand why its so small look at the function 10^10^x, and let x approach negative infinity. As x approaches negative infinity 10^x approaches 0, and therefore 10^10^x approaches 10^0 which equals 1. Stop it at any point however and we find the result has to be greater than 1, since 10^x is always greater than zero, it follows that 10^10^x is always greater than 10^0 or 1. This number begins 1.000000..... with a blasphemorgulus minus one zeroes and is then followed by 230258509299... matching up with ln10 perfectly, up to some point , and then following up with some mysterious digits we may never know.

This is only one example of the weird things that can happen from mixing up the -minex and -plex prefixes. For a full proof of the convergence of the digits as well as a further discussion of some of the interesting properties of these mixed typed googolisms (mixing macro and micro operators) click here.

1 + 1/(E100#^^#100)

one plus the reciprocal of a tethrathoth

Let's begin our exploration of large numbers with a really really REALLY small example. We're in no rush and we have plenty of time to get to the real whoppers. Consider the number "one plus the reciprocal of tethrathoth". This number is inconceivably close to one, yet just ever so slightly larger. It starts out 1.000000... but it's an inconceivably long time until you get anything other than zero, though eventually it must reach ...000000001, on account of the fact that a tethrathoth is just a mind bogglingly large power of 10. You would have to raise this number to a tethrathoth to get a number just shy of e. Despite how small of a large number this is, keep in mind that there is an infinite number of smaller large numbers in the infinitesimal space between 1 and 1+1/(E100#^^#100).

(1+10-10100)G(64)-1

Graham's root of googolminex and one

Let's continue the discussion with a real whopper of a "large number"! This is a number so mind bogglingly large that by raising it to the power of a mere Graham's Number you get the ginormous result of a googolminex and one! What's a googolminex and one? It's 1 plus the reciprocal of a googolplex! More impressed by how close this would be to one than by how "large" it is? That's understandable. It's a stretch to call this large, except to say that it's definitely larger than 1. None the less, 1 to any finite power is still 1, so in comparison this number is quite amendable to exponential growth! And just think of the infinite number of yet smaller large numbers whose Graham's power doesn't even come close to the massive googolminex and one. Even if we raised the previous entry to a godgahlah it would still be way closer to 1 than "Graham's root of googolminex and one"!!!

This was the smallest example I gave on my "Very Small Very Large Numbers" blog entry as an extremely small Large Number. The number must begin with 1.000000000000000000000............ but we can not compute what the first non-zero digit is after the decimal point, just as we can't compute the leading digit of Graham's Number. We also can't compute the exact number of zeroes before the first non-zero digit, though it must be about Graham's Number. All that being said it might be a bit tricky to remember that this number is virtually equal to 1!

Now I know your brain is still reeling from the sheer massiveness of this number, but wait until you see what comes next!

1 + 1/(E100#2)

googolminex and one

This number is MASSIVE! It's is equal to 1 plus the reciprocal of a googolplex. It's a number so staggeringly gargantuan that if you raise it to the miniscule power of a googolplex you get a value just shy of the unfathomably large number e! Consider that up until now, raising the previous entries to a googolplex would not even come close to reaching 1.0000000000000000000000000000000000001 let alone a number larger than 2! Still not impressed?! Hmm, time to bring out the big guns ...

1.0000000000
000000000000000000000000000000000000000000000000
00
0000000000000000000000000000000000000001

one plus the reciprocal of a googol

Alright how about this. Take 1 and add the reciprocal of a googol. This number is so B-I-G that you only have to square it a mere 332 times to get a value exceeding 2! Still not big enough?! Don't worry, we've just gotten started ...

1.000000000000000010903970549325460813650942266345982807...

Time dilation factor of person who is walking versus standing still

This is the time dilation factor in general relativity of a person who is walking verses a person who is standing still. I assume that a walking person travels at about 1.4 m/s which is about 1/214,137,470 the speed of light. One way to think about this is as follows:

Imagine two nigh immortals who can live for billions and billions of years. One decides to go for a walk for the next billion years while the other decides to stand still for just as long. At the end of the billion years when the walker again stops to meet up with the one who stood the whole time the difference in the amount of time that elapsed for them due to time dilation would only differ by about 1/3 of a second! The ratio of the larger elapsed time to the smaller elapsed time will be the value of this entry.

This number is so large you only need to square it 56 times to get a value exceeding 2.

1.0000006931...

millionth root of two

Let's pick up the pace and talk about a real super giant! This is a number so large that if you raise it to the millionth power you get 2! You only need to square this number 20 times to get a value exceeding 2. That's still not large, you say? But just think of all the smaller numbers still greater than 1, whose millionth power doesn't even reach 2. In fact, the millionth power of every previous entry is not even as large as this entry!

1.0013784192...

Ratio of Neutron to Proton mass

How's THAT for a large number! Although both Protons and Neutrons are have more or less the same mass, the Neutron is just slightly heavier than a Proton by a factor of 1.0013784192...etc. To put this in perspective you only need to square this number 9 times to exceed 2.

1.0026654123...

Ratio of troposphere to diameter of the earth

This is the ratio of the radius of the "troposphere" (the lowest part of the earths atmosphere) to the radius of the earth. This means that the earth's radius is only increased by a mere 0.2% by the enveloping troposphere! Our "sky" is little more than a thin film on the earths surface! But that's still bigger than the ratio of the Neutron mass to the Proton mass, so that's got to count for something right?! Also you only need to raise this number to the 261st power to get a number exceeding 2.

1.01

One point oh one

This is the numeric value of a 1% increase. An increase of 1% is considered to be a very small improvement. However, let's consider how large it is. Let's say you had a bank account with a 1% annual interest rate. How many years would you have to wait for you investment to double? It would only take a mere 70 years! Not impressed? Well 70 years might make for a really long term investment, but in other contexts 70 years might be cause for concern. An inflation rate of 1% would mean the price of all commodities would double every 70 years, quadruple every 140 years, etc. If the price of gasoline were to increase exponentially at a 1% annual rate, in only 40 years you'd be spending 50% more on gas. This might seem like a slow rate, but it only takes a few extra percents annually to get a dramatic change in growth rate. See 1.07 as an example.

1.05946409436...

twelve root of two

The average pitch ratio between successive half-steps in western 12-tone music. This number can be computed as the 12th root of 2: 2^(1/12). This is a number large enough that pitches in this ratio can be easily distinguished. One way of looking at this numbers size is that you only have to raise it to the 12th power to get 2. You only need to square it 4 times to get a value exceeding 2.

1.07

one point oh seven

This is the numeric value of a 7% increase. This sounds small as in "the population is increasing by 7% annually". However this value is notoriously deceptive. At a 7% rate it would only take 10 years for the initial value to double! After 20 years the value will be 4 times as large as originally, and after 30 years 8 times as large! All of this from a mere 7% annually.

1.09407190229...

This is the ratio of the radius of the exosphere to the radius of the earth. The exosphere is the very last layer of our planet's atmosphere. Beyond this one enters into true outer space. The exosphere only increases the radius of the planet by a mere 9.4%. That's sizable enough that it would be visible if it were highlighted. However the end of the exosphere can't really be seen clearly since the exosphere is very very thin in comparison to the lower and denser layers of the atmosphere.

1.1

one point one

This is the numeric value of a 10% increase. This is a change significant enough that it is noticeable almost regardless of what is considered. Getting 10% extra for the same price is touted as a bargain. Still too small? We're getting there. Before long we'll be blasting off to the stars. Enjoy the smallish numbers while they last!

1.11178201104...

convergence value of iterated exponentiation of one point one

This value is the limit of the infinite power tower of base 1.1. To approximate it begin with 1.1, and let this be Stage 1. Next compute 1.1^1.1 and let this be Stage 2. Next compute 1.1^1.1^1.1 and let this be Stage 3. For each successive stage just take 1.1 and raise it to the power of the previous result. The limit of this infinite sequence is 1.11178201104...

1.21

one point two one

This is the result of 1.1^2. This is very easy to compute by hand as 1.1+.11 = 1.21. The digits match up with the second row of pascals triangle. They are also the coeffients of x^2+2x+1 which is the product of (x+1)(x+1).

1.331

one point three three one

This is the result of 1.1^3. This is also very easy to compute. Since 1.1^2 = 1.21, 1.1^3 = 1.21+0.121=1.331. Again the digits match up with the third row of pascals triangle and the coefficients of x^3+3x^2+3x+1 which is the result of (x+1)^3.

1.4142135623...

square root of two

1.41421... better known as the square root of two. It's a number "so large" that its square is equal to 2, the first integer after 1. Still feeling underwhelmed?! Tough crowd! Well, we still got a loooooooong way to go. The square root of two is notable for being the first number proven to be irrational. That is, it can be shown that it is not a ratio of integers. It lies between 1 and 2, since 1^2=1 and 2^2=4. It represents the logarithmic half-point between 1 and 2. It is also a definite tipping point on this list as entries begin to accelerate rapidly after this...

1.44466786101...

e to the e to the negative one

This is the number e^e^-1. It has a number of interesting properties. It is a solution of x^e = e, which seems impossible. But notice: (e^e^-1)^e = (e^e^0) = e^1 = e. This is the largest real, greater than 1 for which an infinite power tower converges. Interestingly, an infinite power tower of this number has the value of e. Another important property, for the study of large numbers, is that b^x > x for all real x, provided b > e^e^-1. Thus 2^3 > 3, but 1.1^3 < 3 because 1.1^3 = 1.331. Another way of thinking about it is that b^^n will grow without bound provided b > e^e^-1. You can think of it as a number, "so large" that a power tower with a base of anything higher will grow without bound as the number of terms increase.

1.4641

one point four six four one

This is equal to 1.1^4. Since 1.1^3 = 1.331, it follows that 1.1^4 = 1.331+0.1331 = 1.4641. The numbers 1,4,6,4, and 1 are the numbers on the 4th row of pascals triangle, and these are also the coefficients of x^4+4x^3+6x^2+4x+1, the result of (x+1)^4.

1.5

one and a half

Better known as 3/2. This number is the sum of the first two terms of the harmonic series. This makes it the 2nd Harmonic number. That is, it is equal to 1/1+1/2. It's is also the number exactly half way between 1 and 2. This halfway point is larger than the square root of two, which is the logarithmic half point.

1.5 might seem small but consider this: if you saw someone 50% taller than you, you'd think they were tall regardless of their actual size.

1.6180339887...

golden ratio

This number is known as the golden ratio. One way to explain it is as follows: cut a line segment such that the ratio of the larger part to the smaller is the same as the larger to the whole. It turns out that there is a solution to this problem, and that solution is the golden ratio. If the ratio between the larger and smaller part is the golden ratio, then so will be the ratio between the larger and the whole. The golden ratio can be expressed exactly as [sqrt(5)+1]/2. That is, half the sum of the square root of 5 and 1. The golden ratio has a rather unusual property. It's square is exactly one more than itself. The golden ratio is a number "so large" that the distance between itself and its square is 1!

1.83333333333...

third harmonic number

This is 11/6 or 1+5/6. It is the sum of the first 3 terms of the harmonic series. ie. 1/1+1/2+1/3.

Palpable Epoch
[2,100]
Entries: 38

2

two

I like to say that "two" is the very first large number, since it's at least larger than one. In truth 2 is a number that rests comfortably in the mind. It is actually one of only a handful of truly small whole numbers. By "truly small" I mean that these are the few whole numbers that psychologically seem small to us. These are numbers that we can grasp with our innate number sense...(READ MORE)

2.08333333333...

fourth harmonic number

This the value of the 4th harmonic number, the sum of 1/1+1/2+1/3+1/4. It's notable for being the smallest harmonic number to exceed 2. The harmonic series grows perpetually slower. Getting to 3 in this manner proves to be a bit more difficult.

2.283333333333...

fifth harmonic number

The sum of 1/1+1/2+1/3+1/4+1/5.

2.45

sixth harmonic number

This number is notable for reducing to a relatively simple terminating decimal. It is equal to 1/1+1/2+1/3+1/4+1/5+1/6. This sum equates to 147/60 or 2 + 9/20.

2.718281828459...

e

This is the transcendental constant, e. The value can be obtained by taking the sum of the inverses of the factorials:

e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + ...

Another important definition is:

e = lim n-->oo (1+1/n)^n

It is the base of the natural logarithm. The derivative of the function e^x is e^x. This means that the slope at any point along the curve is equal to the x coordinate of that point. An interesting consequence of this is that the value of e^n, where n is a counting number, must be larger than the nth triangular number plus 1. ie. e^1 > 2, e^2 > 4, e^3 > 7 etc. "e" is really quite a small magnitude, lying between 2 and 3. "e" comes into play in the large number field in the definition of Skewes' Number, Ballium's Number, studying the infinite power tower, etc.

3

three

M2

"3" is the 2nd triangular number. It is one of the few numbers that can be understood with our number sense alone. It is one of the larger examples of a "truly small number".

3 is also the 2nd prime prime, and the smallest mersenne prime (primes of the form 2^p-1 where p is prime. The next mersenne prime is 7). 3 is the sum of 1+2, 2+1, the product 1*3, 3*1, and also is equal to the expressions 3^1, 3^^1, 3^^^1, 3^^^^1, etc.

3.01987734488...

eleventh harmonic number

This is the smallest harmonic number greater than 3.

3.141592653589...

π

A transcendental constant defined as the ratio of the circumference of a circle to its diameter. It is usually denoted "pi", and it lies somewhere between 3 and 4, making it a rather small quantity. It can be approximated as 22/7, but it can not be represented as the ratio of integers. It sometimes crops up mistakenly in large number discussions as an example of a large number. This is because the sequence of digits is never ending, and so it is assumed to have "more digits" than any other number. Usually someone will point out sarcastically that 3.2 is bigger, and an even more impressive number would be 4! A more clever use of pi for a large number might be to say "the largest number is pi with the decimal point removed". However such a number wouldn't be finite and therefore would not even count as a legitimate number (googologist's ban infinities from the discussion as it tends to be a game-breaker, and for other considerations of well-foundedness). Other than that pi doesn't come up too frequently in googology, though it is part of the definition of Ballium's Number.

4

four

This number is probably the largest number that most people can perceive directly with their number sense. It crops up over and over again with the operators. For example: 2+2 = 4, 2x2 = 4, 2^2 = 4. This pattern continues to the hyper-operators with 2^^2=4, 2^^^2=4, 2^^^^2=4 etc. A set of 4 things often has a completeness to it. There are the 4 seasons, the 4 directions (north, west, south, east), the 4 corners of a square, and so on.

Four is also the 2nd tetrahedral number. Just as a nth triangular number is the sum of the first n positive integers, the nth tetrahedral number is the sum of the first n triangular numbers. Since the first and second triangular numbers are 1 and 3, it follows that the 2nd tetrahedral number = 1+3 = 4.

Four is the 2nd square number. It is also the 2nd Busy Beaver number, since BB(2) = 4. It's equal to the sums of 1+3, 2+2, and 3+1. It's also equal to the products 4*1, 2*2, and 1*4, and the exponential expressions 2^2, and 4^1. It is also equal to 4^^1, 4^^^1, 4^^^^1 etc.

5

five

This number is large enough that it is difficult to perceive at once. For example:

ooooo

It is difficult to tell there are 5 o's above without counting them. 5 is usually perceived as 2 and 3 or sometimes 4 and 1. 5 is also a number large enough that it takes about a second to count to. In certain contexts, 5 can be a lot. Having 5 children is a lot, eating 5 pancakes would make you pretty full, etc.

5 is the third prime number, the count of fingers on a single human hand, the sums of 1+4 , 2+3, 3+2, and 4+1. It's equal to the products 1*5 and 5*1, and the exponential expression 5^1. It is also equal to 5^^1, 5^^^1, 5^^^^1 etc.

23-2

6

six

Six is the 3rd triangular number since 6 = 1+2+3. It is also the smallest perfect number since it is equal to the half the sum of it's factors, 6 = (1+2+3+6)/2. The next perfect number is 28.

6 is equal to the sums 1+5, 2+4, 3+3, 4+2, 5+1, the products 1*6, 2*3, 3*2, 6*1, and the expressions 6^1, 6^^1, 6^^^1, 6^^^^1 etc.

7

seven

M3

Seven is the fourth prime number. It is also the 2nd Mersenne Prime, and is equal to 2^3-1. (The next mersenne prime is 31).

It's equal to the sums 1+6, 2+5, 3+4, 4+3, 5+2, 6+1, the products 1*7, 7*1, and the expressions 7^1, 7^^1, 7^^^1, 7^^^^1 etc.

8

eight

2^3. The 2nd Cubic number.

9

nine

Nine is the 3rd Square number.

10

ten

The count of all the fingers on both hands. This is a number which lies outside of our ability to perceive directly, and begins to look like a multitude. It is notable for being the base of our decimal system. It is also the number of digits: 0,1,2,3,4,5,6,7,8, and 9.

10 is also the smallest number legally expressible in Hyper-E Notation as E1.

11

eleven

Eleven is the 5th prime number. It's the smallest positive integer greater than 10. It's name means literally "one left"

12

twelve

Twelve is a popular number in numerology. There are twelve astrological signs, there were twelve disciples of Jesus, twelve tribes of Israel, twelve hours on a clock, etc. Twelve is a highly factorable number, which probably accounts for it's special cult status. It's factors are 1,2,3,4,6, and 12. It's equal to the products 1*12, 2*6,3*4,4*3,6*2, and 12*1.

It is common to package things in groups of 12. When a pack contains twelve it is said to contain a "dozen" such items. For example a dozen eggs, is a pack of twelve eggs. The meaning of dozen is not exactly twelve. Twelve is a number, where as a dozen is a noun (it is always assumed by a dozen that you have a dozen of something. So it is less abstract than the concept of twelve). However informally we can use "dozen" as a synonym for "twelve".

The etymology of twelve is "two left". In other words, there is exactly 2 left over after 10.

13

thirteen

Thirteen has a connotation of bad luck in western culture. It's the 6th prime.

14

fourteen

Fourteen is equal to 7*2 and 2*7. It's only factors are 1,2,7,14.

15

fifteen

Fifteen is equal to 5*3 and 3*5. It is both the 5th triangle number and the 3rd pentachoral number.

16

sixteen

Sixteen is the 4th square. It is also the 2nd tesseract (4-d cube), since 2^4=16. This number crops up in a few places. For example it's 4^2, 2^4, and 2^^3 = 2^2^2 = 16.

17

seventeen

Seventeen is the 7th prime number.

18

eighteen

Eighteen is 9*2, 2*9, 6*3, and 3*6. It's factors are 1,2,3,6,9,18, and it's prime factorization is 2*3*3.

19

nineteen

Nineteen is the 8th prime number

20

twenty

Twenty is a large number that has occasionally been used as the base in numeration systems. For example, the Mayan's used a mixed base twenty system. The choice of twenty may be motivated by the fact that we have twenty fingers and toes combined. Twenty is 1*20,20*1,2*10,10*2,4*5, and 5*4. It's factors are 1,2,4,5,10,20, and it's prime factorization is 2*2*5.

21

twenty-one

Twenty-one is the first number name which is a composite of existing names. 21= 7*3 = 3*7. The factors of 21 are 1,3,7,21 and it's prime factorization is 3*7.

25

twenty-five

5^2. The 5th square.

27

twenty-seven

3^3. The 3rd Cube. Also 3^^2 = 3^3 = 27.

25-22

28

twenty-eight

28 is the 2nd Perfect number. (See 6 and 496)

31

thirty-one

M5

31 is the 3rd mersenne prime, expressible as 2^5-1. It is also the 11th prime number. The next mersenne prime is 127.

33

thirty-three

This number can be expressed as (10^2-1)/3. This is the smallest member of numbers of the form (10^(p-1)-1)/p, where p is a prime not equal to 2 or 5. (See 142,857).

36

thirty-six

6^2. The 6th Square.

48

forty-eight
48 is the smallest number with more possible prime factorization trees then itself. Let arbor(n) be the number of trees possible with n as the seed. Arbor (48) = 70. However if n < 48 arbor (n) <= n. To make this precise we define a prime factorization tree as: (1) having positive integers greater than 1 at all nodes (2) the root node has n (3) every leaf node must have a prime number (4) every node except the leaf nodes must have exactly 2 children (5) the product of a parent node's children must equal the parent and (6) every node except the leaves has a left child and a right child. Two trees are identical if and only if there exists a one-to-one correspondence between their nodes such that each pair of corresponding nodes has the same number and each relationship between nodes is preserved. For example , 6 has two possible trees under these restraints, namely, (2,3) and (3,2) thus arbor(6) = 2. arbor(1) is undefined as condition (1) can not be satisfied. arbor(p)=1 for prime , p, as the root node will also be the only leaf node. Usu. The number of trees is much smaller than the number itself, but 48 can be factored in 70 different ways (see here). The next number with this property is 72.

49

forty-nine

7^2. The 7th Square.

64

sixty-four

8^2. The 8th Square. 4^3, the 4th Cube.

66

Sixty-six

There are 66 "books" in the christian bible. There are 39 old testament books, and 27 new testament books, for a grand total of 66. 66 can therefore be treated as having some numerological significance.

81

eighty-one

9^2. The 9th Square. This is also the 3rd Tesseract (3^4).

100

One hundred

10^2 or E2. The 10th Square. This number is notable for being one of the first numbers we learn about that is introduced as a "large number". The googology wiki defines a large number as any number equal to or greater than 100. 100 is a classic benchmark of large numbers, and is used in the construction of countless large numbers, such as the googol, googolplex, etc.

Terrestrial Epoch
(100,106)
Entries: 62

121

one hundred twenty-one

11^2. The 11th Square.

125

one hundred twenty-five

5^3, the 5th Cube.

127

one hundred twenty-seven

M7

127 is the 4th mersenne prime, expressible as 2^7-1. The next mersenne prime is 8191.

144

one hundred forty-four

12^2. The 12th Square. When a package contains a dozen dozen's (12 packs of 12-packs) it is called a gross. Thus gross can be used as a synonym for one-hundred-forty-four.

169

one hundred sixty-nine

13^2. The 13th Square.

196

one hundred ninety-six

14^2. The 14th Square.

216

two hundred sixteen

6^3, the 6th Cube.

220

two hundred twenty

This is the smallest amicable number. Amicable numbers are pairs of numbers such that the sum of their proper divisors is the other member of the pair. The smallest pair is 220 and 284. The prime factorization of 220 is 2*2*5*11. From this we can obtain it's factors which are 1,2,4,5,10,20,11,22,44,55,110, and 220. This can be computed more quickly as (1+2+4)(1+5+11+55) = 7*72 = 504. Removing the factor "220" which is not a proper divisor we get 504-220 = 284. The prime factorization of 284 is 2*2*71. It's factors are therefore 1,2,4,71,142,284. This can be quickly summed as (1+2+4)(1+71) = 7*72 = 504. Removing the factor "284" which is not a proper divisor we get 504-284 = 220.

It should be noted that this is an example of how a moderate sized number can be the smallest member of a sequence. This means large numbers can sometimes be defined by simple properties.

Amicable numbers are exceedingly rare in the world of number theory, and no general formula for all amicable numbers has been found.

225

two hundred twenty-five

15^2. The 15th Square.

256

two hundred fifty-six

16^2. The 16th Square. This is also the 4th Tesseract (4^4). 4^^2 = 4^4 = 256.

284

two hundred eighty-four

This is the partner of the smallest amicable number 220. (See 220). A consequence of the definition of amicable numbers, is that the smaller of the pair is necessarily abundant while the larger one is deficient.

289

two hundred eighty-nine

17^2. The 17th Square.

324

three hundred twenty-four

18^2. The 18th Square.

343

three hundred forty-three

7^3, the 7th Cube.

361

three hundred sixty-one

19^2. The 19th Square.

400

four hundred

20^2. The 20th Square.

29-24

496

four hundred ninety-six

496 is the 3rd Perfect Number. (See 28 and 8128).

512

five hundred twelve

8^3, the 8th Cube.

625

six hundred twenty-five

The 5th Tesseract (5^4).

656

six hundred fifty-six

656 is the last 3 digits of Mega.

666

six hundred sixty-six

The "number of the beast", according to revelation. 666 is also the 36th triangle number.

729

seven hundred twenty-nine

9^3, the 9th Cube.

945

nine hundred forty-five

This is the smallest odd-abundant number. An abundant number is a positive integer whose factors have a sum greater than 2 times the original number. The first few abundant numbers are 12,18,20,24,30,36,40,42 ...etc. Notice that the first few are all even numbers. A natural question is : are there any odd-abundant numbers, and if so which is the smallest one? This question is partially "googological" in nature because it could possibly lead to a large finite number as the answer. The answer to the first question is, yes, there are odd-abundant numbers, an infinite number of them in fact. Therefore there is a smallest odd-abundant, and that number is 945. What's interesting is that this simple property leads to a naturally occurring relatively large number (greater than 100).

1000

one thousand

10^3, the 10th Cube. Also can be written E3.

1024

one thousand and twenty-four

2^10, the number of bytes in a standard kilobyte.

1296

one thousand two hundred ninety-six

The 6th Tesseract (6^4).

1331

one thousand three hundred thirty-one

11^3, the 11th Cube.

1728

one thousand seven hundred twenty-eight

12^3, the 12th Cube. When a package contains a dozen gross (12 packs of 12-packs of 12-packs) it is called a great gross. Thus a great gross is a synonym of one-thousand-seven-hundred-twenty-eight.

2047

two thousand and forty-seven

This is the first mersenne number that is a counter-example to a prime exponent leading to a prime number. Mersenne numbers are of the form 2^n-1. When n is composite the mersenne number is composite. When n is prime, the mersenne may or may not be prime. The first 4 primes result in the first 4 mersenne primes, namely: 2^2-1 = 3 , 2^3-1 = 7 , 2^5-1 = 31 , and 2^7-1 = 127. But the next prime, 11, gives us a composite number: 2^11-1 = 2047 = 23*89.

2197

two thousand one hundred ninety-seven

13^3, the 13th Cube.

2401

two thousand four hundred one

The 7th Tesseract (7^4).

2656

two thousand six hundred fifty-six

The last 4 digits of Mega are 2656.

2744

two thousand seven hundred forty-four

14^3, the 14th Cube.

3125

three thousand one hundred twenty-five

This is equal to 5^^2 = 5^5 = 3125. It's an example of a very small tetrational number.

3375

three thousand three hundred seventy-five

15^3, the 15th Cube.

4096

four thousand ninety-six

16^3, the 16th Cube. Also the 8th Tesseract (8^4).

4913

four thousand nine hundred thirteen

17^3, the 17th Cube.

5832

five thousand eight hundred thirty-two

18^3, the 18th Cube.

6561

six thousand five hundred sixty-one

The 9th Tesseract (9^4).

6859

six thousand eight hundred fifty-nine

19^3, the 19th Cube.

8000

eight thousand

20^3, the 20th Cube.

213-26

8128

eight thousand one hundred twenty-eight

8128 is the 4th Perfect number. This was the largest perfect number known in antiquity, of which only 4 were known (the other three were 6,28, and 496). See 33,550,336.

8191

eight thousand one hundred ninety-one

M13

8191 is the 5th mersenne prime, expressible as 2^13-1. The next mersenne prime is 131,071.

9000

nine thousand

This number is part of the famous "It's over 9000!" meme. A little more relevant to googology, this is the very first valid entry in the "My Number is Bigger" xkcd thread, a very famous thread in googology circles which is perhaps the best large number contest ever hosted on the internet!

10,000

10^4 or E4. It was known as the myriad in ancient greece and used as the basis of their large numbers, just as a thousand is used as the basis for ours. This is also the 10th Tesseract.

14,641

fourteen thousand six hundred sixty-one

The 11th Tesseract (11^4).

20,736

twenty thousand seven hundred thirty-six

The 12th Tesseract (12^4).

Andre Joyce refers to this number as a great great gross, based on a continuation of the sequence, dozen, gross, great gross.

28,561

twenty-eight thousand five hundred sixty-one

The 13th Tesseract (13^4).

38,416

thirty-eight thousand four hundred sixteen

The 14th Tesseract (14^4).

46,656

forty-six thousand six hundred fifty-six

6^^2 = 6^6 = 46,656.

50,625

fifty thousand six hundred twenty-five

The 15th Tesseract (15^4).

65,536

sixty-five thousand five hundred thirty-six

The 16th Tesseract (16^4). Also: 2^^4 = 2^2^2^2 = 2^2^4 = 2^16 = 65,536. So this is also an extremely small tetrational number.

83,521

eighty-three thousand five hundred twenty-one

The 17th Tesseract (17^4).

100,000

one hundred thousand

10^5 or E5. A number notable for the fact that it rests at a borderline between just too large to fathom, and just large enough to still understand. This is also the number of zeroes in a googolgong.

104,976

one hundred four thousand nine hundred seventy-six

The 18th Tesseract (18^4).

130,321

one hundred thirty thousand three hundred twenty-one

The 19th Tesseract (19^4).

131,071

one hundred thirty-one thousand seventy-one

M17

131,071 is the 6th mersenne prime, expressible as 2^17-1. The next is 524,287.

142,857

(10^6-1)/7

integral-megaseptile

The smallest of Joyce's googolism's. This number is modeled on Fermat's Little Theorem. If we take any prime number, other then 2 or 5, call it p, then the number (10^(p-1)-1)/p is a positive integer. The smallest value of this form is actually (10^2-1)/3 or 33. This is the 2nd smallest number of this form.

160,000

one hundred sixty thousand

The 20th Tesseract (20^4).

248,832

great great great gross

or

two-hundred-forty-eight-thousand-eight-hundred-thirty-two

12^5. Called great great great gross by Andre Joyce.

524,287

five hundred twenty-four thousand two hundred eighty-seven

M19

524,287 is the 7th mersenne prime. This is conventionally denoted as M19 = 219-1. The next mersenne prime is 2,147,483,647.

823,543

eight hundred twenty-three thousand five hundred forty-three

7^^2 = 7^7 = 823,543.

Astronomical Epoch
[106,101000)
Entries: 103

1,000,000

million

10^6 or E6. A classic benchmark of large numbers. In some sense, a million may be treated as one of the smallest large numbers. It's name means "great thousand", and it is equal to a thousand thousand. Counting to a million is a task that can take about a year, realistically, allowing time for sleeping, eating, and all the ordinary activities of life, using only spare time for counting.

1,048,576

one million forty-eight thousand five hundred and seventy-six

2^20. Also the number of Bytes in a standard MB.

2,985,984

great great great great gross

or

two-million-nine-hundred-eighty-five-thousand-nine-hundred-eighty-four

12^6. Called a great great great great gross by Andre Joyce.

16,777,216

sixteen million seven hundred seventy-seven thousand two hundred sixteen

8^^2 = 8^8 = 16,777,216.

225-212

33,550,336

Fifth Perfect Number

33,550,336 is the 5th Perfect number. It was first correctly identified around 1461. There is a noticeable jump from the first 4 perfect numbers (6,28,496,8128). The fast growing nature of the sequence derives from the fact that the even perfect numbers are given by the formula 2^(p-1)(2^p-1), where p is a prime and 2^p-1 is also prime. Consequently the first 4 perfect numbers are created by the first 4 prime numbers:

2(2^2-1) = 2*3 = 6
2^2*(2^3-1) = 4*7 = 28
2^4*(2^5-1) = 16*31 = 496
2^6*(2^7-1) = 64*127 = 8128

The formula then fails for p=11. It works for p=13. The result is 2^12*(2^13-1) = 4096*8191 = 33,550,336. Note that this formula is roughly exponential in nature. It grows a little faster since primes become increasingly sparse, and also since not every prime will produce a perfect number. The next perfect number is 8,589,869,056.

35,831,808

five-ex-great gross

or

thirty-five-million-eight-hundred-thirty-one-thousand-eight-hundred-and-eight

12^7. Called five-ex-great gross by Andre Joyce.

100,000,000

10^8 or E8. This number was called the myriad myriad by the greeks, and Archimedes called his number the "octad". Generally speaking a myriad myriad is usually what passed for very large in antiquity. In the bible it is said that there are a myriad myriad angels in heaven, which if not meant literally, clearly was meant only to impress people with the vastness of Gods kingdom. In fact a myriad myriad is the largest definite number appearing in the bible. Other religious traditions got a lot further.

387,420,489

three hundred eighty-seven million four hundred twenty thousand four hundred eighty-nine

9^^2 = 9^9 = 387,420,489. Also the number of counting numbers less than one billion with exactly 9 non-zero digits (See 1,114,063,345).

429,981,696

six-ex-great gross

or

four-hundred-twenty-nine-million-nine-hundred-eighty-one-thousand-six-hundred-ninety-six

12^8. Called six-ex-great gross by Andre Joyce.

909,090,909

(10^10-1)/11

This is the third smallest number of the form (10^(p-1)-1)/p.

1,000,000,000

billion
/ milliard

10^9 or E9. A very large number equal to a thousand millions. Counting to this number is nigh impossible (See 1,114,063,345).

1,114,063,345

Highest Number a Human could Count to in a Lifetime

This is the absolute highest number a human could "feasibly" count to in a lifetime, and I'm using feasibly VERY loosely here. In fact this is more like an upper bound on the highest number a person could ever count to. To compute it I assumed that the person lived as long as the longest recorded living human, who lived to the age of 122 years and 164 days! Allowing for 8 hours of sleep per day, and assuming all other time is spent counting, you'd have 2,576,131,200 seconds of available time to devote to counting. Figuring out exactly how far you could count is complicated somewhat by the fact that different numbers take different amounts of time to say. A very good approximation can be made however, by assuming that the length of time for saying any given number is determined by the number of non-zero digits it contains. There are 387,420,489 numbers less than a billion with 9 non-zero digits. I estimated that 9 digit numbers take about 2.42 seconds to say. There is also just that many numbers less than a billion with 8 non-zero digits. I estimate it would take about 2.34 seconds to say an 8 digit number. Numbers with 6 or more non-zero digits account for about 99% of all numbers less than a billion, so we can ignore simpler numbers and still get a good approximation. According to my calculations you would theoretically be able to reach a billion in your 109th year of your life (assuming you started from birth!). On your death bed at 122 you would reach somewhere around 1 billion 114 million

This should prove unequivocally that no one can count to a trillion, no matter how hard they try! To make it within a reasonable life time you'd have to count at a thousand times faster than humanly possible! Even counting to a billion is really quite a stretch. It probably can't be done for a number of practical reasons. Firstly, how can someone count from birth? Obviously we'd have to give a few years (5 at least) so that the person could learn how to count. Then someone would have to be willing to count for the remainder of their very long life. Lastly, the human voice would probably wear out after a short time, perhaps after the first few months or years. In short, this is an impossible task. Nobody living today can count to 1,114,063,345. Therefore this is an extreme upper bound on what a human being can actually count to. Of coarse if you want to prove me wrong by example, better get started ... unless you were just born today.

In any case I'll define this as the largest feasibly countable number.

2,147,483,647

8th Mersenne Prime

M31

Also known as M31 , this is the 8th mersenne prime. It can be expressed compactly as 231-1. The next one is 2^61-1.

3,864,196,800

Age in seconds of oldest person who ever lived

The oldest known person who ever lived was Jeanne Calment who reached the extremely advanced age of 122 years and 164 days when she died. Assuming a year to be roughly 365.25 days, converting her final age into seconds, we can say that she lived for 3,864,196,800 seconds. Amazingly, this number is really quite astronomical. The average human life span is about 75 years, but even this amounts to 2,365,200,000 seconds. So we can honestly say that humans live anywhere from about 2 to 4 billion seconds. Hopefully a lifetime seems a lot longer now!

233-216

8,589,869,056

Sixth Perfect Number

8,589,869,056 is the 6th Perfect Number. It was discovered in 1588 by the italian mathematician Pietro Cataldi. This number is the product of 2^16 and the 6th mersenne prime. It can be expressed as 2^16*(2^17-1). The next prefect number is 137,438,691,328.

10,000,000,000

Ten billion / Ten milliard / dialogue

10^^2 = 10^10 = 10,000,000,000. Can be written as E1#2 or E10.

76,923,076,923

(10^12-1)/13

This is the 4th smallest integer of the form (10^(p-1)-1)/p.

100,000,000,000

hundred billion / hundred milliard / ten dialogue

237-218

137,438,691,328

Seventh Perfect Number

137,438,691,328 is the 7th Perfect Number. 2^18*(2^19-1). The next one is 2,305,843,008,139,952,128.

285,311,670,611

11^11

11^^2 = 11^11 = 285,311,670,611. This is the largest member of the sequence S(n) = n^^2 which is less than 3^3^3.

1,000,000,000,000

10^12 or E12. The largest -illion the average person is aware of. It's called a trillion in the short scale, but if referred to as a "billion" in the long scale. To distinguish between the long and short scale, I use the following suffixes:

n-illion = 10^(3n+3)

n-illiard = 10^(6n+3)

7,625,597,484,987

3^3^3

3^^3 = 3^3^3 = 3^27 = 7,625,597,484,987. This is the result of computing 3 tetrated to the 3rd. This number can actually be computed by hand relatively easily, though it does take some time. Here is one way to go about computing it:

3^^3 = 3^3^3 = 3^(3*3*3) = 3^(3*9) = 3^27 =

((3*3*3)
(3*3*3)(3*3*3))((3*3*3)(3*3*3)(3*3*3))((3*3*3)(3*3*3)(3*3*3)) =

((3*9)(3*9)(3*9))((3*9)(3*9)(3*9))((3*9)(3*9)(3*9)) =

(27*27*27)
(27*27*27)(27*27*27) =

(27*729)
(27*729)(27*729) =

19,683*19,683*19,683 =

19,683*387,420,489 =

7,625,597,484,987

In truth this isn't generally the best way to go when computing it by hand as multiplying numbers with more than 3 digits can get confusing. A better approach is simply to create a table of powers of 3 up to the 27th power. If you know one power of 3 the next can simply be found by multiplying each digit by 3 and carrying over as necessary. This is probably one of the very few large numbers you can actually compute without the aid of a calculator.

8,916,100,448,256

12^12

12^^2 = 12^12. This is the smallest member of the sequence S(n) = n^^2, larger than 3^3^3.

588,235,294,117,647

(10^16-1)/17

integral-dekapetaseptemdecile

This is the 2nd smallest of Joyce's googolism's of the form (10^(p-1)-1)/p. This is however the 5th smallest integer of the form (10^(p-1)-1)/p where p is a prime not equal to 2 or 5.

1,000,000,000,000,000

10^15, or E15. The -illion after a trillion.

52,631,578,947,368,421

(10^18-1)/19

integral-exaundevigintile

Joyce's 3rd smallest googolism of the form (10^(p-1)-1)/p. Also the 6th smallest integer of the from (10^(p-1)-1)/p. The number contains 17 digits and is approximately 52 quadrillion.

1,000,000,000,000,000,000

A quintillion is 1 followed by 18 zeroes. In the long scale it's called a trilliad. It can be written compactly as 10^18 or E18.

261-230

2,305,843,008,139,952,128

Eighth Perfect Number

This is the 8th perfect number. It is equal to 2^61-2^30. Incidentally this number is only slightly smaller than the 9th mersenne prime. They share the same 9 leading digits. The next perfect number is 2^121-2^60.

2,305,843,009,213,693,951

Pervushin's Number / 9th Mersenne Prime

M61

This is the 9th mersenne prime. It is traditionally denoted M61. It can be expressed compactly as 2^61-1. It was first discovered by Ivan Mikheevich Pervushin in Novemeber of 1883. For this reason it is sometimes called Pervushin's Number. At the time of it's discovery it was the 2nd largest known prime. It remain as such until 1911. This number is roughly 2.3 quintillion in the short scale, or 2.3 trilliad in the long scale. The next mersenne prime is M89 which contains 27 digits.

434,782,608,695,652,173,913

(10^22-1)/23

integral-dekazettatrevigintile

A Joycian googolism formed using Fermat's Little Theorem.

1,000,000,000,000,000,000,000

sextillion / trilliard

A sextillion is 1 followed by 21 zeroes. In the long scale it's called a trilliard. It can be written concisely as 10^21 or E21.

6.02214xE23

Avogadro's Number is a large constant used in chemistry. Formally it can be defined as the number of carbon-12 atoms it would take to add up to 12 grams of matter. Approximately it's the number of protons it would take to add up to 1 gram of mass. It is therefore very close to the reciprocal of the protons mass as measured in grams. This is a ridiculously large number in comparison even with the millions, billions, and trillions we are used to in the modern world. It's quite tiny however compare to even modest numbers that crop up in pure mathematics such as the mersenne primes.

1,000,000,000,000,000,000,000,000

A septillion is 1 followed by 24 zeroes. In the long scale it's a quadrilliad. It can be written concisely as 10^24 or E24.

344,827,586,206,896,551,724,137,931

(10^28-1)/29

integral-myriayottaundetrigintile

Joycian Googolism. It's larger than Avogadro's Number but smaller than M89.

618,970,019,642,690,137,449,562,111

10th Mersenne Prime

M89

M89 is the 10th mersenne prime. It can be expressed compactly as 2^89-1. This number was first proven prime by Ralph Ernest Powers in 1911. When first discovered it stole M61's (Pervushin's Number) place for 2nd largest known prime, bumping Pervushin's Number down to 3rd. (See M107 and M127). The next mersenne prime is M107 with 33 digits.

1,000,000,000,000,000,000,000,000,000

A octillion is 1 followed by 27 zeroes. In the long scale it's a quadrilliard. It can be written concisely as 10^27 or E27.

1,000,000,000,000,000,000,000,000,000,000

A nonillion is 1 followed by 30 zeroes. In the long scale it's a quintilliad. It can be written concisely as 10^30 or E30.

162,259,276,829,213,363,391,578,010,288,127

11th Mersenne Prime

M107

M107 is the 11th mersenne prime. It can be expressed compactly as 2^107-1. This number was first proven prime by Ralph Ernest Powers in June of 1914. At the time it held 2nd place for largest known prime, following by M89 in 3rd place, and M61 (Pervushin's Number) in 4th place. The next mersenne prime is M127.

1,000,000,000,000,000,000,000,000,000,000,000

decillion / quintilliard

A decillion is 1 followed by 33 zeroes. In the long scale it's a quintilliard. It can be written concisely as 10^33 or E33. It is a personal favorite of mine, along with the centillion.

1,000,000,000,000,000,000,000,000,000,000,000,000

A undecillion is 1 followed by 36 zeroes. In the long scale it's a sextilliad. It can be written concisely as 10^36 or E36.

2121-260

2,658,455,991,569,831,744,654,692,615,953,842,176

Ninth Perfect Number

This is the 9th perfect number. It's approximately 2.6 undecillion, and contains 37 digits, none of them "0" incidentally. It is also fairly close to the 12th mersenne prime. mersenne primes and even perfect numbers are closely related because the nth even perfect number always has the nth mersenne prime as a factor. Every perfect number may be expressed as 2^(p-1)*(2^p-1) where p is prime and 2^p-1 is a mersenne prime. Consequently the corresponding perfect number for each mersenne number is about it's square. The next perfect number is 2^177-2^88.

170,141,183,460,469,231,731,687,303,715,884,105,727

12th Mersenne Prime

M127

M127 is the 12th mersenne prime. It was first proven prime by Edouard Lucas on January 10th of 1876. It held the record for largest known prime from it's discovery until about 1951. (See 180(M127)2+1). The next mersenne prime is a huge leap forward at M521 with 157 digits.

1,000,000,000,000,000,000,000,000,000,000,000,000,000

duodecillion / sextilliard

A duodecillion is 1 followed by 39 zeroes. In the long scale it's a sextilliard. It can be written concisely as 10^39 or E39.

1,000,000,000,000,000,000,000,000,000,000,000,000,000,000

A tredecillion or septilliad is 1 followed by 42 zeroes. It can be written concisely as 10^42 or E42.

212,765,957,446,808,510,638,297,872,340,425,531,914,893,617

(10^46-1)/47

Joycian googolism. This number contains exactly 45 digits.

1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

quattuordecillion / septilliard

A quattuordecillion or septilliard is 1 followed by 45 zeroes. It can be written concisely as 10^45 or E45.

1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

A quindecillion or octilliad is 1 followed by 48 zeroes. It can be written concisely as 10^48 or E48.

1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

sexdecillion / octilliard

A sexdecillion or octilliard is 1 followed by 51 zeroes. It can be written concisely as 10^51 or E51.

2177-288

191,561,942,608,236,107,294,793,378,084,303,638,130,997,321,548,169,216

Tenth Perfect Number

This is the 10th perfect number. It equal to exactly 2^177-2^88 and has 54 digits. It lies between a sexdecillion and a septendecillion in the short scale. The next perfect number is 2^213-2^106.

1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

E54

A septendecillion or nonilliad is 1 followed by 54 zeroes. It can be written concisely as 10^54 or E54.

1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

E57

octodecillion / nonilliard

A octodecillion or nonilliard is 1 followed by 57 zeroes. It can be written concisely as 10^57 or E57

1,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000

E60

A novemdecillion or decilliad is 1 followed by 60 zeroes. It can be written concisely as 10^60 or E60.

1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

E63

vigintillion / decilliard

A vigintillion or decilliard is 1 followed by 63 zeroes. It can be written concisely as 10^63 or E63. A vigintillion is the largest official -illion besides a centillion. There is no cannonical -illions between a vigintillion and a centillion.

2213-2106

13,164,036,458,569,648,337,239,753,460,458,722,910,
223,472,318,386,943,117,783,728,128

11th Perfect Number

This is the 11th perfect number. It contains 65 digits. The next perfect number is 2^253-2^126.

1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

E66

unvigintillion / viginti-untillion

According to Conway's system a unvigintillion is 1 followed by 66 zeroes. I use the term viginti-untillion since the order of terms should reverse after the 20th illion.

1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

E69

duovigintillion / viginti-deutillion

According to Conway a duovigintillion is 1 followed by 69 zeroes. I call it a viginti-deutillion.

1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

E72

trevigintillion / viginti-tretillion

10^72 or E72. A trevigintillion or viginti-tretillion is 1 followed by 72 zeroes

1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

E75

quattuorvigintillion / viginti-quattillion

10^75 or E75

2253-2126

14,474,011,154,664,524,427,946,373,126,085,988,481,573,677,491,474,
835,889,066,354,349,131,199,152,128

12th Perfect Number

This is the 12th perfect number. It has 77 digits. The next perfect number is 2^1041-2^520, which has 314 digits!

1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

E78

quinvigintillion / viginti-quintillion

10^78 or E78

5,210,644,015,679,228,794,060,694,325,390,955,853,335,898,483,908,056,458,352,183,851,018,372,555,735,221

180(2127-1)2+1

From 1951 to January of 1952 the number 180(2^127-1)^2+1 (approx. 5.2106xE78) briefly held the title for largest known prime. It was discovered by Miller and Wheeler in July of 1951. It's notable for being the first record set with the aid of an electronic computer. Interestingly this is not a mersenne prime. Currently the top ten largest known primes are all mersenne primes. Incidently this short lived record holder was quickly eclipsed by the discovery of the 13th and 14th mersenne primes.

15,747,724,136,275,002,577,605,653,961,181,555,468,044,717,914,527,116,
709,366,231,425,076,185,631,031,296

136*2256

Eddington Number

In 1938 astrophysist Arthur Eddington was the first to propose an exact integer value to the number of protons in the observable universe. For aesthetic and numerological reasons he came up with the exact value 136*2^256 (approx. 1.5747xE79). Robert Munafo notes that this is the largest specific integer thought to have a unique and tangible relationship to the physical world. This number is just below the current popular estimate of 10^80 for the number of particles in the observable universe.

100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000

E80

Number of Particles in the Observable Universe

This is a commonly given figure for the number of particles in the observable universe. It is an unimaginably vast number, and yet still a tiny fraction compared to a googol. However this is only the estimated number of particles in the "observable universe", that is, the portion of the universe that we can see because light has had enough time to travel to our little blue planet. Scientists aren't exactly sure how big the universe is in it's entirety, so there is a possibility that there are actually a googol or even more particles in the entire universe.

E81

sexvigintillion / viginti-sextillion

10^81 or E81

E84

septenvigintillion / viginti-septillion

10^84 or E84

E87

octovigintillion / viginti-octillion

10^87 or E87

E90

novemvigintillion / viginti-nonillion

10^90 or E90

E93

trigintillion

A trigintillion is 1 followed by 93 zeroes. It is the 30th illion, equal to 10^93 or E93.

E96

untrigintillion / triginti-untillion

E99

duotrigintillion / triginti-deutillion

10,000,000,000,000,000,000,000,000,000,000,000
,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

10100

googol

or

ten duotrigintillion / ten triginti-deutillion

This is 1 followed by 100 zeroes, best known as a "googol". It can also be given the more "technical" name of "ten duotrigintillion". This is the number that really started it all and began the large number trend and it's bizarre naming conventions. Many large numbers are built around the pattern established by this number. In some sense this IS the smallest googolism. Both Jonathan Bowers and myself have created extended systems based on this number. This is incidentally the smallest googolism mentioned on Jonathan Bowers' infinity scraper page. The infamous googolplex is much larger, and you won't see it until a little later in the list.

10101

This is the smallest of the 4 possible interpretations of Andre Joyce's great googol. Joyce states that if (n) = b^a then great-(n) = b^(a+1). By this reasoning since a googol = 10^100, it should follow that great googol = 10^101. The problem is that such a definition is actually ambiguous because "n" is not always a uniquely defined power. For example googol = 100^50, therefore great googol = 100^51 = 10^102. Furthermore we also have googol = 100,000^20, which gives great googol = 100,000^21 = 10^105, or googol = 10,000,000,000^10 which gives great googol = 10,000,000,000^11 = 10^110, or googol = (10^20)^5 which gives great googol = (10^20)^6 = 10^120, etc. (See also 10^150, 10^300 , and 10^1000).

E102

tretrigintillion / triginti-tretillion

E123

10150

This is one of the 4 possible interpretations of the Joycian great googol. Joyce defines it as (10^100)^(3/2) in one place, believing this to be equivalent to 10^100^(3/2) = 10^1000. Instead we get that (10^100)^(3/2) = 10^(100*3/2) = 10^150. (Also see 10^101 , 10^300, and 10^1000).

E153

quinquagintillion

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

4^4^4

4 Tetrated to the 3rd. This value is notable for being larger than a googol. This number contains exactly 155 digits and is approximately equal to 1.34x10^154. It can also be approximated as thirteen quinquagintillion. This is a pretty big number by ordinary standards, but small by tetrational standards. It doesn't even really clear the high astronomical range.

2521-1

6,864,797,660,130,609,714,981,900,799,081,393,217,269,435,300,143,305,409,
394,463,459,185,543,183,397,656,052,122,559,640,661,454,554,977,296,311,391,
480,858,037,121,987,999,716,643,812,574,028,291,115,057,151

13th Mersenne Prime

M521

This is the 13th mersenne prime (approx. 6.8647xE156). It was first discovered by Raphael Robinson on January 30th of 1952. It was the first prime discovered with at least 100 digits. There is a noticeably drastic gap between the 12th and 13th mersenne prime. It's around here that the mersenne primes begin to grow in size quite rapidly. On the same day Raphael Robinson also discovered the 14th mersenne prime, M607.

2607-1

531,137,992,816,767,098,689,588,206,552,468,627,329,593,117,727,031,923,199,
444,138,200,403,559,860,852,242,739,162,502,265,229,285,668,889,329,486,246,
501,015,346,579,337,652,707,239,409,519,978,766,587,351,943,831,270,835,393,
219,031,728,127

14th Mersenne Prime

M607

This is the 14th mersenne prime (approx. 5.3113xE182). It was first discovered by Raphael Robinson on January 30th of 1952, the same day the 13th mersenne prime was found. For just a few months this number held the record for largest known prime. It would subsequently be trumped three more times as Robinson cranked out the 15th,16th, and 17th mersenne prime all in the same year! The next mersenne prime is M1279.

E183

sexagintillion

3.566*E185

Volume of Observable Universe in Planck Volumes

Who said that there isn't a googol of anything. Okay perhaps there isn't a googol objects, but there is more than a googol spaces. Recall that the Planck Length is only 10^-35 meters. Now imagine a cube with side length of 1 Planck Length. That's a Planck Volume. 10^105 of these fit in a cubic meter! The observable universe has a radius of 46.5 billion light years. Figure out the volume of the sphere with this radius, divide it by the Planck Volume, and you get the massive figure of 10^185. This proves that a googol actually still has some physically tangible meaning!

10200

gargoogol

This number is derived from Kieran's gar- prefix. It was first used to define a gargoogolplex as a googolplex googolplexes, namely gargoogolplex is googolplex^2. From this we extrapolate that gar-(n) = n^2, with the caveat that "gar", a prefix, should be applied after all other suffixes the number possesses so that gargoogolplex is understood as gar(plex(googol)) not plex(gar(googol)) (see gargoogolplex and gargoogol-plexed for disambiguation). In any case, there is no ambiguity in regards to a gargoogol, which would be googol^2. It turns out to be a nifty name for 1 followed by 200 zeroes, in case, you know, we might have some use for that :p

E213

septuagintillion

E243

octogintillion

E273

nonagintillion

10300

thrargoogol

This is another value that can be derived for Andre Joyce's great googol. Joyce says that n-ex-great googol = (10^100)^(n+2). By this reasoning great googol = (10^100)^(1+2) = (10^100)^3 = 10^300. (See also 10^101 , 10^150, and 10^1000 )

I offer a better name for this number: thrargoogol, a corruption of gargoogol. thrar- is a prefix formed from combining "three" with "gar". Let thrar-(n) = n^3.

E303

centillion

The 100th member of the -illion series in the short scale system. This was the largest officially recognized large number name that I knew about as a kid (Ironically I didn't learn about the googol and googolplex until much later). It was my favorite number for a time, along with the googolgong (seen a little later).

21041-2520

`2356272345726734706578954899670990498847754785839260071014302759750633728317862223973036553960260056136025556646250327017505289257804321554338249842877715242701039449691866402864453412803383143979023683862403317143592235664321970310172071316352748729874740064780193958716593640108741937564905791854949216055564697613th Perfect NumberThis is the 13th perfect number. It has 314 digits. This makes it larger that a centillion in the short scale.The next perfect number is 2^1213-2^606.`
21213-2606

`141053783706712069063207958086063189881486743514715667838838675999954867742652380114104193329037690251561950568709829327164087724366370087116731268159313652487450652439805877296207297446723295166658228846926807786652870188920867879451478364569313922060370695064736073572378695176473055266826253284886383715072974324463835300053138429460296575143368065570759537328128`

14th Perfect Number

This is the 14th perfect number. It has 366 digits. The next perfect number is 2^2557-2^1278.

21279-1

10,407,932,194,664,399,081,925,240,327,364,085,538,615,262,247,266,704,805,
319,112,350,403,608,059,673,360,298,012,239,441,732,324,184,842,421,613,954,
281,007,791,383,566,248,323,464,908,139,906,605,677,320,762,924,129,509,389,
220,345,773,183,349,661,583,550,472,959,420,547,689,811,211,693,677,147,548,
478,866,962,501,384,438,260,291,732,348,885,311,160,828,538,416,585,028,255,
604,666,224,831,890,918,801,847,068,222,203,140,521,026,698,435,488,732,958,
028,878,050,869,736,186,900,714,720,710,555,703,168,729,087

15th Mersenne Prime

M1279

The 15th mersenne prime (approx. 1.0407xE385). It was first discovered by Raphael Robinson on June 25th of 1952, only 5 months are he had discovered the last record prime, the 14th mersenne prime. The next mersenne prime is M2203.

10500

googolding

This is a relatively small googolism I created as an extension of the googolgong. -ding is a modifier that takes the base number and raises it to the 5th power, or when operating on much larger numbers, if N is expressed as f(n) for some function, f, then (N)-ding will mean f(5n). The next larger one is googolchime, for 101000.

32507925092532526327561017283413824652374863824571
23648312068325872634187254045013845321064343218561
32840151283461328410827545125763218561032841803247
63218065230850123857632105612086532150632150623199
23165991236598123649823645514560723570321652371652
13078423145213748632804238451246109823468458364128
43210612734081642309452139562982309126570932156013
29754123906012934632109746231705231650923615076120
93753209843261897432980213647126509243798107561029
81276576321095621359762315078961235981723476321756
2310947231984654761253908216507486502937521093865

Twasbrillig's Digit Wall

This was the 2nd valid entry in the "My Number is Bigger" thread, right after 9000. This number is a pretty big leap forward, being much larger than the number of particles in the observable universe (1080) or even the number of plank volumes (10175), but it's classified here as part of the Astronomical Epoch since it is quite conceivable that the universe as a whole might be astronomically larger than the observable universe.

This number has exactly 549 digits. It was posted by Twasbrillig.

568390125739205684705612809352167456489132749013265712367432
718953216987051326795312659012367567218920165701897342905621
746312089234798162348902357390216705163290561325071326479012
36439210609321457923106512390756219032892659312549032461804
372160123482146385486432890164215483240823684731254132487031
256173256123075327065415546328946321895632199561329913260512
3605123568021650123675832105803256081236742308148230165812367
5215457280148231643821510482316581234346012354831054045278143
627852386021384632175428368473256428314382710165723625235290
52970523

Crazyjimbo's Response

Crazyjimbo's response is the 3rd valid entry in the "My Number is Bigger" thread. It is simply Twasbrillig's Digit Wall but with the digits reversed. Thus it also has exactly 549 digits. Interestingly the ratio between the 1st and 2nd entry is huge, but the ratio between the 2nd and 3rd is less than 2.

568390125739205684705612809352167456489132749013265712367432
718953216987051326795312659012367567218920165701897342905621
746312089234798162348902357390216705163290561325071326479012
36439210609321457923106512390756219032892659312549032461804
372160123482146385486432890164215483240823684731254132487031
256173256123075327065415546328946321895632199561329913260512
3605123568021650123675832105803256081236742308148230165812367
5215457280148231643821510482316581234346012354831054045278143
627852386021384632175428368473256428314382710165723625235290
529705231

Twasbrillig's Rebuttal

Twasbrillig responded to Crazyjimbo's Number by simply appending a 1 to it. This number thus has exactly 550 digits. Despite the fact that this seems to be only a miniscule improvement this number is roughly 10 times larger than the previous entry. This is the 4th valid entry in the "My Number is Bigger" thread.

E600

This is the centillion in the long scale system. To distinguish it from 10^303 I call it the centilliad. The centilliad is the 100th power of a million. This number is ridiculously large and is already much larger than number you even encounter in astronomy! So it a real sense this number is almost post-astronomical. The boundary for astronomical numbers is a little vague since we actually don't know how large the entire universe is, or even if it's finite.

E603

centilliard

This would be the largest cannonical -illion in the long scale.

12623830496605862226841748706511699984548477605357610950050916182626818413620269880155156801376138071753405453485116413864890452793160
5160527688095259563605939964364716019515983399209962459578542172100149937763938581219604072733422507180056009672540900709554109516816573779593326332288314873251559077853068444977864803391962580800682760017849589281937637993445539366428356761821065267423102149447628375691862210717202025241630303118559188678304314076943801692528246980959705901641444238894928620825482303431806955690226308773426829503900930529395181208739591967195841536053143145775307050594328881077553168201547775

22040-1

This is the largest number that can be stored on the TI-89 exact mode. Exact mode allows you to manipulate integers directly, among other things. Unlike approximate mode there is no rounding off of numbers and therefore no rounding error. If you type in 2^2040-1 into the TI-89 in exact mode it will not return this number but will return "infinity" as the answer. The reason for this is it must first compute 2^2040 before subtracting 1. Since 2^2040 is just over it's limit it will return "infinity" for the rest of the calculation. In order to get the TI-89 to display this number in full you must obtain this number in a calculation that does not involve overflow at any step. One way to do this is to ask the TI-89 to compute 2(2^2039-1)+1. This is equivalent to 2^2040-1. When you add 1 to this value you'll immediately get an overflow, proving that 2^2040-1 is the largest possible integer in can work with in exact mode. (See 10^1000-10^986).

22203-1

1475979915214180235084898622737381736312066145333169775147771216478
5702978780789493774073370493892893827485075314964804772812648387602
5919181446336533026954049696120111343015690239609398909022625932693
5025281409614983499388222831448598601834318536230923772641390209490
2318364468996082107954829637630942366309454108327937699053999824571
8632294472963641889062337217172374210563644036821845964963294853869
6905872650486914434637457507280441823676813517852099348660847172579
4084223166780976702240119902801704748944874269247421088235368084850
7250224051945258754287534997655857267022963396257521263747789778550
1552646522609988869914013540483809865681250419497686697771007

16th Mersenne Prime

M2203

This is the 16th mersenne prime (approx. 1.4759xE663). It was first discovered by Raphael Robinson on October 7th of 1952, setting yet another record for largest prime, just about 4 months after setting the previous record (the 15th mersenne prime). Two days later Robinson would find yet a slightly larger prime, the next mersenne prime, M2281.

22281-1

446087557183758429571151706402
101809886208632412859901111991
219963404685792820473369112545
269003989026153245931124316702
395758705693679364790903497461
147071065254193353938124978226
307947312410798874869040070279
328428810311754844108094878252
494866760969586998128982645877
596028979171536962503068429617
331702184750324583009171832104
916050157628886606372145501702
225925125224076829605427173573
964812995250569412480720738476
855293681666712844831190877620
606786663862190240118570736831
901886479225810414714078935386
562497968178729127629594924411
960961386713946279899275006954
917139758796061223803393537381
034666494402951052059047968693
255388647930440925104186817009
640171764133172418132836351

17th Mersenne Prime

M2281

This is the 17th mersenne prime (approx. 4.4608xE686). It was first discovered by Raphael Robinson on October 9th of 1952. This remained the largest known prime until 1957 when the 18th mersenne prime, M3217, was discovered.

22557-21278

`54162526284365847412654465374391316140856490539031695784603920818387206994158534859198999921056719921919057390080263646159280013827605439746262788903057303445505827028395139475207769044924431494861729435113126280837904930462740681717960465867348720992572190569465545299629919823431031092624244463547789635441481391719816441605586788092147886677321398756661624714551726964302217554281784254817319611951659855553573937788923405146222324506715979193757372820860878214322052227584537552897476256179395176624426314480313446935085203657584798247536021172880403783048602873621259313789994900336673941503747224966984028240806042108690077670395259231894666273615212775603535764707952250173858305171028603021234896647851363949928904973292145107505979911456221519899345764984291328`

15th Perfect Number

This is the 15th perfect number. It has 770 digits. The next perfect number is 2^4405-2^2202.

23217-1

259117086013202627776246767922
441530941818887553125427303974
923161874019266586362086201209
516800483406550695241733194177
441689509238807017410377709597
512042313066624082916353517952
311186154862265604547691127595
848775610568757931191017711408
826252153849035830401185072116
424747461823031471398340229288
074545677907941037288235820705
892351068433882986888616658650
280927692080339605869308790500
409503709875902119018371991620
994002568935113136548829739112
656797303241986517250116412703
509705427773477972349821676443
446668383119322540099648994051
790241624056519054483690809616
061625743042361721863339415852
426431208737266591962061753535
748892894599629195183082621860
853400937932839420261866586142
503251450773096274235376822938
649407127700846077124211823080
804139298087057504713825264571
448379371125032081826126566649
084251699453951887789613650248
405739378594599444335231188280
123660406262468609212150349937
584782292237144339628858485938
215738821232393687046160677362
909315071

18th Mersenne Prime

M3217

This is the 18th mersenne prime (approx. 2.5911xE968). It was first discovered by Han Riesel on September 9th of 1957 and was the largest known prime until 1961. The next mersenne prime is M4253 with 1281 digits.

E999

trecenti-triginti-deutillion / centi-sexaginti-sextilliard

This is the largest -illion or integer power of a thousand less than E1000. I've written both the short and long scale in my -illion scheme.

101000-10986

Largest Number possible on TI-89

This is the largest number that can be stored on the TI-89 approximate mode. It is displayed as 9.9999999999999xE999. This number can only be seen in the "equation display". In the answer display it rounds it to 12 decimal places of precision. However in the background the TI-89 actually holds 14 digits of precision. This can be detected by certain anomalies in calculations. For example you can add 1xE985 to this number an infinite number of times without ever changing it's value because it rounds off the addend to 0. Add 5xE985 just once however and you'll instantly get the result "infinity", implying there is an overflow.

Although this marks the limit of TI-89s hardwired number crunching abilities, it is possible to use it to perform computations for much much larger numbers by means of estimation with logarithms.

Super-Astronomical Epoch
[101000,101,000,000)
Entries: 57

101000

googolchime

Andre Joyce, the same guy who coined the term "googology", coined the googolism "great googol" to stand for the number 10^1000, based on the idea that since a gross is 12^2 and a great gross is 12^3, it should follow that if a googol is 10^10^2, then a great googol should be 10^10^3. Unfortunately he provides no less than 4 conflicting definitions for this number in his own writing. This is the value most commonly cited (See also 10^101 , 10^150 , 10^300).

I call this number a googolchime. I use the name "great googol" for a much larger value (See E100##1#2 ). The name is formed by following the theme established by a googolgong of using things which can producing ringing sounds. (See also googolbell, googoltoll, googolgong, googolbong, googolthrong, etc.).

This number works nicely as a bench mark for passing beyond the merely astronomical and entering into the hyper-astronomical. Eventually we reach numbers so large that to call them "astronomical" is insulting since astronomy doesn't even use such big numbers. At that point we reach what I like to call the hyper-exponential numbers, which are numbers so huge that their number of digits is itself of astronomical proportions!

24253-1

190797007524439073807468042969
529173669356994749940177394741
882673528979787005053706368049
835514900244303495954950709725
762186311224148828811920216904
542206960744666169364221195289
538436845390250168663932838805
192055137154390912666527533007
309292687539092257043362517857
366624699975402375462954490293
259233303137330643531556539739
921926201438606439020075174723
029056838272505051571967594608
350063404495977660656269020823
960825567012344189908927956646
011998057988548630107637380993
519826582389781888135705408653
045219655801758081251164080554
609057468028203308718724654081
055323215860189611391296030471
108443146745671967766308925858
547271507311563765171008318248
647110097614890313562856541784
154881743146033909602737947385
055355960331855614540900081456
378659068370317267696980001187
750995491090350108417050917991
562167972281070161305972518044
872048331306383715094854938415
738549894606070722584737978176
686422134354526989443028353644
037187375385397838259511833166
416134323695660367676897722287
918773420968982326089026150031
515424165462111337527431154890
666327374921446276833564519776
797633875503548665093914556482
031482248883127023777039667707
976559857333357013727342079099
064400455741830654320379350833
236245819348824064783585692924
881021978332974949906122664421
376034687815350484991

19th Mersenne Prime

M4253

This is the 19th mersenne prime (approx. 1.9079xE1280). It was first discovered by Alexander Hurwitz on November 3rd of 1961. On the same day Hurwitz also discovered the 20th mersenne prime, M4423.

24405-22202

`108925835505782933769822527352204898195710845430260806731890661850847015529861699629194096185890137954618268553122005576278075934240749906604670418208308712462692637816441093145096882635520557367167162420268663336080712310947045266837153759966279748493435903977995421366659882029950136638016461908026040323522955673055416399230300975265135032061993056367369528015302304949846869661814407202137283142596370146050560637811924584138655260014538407298330971714195008549808570967138705486832047797229905527391479844693621414786070688705210731238006707260231700942280931477479189470076989100981874316930302815430329007119939298429294028385221780016662922915711026408059929401645248302852815333111952344142315961493414026555024236000785821593679848950072719634751638604424172198470655832936427799590310229203462062808075234242290640128302703464967144556932428194685962217756664337548971567845131179267593598101035556288797194856901606003533460787935977037184650765997060161699831198387815042076330628949088642990048178649953764537983936521272549444151193277218276814994365984900745724698386155826514482319136775835034152778077022155694527556650483163656485683150255607805813304340005565354041331326603463935520283400612690549156956054248955102320738227613735266571701826151960481741711257652641053532399150005874999624758083445378252816th Perfect Number`

This is the 16th perfect number. It has 1327 digits. The next perfect number is 2^4561-2^2280.

24423-1

285542542228279613901563566102
164008326164238644702889199247
456602284400390600653875954571
505539843239754513915896150297
878399377056071435169747221107
988791198200988477531339214282
772016059009904586686254989084
815735422480409022344297588352
526004383890632616124076317387
416881148592486188361873904175
783145696016919574390765598280
188599035578448591077683677175
520434074287726578006266759615
970759521327828555662781678385
691581844436444812511562428136
742490459363212810180276096088
111401003377570363545725120924
073646921576797146199387619296
560302680261790118132925012323
046444438622308877924609373773
012481681672424493674474488537
770155783006880852648161513067
144814790288366664062257274665
275787127374649231096375001170
901890786263324619578795731425
693805073056119677580338084333
381987500902968831935913095269
821311141322393356490178488728
982288156282600813831296143663
845945431144043753821542871277
745606447858564159213328443580
206422714694913091762716447041
689678070096773590429808909616
750452927258000843500344831628
297089902728649981994387647234
574276263729694848304750917174
186181130688518792748622612293
341368928056634384466646326572
476167275660839105650528975713
899320211121495795311427946254
553305387067821067601768750977
866100460014602138408448021225
053689054793742003095722096732
954750721718115531871310231057
902608580607

20th Mersenne Prime

M4423

The 20th mersenne prime (approx. 2.8554xE1331). It was first discovered by Alexander Hurwitz on November 3rd of 1961. This  number held the record for largest known prime until 1963. The next mersenne prime is M9689.

24561-22280

`99497054337086473442435202604522816989643863571126408511774020575773849326355529178686629498151336416502516645641699516813140394897940636561646545947753232301453603583223268085613647233768081645727669037394385696522820301535888041815559513408036145123870584325525813950487109647770743827362571822870567643040184723115825645590386313377067112638149253171843914780065137373446222406322953569124771480101363180966448099882292453452395428270875732536311539266115116490704940164192417744919250000894727407937229829300578253427884494358459949535231819781361449649779252948099909821642207485514805768288115583409148969875790523961878753124972681179944234641016960011815788847436610192704551637034472552319820336532014561412028820492176940418377074274389149924303484945446105121267538061583299291707972378807395016030765440655601759109370564522647989156121804273012266011783451102230081380401951383582987149578229940818181514046314819313206321375973336785023565443101305633127610230549588655605951332351485641757542611227108073263889434409595976835137412187025349639504404061654653755349162680629290551644153382760681862294677414989047491922795707210920437811136712794483496437355980833463329592838140157803182055197821702739206310971006260383262542900044072533196137796552746439051760940430082375641150129817960183028081010978780902441733680977714813543438752546136375675139915776`

17th Perfect Number

This is the 17th perfect number. It has 1373 digits. The next perfect number is 2^6433-2^3216.

999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999
999999999999999999999999999999

Une See's Wall'O'Nines

This is the 5th valid entry in the "My Number is Bigger" competition. The number was entered by user "Une See". The number contains 1440 nines. This makes it larger than the 20th Mersenne Prime, but less than 5^^3.

101500

Odd Perfect Threshold

It has been proven, if there are any odd perfect numbers, they must be largest than 101500 (see paper here). Combining this with the Euclid-Euler Theorem, which states that all even perfect numbers are of the form 2^(p-1)(2^p-1), where p is prime and 2^p-1 is a mersenne prime, as well as the exhaustive search of the mersenne primes below 101500 gives us the definite result that the first 17 known perfect numbers, are in fact the 17 smallest perfect numbers. That is, there are no other perfect numbers less than 101500.

This number is an example of a fairly large number occurring in a mathematical paper, though it's far from the largest in a mathematical paper (see Graham's Number and TREE(3) for much much larger examples).

101500 also occurs in theoretical cosmology. It is estimated that in 101500 years we will enter the age of the iron stars; an age of incredible darkness in which almost all the matter in the universe will be reduced to black holes or stars that emit no light called "iron stars", that are just cold spheres of iron. Humanity, and probably any form of life, or even machine intelligence, will probably have long since past away by this time. Imagine that kind of darkness and silence ... THAT is the eternity that awaits us ... grim stuff, eh?

9*(16^1441-16)/15+10

Twasbrillig's Return

This is the 6th valid entry in the "My Number is Bigger" competition. Twasbrillig responds to Une See's new entry by interpreting the wall of 9's as hexadecimal and appending an "A". This can be described quite succinctly as 9*(16^1441-16)/15+10. Despite the clever change in base this number is not a big jump from before. It has roughly 1735 digits, vs. 1440. That's only a jump of 295 orders of magnitude. There were more orders of magnitude jumped from the 1st to the 2nd valid entry, though Une See's Wall'O'Nines was the most drastic jump in order-of-magnitude so far. We are still only skirting the lower end of the Super-Astronomical Epoch, and the contestants are still only relying on decimal notation which is fairly weak googologically speaking.

10*16^1440+6*(16^1440-1)/15

Blatm's First Entry

This is the 7th valid entry in the "My Number is Bigger" competition. Blatm, who was has been standing by the sidelines up until this point makes his first entry. He turns Twasbrillig's entry upside-down, turning the 9's into 6's, the A unaffected, but now in the beginning instead of the end of the number. This number is still in hexadecimal and still only has about 1735 digits in decimal.

11*16^1440+6*(16^1440-1)/15

Twasbrillig's Rebuttal II

This is the 8th valid entry in the "My Number is Bigger" competition. Twasbrillig jumps back in the race by changing Blatm's Number by swapping the "A" with "B". This number is still not much bigger and we are still stuck around 1735 digits. This is the last time writing out digits does any good at all, because the next entry completely mops the floor with this sort of thing. (See 10^10^10)

26433-23216

`33570832131986724437010877211080384841138028499879725454996241573482158450444042882048778809437690388449535774260849885573694759906173841157438424730130807047623655942236174850509108537827658590642325482494761473196579074656099918600764404702181660294469121778737965822199901663478093006075022359223201849985636144177185925402078185073015045097727084859464743635537781500284915880244886306461785982956072060013474955617851481680185988557136609224841817877083608951191123174885226416130683197710667392351007374503755403352531476227943590071651702697594241031955529898971218001214641774673134944471562560957179657881556419122102935450299751813340515170956167951095453649485576150660101689160658011770193274226308280507786835049549112576654510119670456745939890194205255175384484489909328967646988163155982471564998196261632751283127879509198074253193409580454562488664383465379885002735506153988851506645137759275553988219425439764732399824712438125054117523837438256744437055019441051006489972341609117978404563794992004873057518455748701444951238377139620494287982489529827233140637014837408856156199515457669607964052126908149265601786094447595560440059050091763547114092255371397425807867554352112542194784815494784276201170845949274674632985210421075531784918358926690395463649721452265405713484388043911634485432358638806645313826206591131266232422007835577345584225720310518698143376736219283021119287617896146885584860065048876315701088796219593640826311622273328035603309475642390804499460156797855361018246696101253922254567240908315385468240931846166962495983407607141601251889544407008815874744654769507268678051757746956891212485456261121386667407711139619071530923355823178662705374393035049022603882479742334799407130280148769298597743778193050348749740786928096033906295910199238181338557856978191860647256209708168229116156300978059197026855726877649767072684960463452763160384093838292277544911857859658328888332628525056`

18th Known Perfect Number

This is the 18th known perfect number. It has 1937 digits. I say "known" rather than 18th perfect number because there is still the possibility of "odd perfect numbers". If an odd perfect number exists, it is greater than 101500, below this threshold we can be sure they don't exist. Combining this with the Euclid-Euler Theorem that states that every even perfect number is of the form 2^(p-1)(2^p-1) where both p and 2^p-1 is prime, we can be sure that the first 17 perfect numbers, obtained from the first 17 mersenne primes are indeed the only perfect numbers from 1 to 101500. Beyond this point however there is the chance, admittedly small, that there may be odd perfects between the known even cases. For this reason perfect numbers beyond this point will be numbered by sorting the known ones in size order. So this is the 18th *known* perfect number. The next perfect number is 2^8505-2^4252.

191101259794547752035640455970
396459919808104899009433713951
278924652053024261580301205938
651973985026558644015579446223
535921278867380697228841014691
598660208796189675719570183928
166033804761122597553362610100
148265112341314776825241149309
444717696528275628519673751439
535754247909321920664188301178
716912255242107005070906467438
287085144995025658619446154318
351137984913369177992812743384
043154923685552678359637410210
533154603135372532574863690915
977869032826645918298381523028
693657287369142264813129174376
213632573032164528297948686257
624536221801767322494056764281
936007872071383707235530544635
615394640118534849379271951459
450550823274922160584891291094
518995994868619954314766693801
303717616359259447974616422005
088507946980448713320513316073
913423054019887257003832980124
605019701346739717590902738949
392381731578699684589979478106
804282243609378394633526542281
570430283244238551508231649096
728571217170812323279048181726
832751011274678231741098588868
370852200071173349225391332230
075614718042900752767779335230
620061828601245525424306100689
480544658470482065098266431936
096038873625851074707434063628
697657670269925864995355797631
817390255089133122329474393034
395616132833407283166349825814
522686200430779908468810380418
736832480090387359621291963360
258312078167367374253332287929
690720549059562140688882599124
458184237959786347648431567376
092362509037151179894142426227
022006628648686786871018298087
280256069310194928083082504419
842479679205890881711232719230
145558291674679519743054802640
464685400273399386079859446596
150175258696581144756851004156
868773090371248253534383928539
759874945849705003822501248928
400182659005625128618762993804
440734014234706205578530532503
491818958970719930566218851296
318750174353596028220103821161
604854512103931331225633226076
643623668829685020883949614283
048473911399166962264994856368
523471287329479668088450940589
395110465094413790950227654565
313301867063352132302846051943
438139981056140065259530073179
077271106578349417464268472095
613464732774858423827489966875
505250439421823219135722305406
671537337424854364566378204570
165459321815405354839361425066
449858540330746646854189014813
434771465031503795417577862281
1776585876941680908203125

5^5^5

This value, 5^^3, can be approximated as 1.911x10^2184. It's lies between 10^2184 and 10^2185, and thus it is between a centillion and a millillion. This number is still small enough that it can actually be computed.

28505-24252

19th Known Perfect Number

The 19th known perfect number. It has 2561 digits. The next perfect number is 2^8845-2^4422.

28845-24422

20th Known Perfect Number

This is the 20th known perfect number. It has 2663 digits. The next known perfect number is 2^19,377-2^9688.

29689-1

21st Mersenne Prime

This is the 21st mersenne prime. It has 2917 digits. It was first discovered by Donald B. Gillies on May 11th of 1963. It held the title for largest known prime for a mere 5 days! The next mersenne prime is M9941.

29941-1

22nd Mersenne Prime

This is the 22nd mersenne prime. It contains 2993 digits. It was first discovered by Donald B. Gillies on May 16th of 1963. It held the title for largest known prime for about 2 weeks. The next mersenne prime is M11213.

103003

millillion

The 1000th member of the short scale -illion series. Although not "official" a variant of it is endorsed in Conway & Guys "Book of Numbers", and has become nigh canon in the googology community.

Bowers' mentions this number on his -illions page, and says that "he found out he was not the only one to give this number this name", implying that Bowers' independently coined this -illion. It should be noted that the actual name for this number in Conway & Guys system is actually millinillion, not the arguably more sensible millillion. The name millillion can therefore be thought of as a very small Bowerism. It's one of the smallest to which we can attribute to him, although there are a few smaller.

211,213-1

23rd Mersenne Prime

This is the 23rd mersenne prime. It has 3376 digits. It was first discovered by Donald B. Gillies on June 2nd of 1963. It held the title for largest known prime for about 8 years from 1963 to 1971. The next mersenne prime is M19937.

105000

googolbell

Part of a series of modification on a googol, including googolding, and googolchime. The next largest one is googoltoll.

219,377-29688

21st Known Perfect Number

This is the 21st known perfect number. It has exactly 5834 digits.
The next known perfect number is 2^19,881-2^9940.

219,881-29940

22nd Known Perfect Number

This is the 22nd known perfect number. It has 5985 digits.
The next known perfect number is 2^22,425-2^11,212.

219,937-1

24th Mersenne Prime

This is the 24th mersenne prime. It has 6002 digits. It was first discovered by Bryant Tuckerman on March 4th of 1971. It held the title for largest known prime for 7 years from 1971 to 1978. The next mersenne prime is M21701.

221,701-1

25th Mersenne Prime

This is the 25th mersenne prime. It contains 6533 digits. It was first discovered by Landon Curt Noll and Laura Nickel on October 30th of 1978. It held the title for largest known prime for about 4 months. The next mersenne prime is M23209.

222,425-211,212

23rd Known Perfect Number

This is the 23rd known perfect number. It has 6751 digits.
The next known perfect number is 2^39,873-2^19,936.

223,209-1

26th Mersenne Prime

This is the 26th mersenne prime. It has 6987 digits. It is the largest mersenne prime less than a googoltoll. It was first discovered by Landon Curt Noll on February 9th of 1979. It held the title for largest known prime for only a mere 2 months. The next mersenne prime is M44497.

1010,000

googoltoll

A googoltoll is 1 followed by 10,000 zeroes. I coined this name by extension with the googolgong. It's name is based on the idea that bells "toll" or "ring" in a way similar to a gong. It serves as a very round bench mark number. It can also be written as 10^10^4. It lies between the 26th and 27th mersenne primes

This number is already far too large to comprehend ... but we can try. A googoltoll is the 100th power of a googol. So if we have any concept of what the googol is like, we can imagine dwarfing it by a factor of itself an additional 99 times to get a feel for a googoltoll. It is also the 10th power of a googolchime, already a very large number. Suffice it to say we have yet to encounter anything like a googoltoll in the universe. But you ain't seen nothing yet ... (see googolgong)

239,873-219,936

24th Known Perfect Number

This is the 24th known perfect number. It has 12,003 digits.
The next known perfect number is 2^43,401-2^21,700.

243,401-221,700

25th Known Perfect Number

This is the 25th known perfect number. It has 13,066 digits.
The next known perfect number is 2^46,417-2^23,208.

244,497-1

27th Mersenne Prime

This is the 27th mersenne prime. It has 13,395 digits. It's the first mersenne prime greater than a googoltoll. It was first discovered by Harry Lewis Nelson and David Slowinski on April 8th of 1979. It held the title for largest known prime for about 3 years from 1979 to 1982. The next mersenne prime is M86243.

246,417-223,208

26th Known Perfect Number

This is the 26th known perfect number. It has 13,973 digits.
The next known perfect number is 2^88,993-2^44,496.

2^2^2^2^2

This is 2 tetrated to the 5th. 2^^5 = 2^2^2^2^2 = 2^2^2^4 = 2^2^16 = 2^65,536 ~ 10^19,728.

286,243-1

28th Mersenne Prime

This is the 28th mersenne prime. It contains exactly 25,962 digits! It was first discovered by David Slowinski on September 25th of 1982. It held the title for largest known prime until 1983 when Slowinski found an even larger one! The next mersenne prime is M110503.

288,993-244,496

27th Known Perfect Number

This is the 27th known perfect number. It has 26,790 digits.
The next known perfect number is 2^172,485-2^86,242.

1030,003

decimillillion / myrillion

A myrillion is one of the 433 number names coined by Bowers' and it's one of his smallest googolism's (His smallest is cenuntillion for 10306 ). It comes from myriad for 10,000. It thus translates literally as ten thousandth illion. The myriad however is greek, where as the other prefixes used for illions are usually latin, so it is actually an inconsistent usage. However it is a fairly simple and easy to understand googolism. A more appropriate name for this number is probably decimillillion, which uses the latin prefixes deci- and milli- for ten and thousand respectively.

2110,503-1

29th Mersenne Prime

This is the 29th mersenne prime. It contains 33,265 digits. It was first discovered by Walter Colquitt and Luke Welsh on January 28th of 1988. At the time of it's discovery it was not the largest known prime. Rather it was discovered as a missing mersenne prime between M86243 and M132049. The next mersenne prime is M132,049.

6^^3 = 6^6^6 = 6^46,656 ~ 10^36,305. This number goes beyond ordinary astronomical numbers, and is actually post-astronomical. Yet this is still pretty small as far as tetrational numbers go, and it isn't even large enough for me to really call it hyper-exponential. This number is still small enough that I can actually compute it online using a big number calculator (see decimal expansion via link above), though this is beginning to push the limits of what I can work with directly.

2132,049-1

30th Mersenne Prime

This is the 30th mersenne prime. It has 39,751 digits, making it just a little larger than 6^6^6. It was discovered by David Slowinski on September 19th of 1983. The next mersenne prime is M216091.

2172,485-286,242

28th Known Perfect Number

This is the 28th known perfect number. It has a whopping 51,924 digits. The javascript I wrote to compute the digits of it took about 48 seconds to load. The next known perfect number is 2^221,005-2^110,502.

2216,091-1

31st Mersenne Prime

This is the 31st mersenne prime. It has 65,050 digits. It's the largest mersenne prime less than a googolgong. It was discovered by David Slowinski on September 1st of 1985. It's one of only 4 mersenne primes that were discovered in the 1980s. The next mersenne prime is M756839, that wasn't discovered until 1992.

2221,005-2110,502

29th Known Perfect Number

This is the 29th known perfect number. It has 66,530 digits. The javascript I wrote took about 1 minute and 20 seconds to compute all the digits of this number. The next known perfect number is 2^264,097-2^132,048.

2264,097-2132,048

30th Known Perfect Number

This is the 30th known perfect number. It has 79,502 digits. The javascript I wrote took about 1 minute and 48 seconds to compute all the digits of this number. The next known perfect number is 2^432,181-2^216,090.

2276,709

Hitchhiker's Number

This number comes from the "Hitchhiker's Guide to the Galaxy", the first of a science-fiction book series by Douglas Adams. In the 8th chapter of the first book it is stated that you can survive in the total vacuum of space for about 30 seconds, and that the probability of being picked up by a passing spaceship within that time frame is "two to the power of two hundred seventy-six thousand, seven hundred and nine to one against". This number has sometimes been cited as the largest number appearing in a work of fiction.

The number has exactly 83,298 digits and it begins 511,764,533,051,720,592,987,157,233,954, ... ... and ends with ... ... 483,635,033,435,620,175,872,379,584,512. It can be approximated as E83,297.70907. This makes it larger than 6^6^6 but smaller than a googolgong. This number is way too large to be described as merely astronomical, as numbers this large don't even occur in astronomy! In fact the claimed improbability seems to be way to large. Even if there was only one person and one intergalactic spacecraft in the entire observable universe, the probability that both would be within a 1 meter proximity would still only be about 1078 to one against; vanishingly smaller than the Hitchhiker's Number. Even if the universe we're made much much bigger to account for the high-improbability it still couldn't account for the extremely low density of intergalactic spacecraft. Apparently in the Hitchhiker's series the universe must be an extremely lonely place.

10100,000

googolgong

This was the largest number that I "knew" about as a kid. The father of my best friend had told me about it when I was explaining the centillion to him. He told me that there was some number called a "googolgong" which was 1 followed by 100,000 zeroes that was a number scientists had come up with. I didn't know it at the time, but he was incorrectly explaining the googolplex to me. Because he changed both the name and the definition, I have since appropriated it as my own number, and used it as a base for a whole series of larger numbers.(See also googolbong , googolthrong, googolplexigong).

2432,181-2216,090

31st Known Perfect Number

This is the 31st known perfect number. It has a whopping 130,100 digits! This makes it larger than a googolgong. In fact its the first known perfect number greater than a googolgong. The javascript I wrote to compute it's digits took over 4 minutes to load! The next known perfect number is a huge leap forward with 455,663 digits, and is equal to 2^1,513,677-2^756,838.

2756,839-1

( 227,832 digits )

32nd Mersenne Prime

This is the 32nd mersenne prime. It has 227,832 digits. It begins 174... and ends with ...7. It was discovered by David Slowinski and Paul Gage on February 19th 1992. The next mersenne prime is M859433.

2859,433-1

( 258,716 digits )

33rd Mersenne Prime

This is the 33rd mersenne prime. It was discovered by David Slowinski and Paul Gage on January 4th of 1994. The next mersenne prime is M1,257,787.

M1,257,787

( 378,632 digits )

This is the 34th mersenne prime. It was discovered by David Slowinski and Paul Gage on September 3rd of 1996. This is the last mersenne prime that was found that was not part of the GIMPS (Great Internet Mersenne Primes Search) project. The next mersenne prime, M1398269 was the first found by GIMPS.

2,996,863,034,895*21,290,000-1

( 388,342 digits )

*Largest Known Twin Prime Pair*
( As of 2016)

This is the smaller of the pair of largest known twin primes! Like the mersenne primes, large twin primes are found by distributed computing projects, namely The Twin Prime Search and PrimeGrid. This number is huge and contains 388,342 digits. It's worth noting however that it's quite small compare to the largest known prime number.

2,996,863,034,895*21,290,000+1

( 388,342 digits )

*Largest Known Twin Prime Pair*
( As of 2016 )

This is the larger of the pair of largest known twin primes. This is one of the very rare instances on this list in which consecutive entries are a unit or less apart. Most of the time they are incomprehensibly far from each other!

21,398,269-1

( 420,921 digits )

35th Mersenne Prime

This is the 35th mersenne prime, and the first found by GIMPS. Credit is given to Joel Armengaud who found it on  November 13th of 1996. The next mersenne prime is M2,976,221.

21,513,677-2756,838

( 455,663 digits )

32nd Known Perfect Number

The next known perfect number is 2^1,718,865-2^859,432.

21,718,865-2859,432

( 517,430 digits )

33rd Known Perfect Number

The next known perfect number is 2^2,515,573-2^1,257,786.

~3.7598*E695,974

7^7^7

7^^3 = 7^7^7 = 7^823,543 ~ 10^695,974. This number has nearly a million digits!

22,515,573-21,257,786

( 757,263 digits )

34th Known Perfect Number

The next known perfect number is 2^2,796,537-2^1,398,268.

22,796,537-21,398,268

( 841,842 digits )

35th Known Perfect Number

The next known perfect number is 2^5,952,441-2^2,976,220.

22,976,221-1

( 895,932 digits )

36th Mersenne Prime

This is the 36th mersenne prime. Credit is given to Gordon Spence who found it on August 24th of 1997. The next mersenne prime is M3,021,377.

23,021,377-1

( 909,526 digits )

37th Mersenne Prime

This is the 37th mersenne prime. This is the largest mersenne prime with less than a million digits. Credit is given to Roland Clarkson who found it on January 27th of 1998. The next mersenne prime is M6,972,593.

Hyper-Exponential Epoch
[101,000,000,1010^1,000,000)
Entries: 110

101,000,000

milliplexion

This number is a borderline case of an extremely large number. I've used it as a bench mark for entering into number region I call the "Hyper-Exponential Numbers". These are loosely defined as numbers which have an exponential number of digits. There is a sort of grey area between exponential/astronomical numbers and hyper-exponential Numbers. Is a million an exponential number? I've usually started my exponential range closer to a billion rather than a million, on account of the fact that you can actually count to a million. So if this is a hyper-exponential number, it's a borderline case. It's certainly is too large to be called "Astronomical".

Robert Munafo has used this as the upper-limit of his Class 2 numbers. Munafo's Class 2 numbers roughly correspond to my idea of exponential numbers. Beyond this point we enter Class 3 numbers and the hyper-exponentials.

25,952,441-22,976,220

( 1,791,864 digits )

36th Known Perfect Number

This is the first known perfect number to exceed a million digits. Consequently it's the smallest known hyper-exponential perfect number. The next known perfect number is 2^6,042,753-2^3,021,376.

26,042,753-23,021,376

( 1,819,050 digits )

37th Known Perfect Number

The next known perfect number is 2^13,945,185-2^6,972,592.

26,972,593-1

( 2,098,960 digits )

38th Mersenne Prime

This is the 38th mersenne prime, and one of only a handful in the hyper-exponential range. It has a whopping 2,098,960 digits. Credit is given to Nayan Hajratwala who found it on June 1st of 1999. The next mersenne prime is M13,466,917.

103,000,003

milli-millillion

This was the largest -illion in Prof. Henkle's 1904 proposal. As far as I know prof. Henkle was the first to extend the latin based -illion all the way up to the millionth member. Although the fine points of his system have fallen into disuse in the googological community, John Conway's popular extension follows very closely Henkle's proposal and is in fact a nice improvement. In Conway's system however this number is officially called millinillinillion. This is far more ad hoc and less natural than milli-millillion which is literally means "thousand thousand"-illion. Jonathan Bowers' calls this number micrillion.

213,945,185-26,972,592

( 4,197,919 digits )

38th Known Perfect Number

The next known perfect number is 2^26,933,833-2^13,466,916.

213,466,917-1

( 4,053,946 digits )

39th Mersenne Prime

This is the 39th mersenne prime. It has over 4 million digits! Credit given to Michael Cameron who found it on November 14th of 2001. The next mersenne prime is M20,996,011.

1,000,0001,000,000 = 106,000,000

fzmillion

A fzmillion is a million to the millionth power. It is "slightly" larger than a milliplexion or 1 followed by one million zeroes. fzmillion is the 6th power of milliplexion and is equal to 10^6,000,000.

220,996,011-1

( 6,320,430 digits )

40th Mersenne Prime

This is the 40th mersenne prime. It has over 6.3 million digits! Credit given to Michael Shafer who found it on November 17th of 2003. The next mersenne prime is M24,036,583.

224,036,583-1

( 7,235,733 digits )

41st Mersenne Prime

This is the 41st mersenne prime. It has over 7.2 million digits! Credit is given to Josh Findley who found it on May 15th of 2004. The next mersenne prime is M25,964,951.

225,964,951-1

( 7,816,230 digits )

42nd Mersenne Prime

This is the 42nd mersenne prime. It has over 7.8 million digits! Credit is given to Martin Nowak who found it on February 18th of 2005. The next mersenne prime is M30,402,457.

226,933,833-213,466,916

( 8,107,892 digits )

39th Known Perfect Number

The next known perfect number is 2^41,992,021-2^20,996,010.

230,402,457-1

( 9,152,052 digits )

43rd Mersenne Prime

This is the 43rd mersenne prime. It has over 9 million digits! Credit is given to Curtis Cooper and Steven Boone who found it on december 15th of 2005. The next mersenne prime is M32,582,657.

232,582,657-1

( 9,808,358 digits )

44th Mersenne Prime

This is the 44th mersenne prime. It has over 9.8 million digits! Credit again goes to Curtis Cooper and Steven Boone who found it on September 4th of 2006. The next mersenne prime is M37,156,667.

237,156,667-1

( 11,185,272 digits )

45th Mersenne Prime

This is the 45th mersenne prime. It has 11,185,272 digits. Credit is given to Hans-Michael Elvenich who found it on September 6th of 2008. The next largest known mersenne prime is M42,643,801.

241,992,021-220,996,010

( 12,640,858 digits )

40th Known Perfect Number

The next known perfect number is 2^48,073,165-2^24,036,582.

242,643,801-1

( 12,837,064 digits )

46th Known Mersenne Prime

This is the next largest known mersenne prime after the 45th mersenne prime. It is not strictly known if no mersenne primes lie between M42,643,801 and M37,156,667. Credit for its discovery goes to Odd M. Strindmo who found it on June 4th of 2009. The next largest known mersenne prime is M43,112,609.

243,112,609-1

( 12,978,189 digits )

47th Known Mersenne Prime

This is the next largest known mersenne prime after M42,643,801. Credit for its discovery goes to Edson Smith, Woltman, and Kuroski who found it on August 23rd of 2008. This number remained the largest known prime for almost 5 years before it was taken over by the next largest known mersenne prime. The next largest known mersenne prime is M57,885,161.

248,073,165-224,036,582

( 14,471,465 digits )

41st Known Perfect Number

The next known perfect number is 2^51,929,901-2^25,964,950.

8^8^8

~10^15,151,335

8^^3 = 8^8^8 = 8^16,777,216 ~ 10^15,151,335.

251,929,901-225,964,950

( 15,632,458 digits )

42nd Known Perfect Number

The next known perfect number is 2^60,804,913-2^30,402,456.

257,885,161-1

( 17,425,170 digits )

48th Known Mersenne Prime

On January 25th of 2013 this number became the largest known prime number, as well as the largest known mersenne primeIt was discovered by Dr. Curtis Cooper as part of the GIMPS project. The next largest known mersenne prime is M74,207,281.

260,804,913-230,402,456

( 18,304,103 digits )

43rd Known Perfect Number

The next known perfect number is 2^65,165,313-2^32,582,656.

265,165,313-232,582,656

( 19,616,714 digits )

44th Known Perfect Number

The next known perfect number is 2^74,313,333-2^37,156,666.

274,207,281-1

( 22,338,618 digits )

49th Known Mersenne Prime/

Largest Known Prime Number
(As of 2016)

As of January 2016, this is currently the largest known prime number, as well as the largest known mersenne prime, an extremely rare type of number known since antiquity. There are currently only 49 known mersenne primes, and it is not known whether there are an infinite number of them or not. Regardless of that mersenne primes are exceedingly rare. They appear to grow at a roughly hyper-exponential rate. The exact sequence number for this mersenne prime is not known, but it is at least the 49th mersenne prime in the sequence.

This number has exactly 22,338,618 digits. It isn't difficult to confirm that this number falls between 8^8^8 and 9^9^9. Since 8^8^8 = 2^(3*8^8) = 2^50,331,648 we can see that it's much smaller. On the other hand since 9^9^9 > 8^9^9 = 2^(3*9^9) = 2^1,162,261,467 we can see that this is much much larger. So this number is actually much more close to 8^8^8 than it is to 9^9^9.

274,313,333-237,156,666

( 22,370,543 digits )

45th Known Perfect Number

This is the 45th known perfect number. It has a whopping 22,370,543 digits, making it just slightly larger than the current largest known prime with 22,338,618 digits. I use the term slightly larger loosely here. In actuality this number is 1031,925 times larger!!! They are not close in the ordinary day-to-day sense in which their ratio is something benign. They are close in what we might call the googological sense. That is to say they are in roughly the same vicinity of large numbers. We can informally say two numbers are googologically close if there exists no googologically significant number between them. In other words, numbers that are consecutive to each other on this list are usu. googologically close in some sense. The gaps between googologically significant number however just keep getting more and more insane ...

The next known perfect number is 2^85,287,601-2^42,643,800.

285,287,601-242,643,800

( 25,674,127 digits )

46th Known Perfect Number

The next known perfect number is 2^86,225,217-2^43,112,608.

286,225,217-243,112,608

( 25,956,377 digits )

47th Known Perfect Number

The next known perfect number is 2^115,770,321-2^57,885,160.

2115,770,321-257,885,160

( 34,850,340 digits )

48th Known Perfect Number

The next known perfect number is 2^148,414,561-2^74,207,280

2148,414,561-274,207,280

( 44,677,235 digits )

Largest Known Perfect Number
(As of 2016)

This is the largest known perfect number as of 2016. It has a mind-numbing 44,677,235 digits! Confirmed perfect numbers are extremely rare. There are only 49 confirmed perfect numbers. There may be an infinite number of them, but there is no mathematical proof of this yet. All the known perfect numbers are even, and can be expressed as the difference between two powers of 2. It is not known whether or not odd perfect numbers exist. New mersenne primes, and perfects are being found every few years by the Great Internet Mersenne Primes search. This probably won't be the last perfect number to appear on this list, so stay tuned for updates!

10100,000,000

googolbong

An even larger variation of the googolgong. The "bong" is a sound that gongs make. A googolbong is 1 followed by a hundred million zeroes. This makes it the 1000th power of a googolgong in the same way that a googolgong is the 1000th power of a googol. It is also the 1,000,000th power of a googol. Despite it's vast size, this is still a relatively small hyper-exponential number. ( See googolthrong )

~10^369,693,099

9^9^9

9^^3 = 9^9^9 = 9^387,420,489 ~ 10^369,693,099. This number has a bit of history in large number discussions. It's often said to be the largest value you can write using 3 digits and standard operations. It's also cited for being a number just beyond reach. Since it contains more than 300 million digits you would need something like the encyclopedia britannica just to store all the digits! You could easily store it on a flash drive, but it would take up 147 MiB of space!

101,000,000,000

billiplexion

This is a googolism I coined in a series that combines the popular short scale -illions with the plex suffix. Normally this number is referred to as billionplex. However I find this sounds clumsy. So I reverse the roots plex and -ion. The meaning however is the same. In this way we can create a whole series of googolisms for 10 raised to the power of an -illion number. The next one would be trilliplexion.

10^10^10

trialogue

This number is 10^^3. It is also the 9th valid entry in the "My Number is Bigger" competition. This number was entered by Gmalivak, the guy who began the competition with 9000 and his first competitive response. This also marks an important transition at which entries based on solely on decimal notation will no longer be competitive since the person would have to write out at least 10 billion digits. The upper limit of post length is probably much much smaller than this.

(10^10^10)^2

gartrialogue

A trialogue trialogues. It is 1 followed by 20,000,000,000 zeroes. It can also be expressed as 1020,000,000,000 and it's smaller than a googolthrong. This number was incorrectly given as an expression for a googolplex by Andre Joyce. The implication is that (a^b^c)^d = a^b^(c^d). This is false, and the failure to recognize this shows a lack of mathematical prowess on the part of Joyce. (10^10^10)^2 << 10^10^100. In fact, (10^10^10)^2 < 10^10^11. Cookiefonster gave this number the name gartrialogue by combining by googolism trialogue with the gar- prefix using it's original definition. Although it's smaller than a googolplex it's still a cool number and name in it's own right.

(10^10^10)^3

thrartrialogue

A trialogue cubed because "googology". Here I introduce a nifty new prefix "thrar-" from "three"+"gar" that allows us to cube the root instead of square it:

thrar(n) = n^3

This number is still only 1030,000,000,000 and therefore still smaller than a googolthrong.

10100,000,000,000

googolthrong

A googolthrong is 1 followed by a 100,000,000,000 zeroes. It's the 1000th power of a googolbong and the 10th power of a trialogue.

(794,843,294,078,147,843,293.7+1/30)*e^π^e^π

Ballium's Number

This is a spoof number jokingly called "the largest number" as in "the largest number possible". The joke video in which this number is defined can be found on youtube[2]. In the video mathematician "Samuel Ballium" claims that numbers do not go on forever and that the highest number is "794 quintillion 843 quadrillion 294 trillion 78 billion 147 million 843 thousand 293.7 3 recurring multiplied by e to the power of pi to the power of e to the power of pi".

Disappointingly this supposed "largest number" turns out not to be so big after all. Roughly speaking it would have about a trillion digits. More precisely it has exactly 138,732,019,350 digits. This places it between 10^10^11 and 10^10^12. This places it somewhere between a trialogue and a googolplex. The first few digits of it can be computed . Ballium's Number begins 2040427...

Unfortunately Ballium's Number is still not small enough to compute practically since it would require trillions of operations and the result would take up about a terabyte of information. This however does serve as an example of a typical persons idea of a very large number. It is reminiscent of Skewes' number which uses e in its definition.

If this was the "largest number" I'd be woefully disappointed as a googologist, because it's way way too small. Even a number like Graham's Number is relatively small compared to the numbers googologists have studied.

E297,121,486,765

11^11^11

11^^3 = 11^11^11 = 11^285,311,670,611 ~ 10^297,121,486,765. This is 11 tetrated to the 3rd. It is just above Ballium's Number, as it contains about twice as many digits. It is still less than 3^^4 however.

10^10^12

Size of Hypothetical Inflationary Universe

also

trilliplexion

Now days it's seems even physicists are into some really big numbers. This is the estimated size of the entire universe, assuming an inflationary model in which the universe expanded very rapidly in it's early evolution to account for the relative smoothness in the background radiation. It was Alan Guth's special Inflationary model that lead to this enormous figure. If he's correct, then there should also be about this many particles in the universe. This therefore could be the physical limit of an actualized number of objects!

This number falls just between the cracks of 11^^3 and 3^^4, two moderate sized tetrational numbers. It has roughly three times as many digits as 11^^3, but only a third as many digits as 3^^4, putting it almost dead center hyper-logarithmically.

E(3.63833*E12)

3^3^3^3

3^^4 = 3^3^3^3 = 3^3^27 = 3^7,625,597,484,987 ~ 10^10^12. This is 3 tetrated to the 4th. This number just above 11^^3 since:

11^11^11 < 10^10^12 < 3^3^3^3

As you can see, the base matters significantly less than the "tetrate".

E(9.622*E12)

12^12^12

12^^3 = 12^12^12 ~ 10^10^13. This is 12 tetrated to the 3rd. It is just slightly bigger than 3^^4. Actually it's roughly the cube of 3^^4 since it contains about three times as many digits! At this range of numbers however, that's considered pretty close.

10100,000,000,000,000

googolgandingan

or

googolgandingan is 1 followed by 100,000,000,000,000 zeroes. It can be written as 10^10^14. It's the 1,000,000,000,000th power of a googol and the 1000th power of a googolthrong. The name is derived from "gandingan", a special instrument composed of four gongs in series. Further modifiers can be used. See googolquintigong.

101,000,000,000,000,000

10100,000,000,000,000,000

googolquintigong

This is the first in a series, googolgong, googolbong, googolthrong, googolgandingan, with a formulaic name. We can combine the latin prefixes with -gong to indicate the number of times this is applied. Applying it 5 times to a googol gives us (10^100)^1000^5 = 10^10^17. Next is googolsextigong.

101,000,000,000,000,000,000

quintilliplexion

10^100,000,000,000,000,000,000

or

10^10^20

guppyplex / googolsextigong

This number may be called guppyplex or googolsextigong.

10^10^21

sextilliplexion

10^10^23

googolseptigong

10^10^24

septilliplexion

10^10^26

googoloctigong

In addition to being a googolism constructable with my naming scheme for ExE, this is also a lower bound on little foot.

100000000000000000000000000000000^10000000000000000000000000

(1032)^(1025)

little foot

An AMAZING INCREDIBLE TRULY COLOSSOL NUMBER ... which is nowhere fucking near BIG FOOT. Heck it's not even a contender against a googolplex, go fig.

It can be written more concisely as (1032)^(1025), or as 103.2*10^26 and bounded by 1010^27.

10^10^27

Upper Bound on "little foot"

also

octilliplexion

This is a simple upperbound on little foot that demonstrates its much much less than a googolplex.

10^10^29

googolnonigong

10^10^30

nonilliplexion

10^10^32

googoldecigong

10^10^33

decilliplexion

10^10^35

googol-undecigong

10^10^36

undecilliplexion

10^10^38

googol-duodecigong

10^10^39

duodecilliplexion

10^10^41

googol-tredecigong

10^10^42

tredecilliplexion

10^10^44

googol-quattuordecigong

10^10^45

quattuordecilliplexion

10^10^47

googol-quindecigong

10^10^48

quindecilliplexion

10^21*3^4^3^4

Upper Bound for Ballium's Number

This is an upper bound that can be used to prove that Ballium's Number is much less than a googolplex. Instances of "e" has been replaced with 3 and instances of "pi" has been replaced with 4. The first component of Ballium's Number has been replaced with 10^21. This value is actually a gross overestimate, yet it's still vastly smaller than a googolplex. This upper bound is approximately 10^10^48. (See Ballium's Number).

10^10^50

gogolplex / googol-sexdecigong

A gogol is a diminutive corruption of googol I invented. It's 1 followed by 50 zeroes. So a gogolplex is one followed by 50 zeroes. This number also gets the name googol-sexdecigong from another naming system of mine.

10^10^51

sexdecilliplexion

10^10^53

googol-septendecigong

10^10^54

septendecilliplexion

10^10^56

googol-octodecigong

10^10^57

octodecilliplexion

10^10^59

googol-novemdecigong

10^10^60

novemdecilliplexion

10^10^62

googolvigintigong

10^10^63

vigintilliplexion

10^10^92

googoltrigintigong

Here is a googolism that is "just shy" of a googolplex ... well, if raising a number to the 100,000,000th power to get the larger of the two can be considered close :p

56^56^56

~ 10^10^98.14

This is the largest member of n^^3 less than a googolplex. It's approximately equal to 10^10^98.1411176539.

10^10^100

googolplex

A googolplex is defined as 1 followed by a googol zeroes. A lot of attention has been given to this number do to it's vast size and simple explanation. It is also one of the very few googolism's coined by a professional mathematician, giving it some credentials. As far as Large numbers go however it's not actually that large!

In Hyper-E Notation this number can be written as E100#2 or E2#3.

(10^10^100)^2

gargoogolplex

A gargoogolplex was defined by Kieran (son of Alistair Cockburn) as a googolplex googolplexes. In other words a gargoogolplex is a googolplex squared. In terms of hyper-exponential numbers this isn't too much of an improvement. It turns out to be less than even 10^10^101. A gargoogolplex simply has twice as many zeroes as a googolplex, hence a gargoogolplex is 1 followed by 2 googol zeroes, or E(2E100). It follows E(2E100) < E(10E100) = EE101 = E101#2.

57^57^57

~ 10^10^100.329360333

This is the smallest member of n^^3 bigger than a googolplex. It is approximately 10^10^100.329360333. Interestingly it falls between a gargoogolplex and a thrargoogolplex, meaning it lies somewhere between the square and cube of a googolplex. It can be also be approximated as (10^10^100)^2.13481542964, making it closer to gargoogolplex then thrargoogolplex.

(10^10^100)^3

thrargoogolplex

A googolplex cubed. It can be expressed as 10^(3*10^100), and is equal to 1 followed by three googol zeroes. This number is still much smaller than 10^10^101.

10^10^101

This number might appear to be slightly larger than a googolplex. However the second exponent is very deceptive. In truth 10^10^101 = googolplex^10. In other words this number is a ...

googolplex googolplex googolplex googolplex googolplex googolplex googolplex googolplex googolplex googolplexes

The googolplex itself vanishes to an infinitesimal dot compare to this number! Yet these kind of thing is quite common with this range of numbers!!

Besides being instructive to the nature of hyper-exponential numbers, this number also serves as a lower-bound for the googol-bang.

(10100)!

googol-bang

This is an unusual number I have recently encountered on the internet. I first discovered it on Cantor's Attic, a website about the transfinite numbers that Michael B. brought to my attention. I haven't been able to figure out where this number has come from, but it is clearly pretty new. It only get's 30 hits on google, and it isn't even listed on the googology wiki. None the less it is a well defined number and I've decided to include it on my list.

Just as n-plex is defined as 10^n, n-bang is defined as n!. Thus a googol-bang is the factorial of a googol. One interesting thing about this number is that it turns out to be just "a little larger" than a googolplex. In fact we can get decent bounds on this number without any sophisticated mathematics or trillions of computations!

It turns out that a googol-bang lies between 10^10^101 and 10^10^102. To see the full proof along with a good approximation click here.

10^10^102

This number is equal to googolplex^100. In other words its a ...

googolplex googolplex googolplex googolplex ... ... ... ... googolplex googolplex googolplex googolplexes

where you say googolplex a hundred times. This number serves as an upper-bound on the googol-bang.

10^10^122

10^10^152

googolquinquagintigong

E(8.0723*E153)

4^4^4^4

4^^^2 = 4^^4 = 4^4^4^4 = 4^4^256 ~ 10^10^153. This is 4 tetrated to the 4th, and also 4 pentated to the 2nd. It's a very small pentational number, but a moderately sized tetrational number. It's even larger than a googolplex and googol-bang, but it's still less than a promaxima, so in some sense it's still in the practical number range.

10^10^182

googolsexagintigong

10^10^200

gargoogol-plexed

By adding -ed to the plex operator, it is implied that the -plex suffix should be applied after the gar- prefix, in exception applying gar after all other suffixes. The result is a larger number than gargoogolplex, perhaps counter-intuitively. In any case this serves as a nifty name for 10^10^200.

100^100^100

100^^3 = 100^100^100. This is 100 tetrated to the 3rd. This number is much larger than a googolplex but still much much smaller than a googolduplex. It can be computed exactly as:

100^100^100 = 100^10^200 = 10^(2*10^200)

It can also be directly compared to a googolplex:

10^(2*10^200) = (10^10^100)^(2*10^100)

In other words, 100^^3 is a googolplex raised to the power of two googol. To put that in perspective, if the googolplex were a sphere with volume googolplex, then you'd have to dwarf this sphere by a factor of a googolplex 20,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 times to reach a sphere of volume 100^^3. Googolplex sounds pretty microscopic now doesn't it ... hold on because were just getting warmed up!

10^10^212

googolseptuagintigong

10^10^242

googoloctogintigong

10^10^245

promaxima

Back in 2004, I had made my first large numbers post on the internet, on the "Big Numbers Page". I was trying to come up with an upper bound on the number of possible parallel universes that would theoretically exist, if every possible history were counted as it's own universe. This computation can be made finite by assuming that measurements below the Planck scale are not meaningful. I came up with the figure 10^10^245.

10^10^272

googolnonagintigong

10^10^300

thrargoogol-plexed

1 followed by thrargoogol zeroes, as the name implies.

10^10^302

googolcentigong

10^10^303

ecetonplex

This is a number I called centillionillion as a kid. This was probably the very first large number I ever devised. It was in response to hearing about the googolgong, 1 followed by 100,000 zeroes, that I was inspired to devise an even larger larger number, 1 followed by a centillion zeroes.

~10^10^2184

5^5^5^5

5^^4 = 5^5^5^5 = 5^5^3125 ~ 10^10^2184. This is 5 tetrated to the 4th. This number is larger than an ecetonplex but still less than a googolplexigong.

10^10^3002

googolmilligong

~10^10^36,305

6^6^6^6

6^^4 = 6^6^6^6 = 6^6^46,656 ~ 10^10^36,305.

10^10^100,000

googolplexigong

This number is a result of combing googolplex with my -gong suffix. This number is greater than a googolplex but less than Skewes' Number.

(10^10^100,000)^2

gargoogolplexigong

A googolplexigong squared. At this scale it barely matters.

(10^10^100,000)^3

thrargoogolplexigong

A googolplexigong cubed. Yawn...

10^10^200,000

gargoogolgong-plexed

The order that suffixes and prefixes are evaluated matters. When a base number has only suffixes or only prefixes, then it unambiguously means that we can find its value by taking the base and applying the operators from the inside out. However when a googolism involves both prefixes and suffixes, ambiguity is introduced. In less formal discussions this detail is usually glossed over or not even noticed, but here is my proposed solution. What numbers like gargoogolplex, fzgoogolplex, fugagoogolplex, and megafugagoogolplex suggest is that suffixes are always evaluated first before prefixes. Otherwise these numbers have completely different values. If we want the suffix to be evaluated second (which can sometimes actually lead to a bigger number), we can add -ed to the end of the suffix, implying it is "acting" upon all other operators. So under this ruling we have gargoogolgong-plexed means plex(gar(gong(googol))), unlike gargoogolplexigong which means gar(plex(gong(googol))). The result is a significantly larger number! Applying gar- to googolgong first gives us (10^100,000)^2 = 10^200,000. Then by "plexing" this result we get 10^10^200,000.

Unlike gargoogolplexigong and thrargoogolplexigong are only googolplexigong^2 and googolplexigong^3 respectively, gargoogolgong-plexed is actually googolplexigong^googolgong.

10^10^300,000

thrargoogolgong-plexed

1 followed by googolgong^3 zeroes.

7^2^999,997

~10^10^301,029

Upperbound on P1,000,000

This is a really bad upperbound that one can obtain for the one millionth prime number using some very basic number theory. It might seem that since the primes are "random" that there would be no way to predict how large a given prime number could be. Turns out this is false. We can bound primes both from below and above. For googological reasons, the elementary upperbounds are more interesting. For a full explaination for how to obtain this massive bound click the link here.

~10^10^695,974

7^7^7^7

7^^4 = 7^7^7^7 = 7^7^823,543 ~ 10^10^695,974. This number is larger than a googolplexigong. The difference in size however is deceptive. It's not 6.9 times larger. That's only how much larger it's leading exponent is. It's not even 10^595,000 times larger! No, you have to take a googolplexigong and multiply it by itself 10^595,000 times! That's a big difference! It means that a googolplexigong is dwarfed by a factor of itself, countless times before we reach 7^7^7^7. And we still are just getting started!

Tetranomical Epoch
[1010^1,000,000,10010)
Entries: 48

10^10^1,000,000

milliduplexion

This is one followed by one followed by a million zeroes zeroes.

(10^1,000,000)^(10^1,000,000)

fzmilliplexion

This number was created in a naive attempt to prove how "easy" it is to beat the numbers in googology by simply adding a lot of "zeroes". It can be simplified as 10^10^1,000,006 and it's approximately equal to 10^10^10^6. It's smaller than Skewes' Number but larger than a googolplex. It can't hold a candle to a mega or Graham's Number, let alone TREE(3). Decimal notation and elementary arithmetic are not enough to express numbers of this size. Googology turns out to be much harder than might be surmised at the outset. Incidently, I coined the name fzmilliplexion for this number using already existing googology. So even the construction isn't all that original :/

10^10^3,000,002

googolmilli-milligong

~10^10^15,151,335

8^8^8^8

8^^4 = 8^8^8^8 ~ 10^10^15,000,000.

~10^10^369,693,099

9^9^9^9

9^^4 = 9^9^9^9 ~ 10^10^369,000,000.

10^10^10^10

tetralogue

This is 10^^4. It is also the 10th valid Entry in the "My Number is Bigger" competition. This number was entered by Rodan in response to Gmalivak, but only after a failed attempt to add "infinity" to the competition (only finite numbers are allowed). This number was far bigger than anything previous at this point, but still nameable using elementary arithmetic. It's also only in the tetrational Epoch. It's still smaller than Skewes' Number.

e^e^e^79

Skewes' Number

Skewes' Number is equal to e^e^e^79. It can be approximated as 10^10^10^34, or more accurately as 10^10^10^33.947. Thus it lies between a tetralogue and a googolduplex.

It was first defined by Stanley Skewes in 1933 in a proof involving the distribution of primes. For a time it held the title of "largest number to appear in a serious mathematical proof". It was later trumped by 2nd Skewes' Number in 1955.

10^10^10^34

Skewes' Approxima

This is an approximation usually given for Skewes' Number, since Skewes' Number is about 10^10^10^33.947. This might seem like a good approximation, but this value is actually A LOT LARGER THAN Skewes' Number. How much larger? You'd have to raise Skewes' Number to the power of about 10^10^33 to get 10^10^10^34!

Here's a way to get an idea of what that means. Imagine that you had a sphere containing roughly a Skewes' Number particles. That sphere would be massive, even assuming the particles were tightly packed. Now imagine that sphere being just one amongst a Skewes' Number of such spheres! Imagine all these spheres are contained in A 2nd order "Skewes' sphere". Now imagine that is only one amongst a Skewes' Number of 2nd order Skewes' spheres all contained in a 3rd order Skewes' sphere!! Now keep scaling up to the 4th order, 5th, 6th, 7th, 8th, 9th, 10th, ... 100th, 1000th, millionth, billionth, trillionth, ... centillionth, ... ... ... ... and keep on going until you reach the 10^10^33 order sphere. That sphere will contain roughly 10^10^10^34 particles! Mind boggling! And this is only the difference between 10^10^10^33.947 and 10^10^10^34, and we're still only talking about moderately sized tetrational numbers!!!

10^10^10^100

googolduplex

Can be written E100#3 in Hyper-E. This number is mentioned on Jonathan Bowers' Infinity Scrapers page. It is not known who is credited with coining this googolism however. This number also has two other fairly common names: googolplexplex and googolplexian.

*** Bowers' is the earliest known person to use the term "googolduplex", and it is possible that he may have initiated the trend of inserting greek infixes into googolplex to continue the sequence. Googolduplex therefore might be a bowerism, though this is difficult to confirm as the coinage of googolduplex occurred before the googology community existed and accurate records of coiners was kept track.

10^10^10^303

ecetonduplex

E303#3 in Hyper-E

10^10^10^963

2nd Skewes' Demitto

This is the value usually cited as 2nd Skewes' Number. In truth 2nd Skewes' Number was defined as e^e^e^e^7.705 in the original paper. It turns out that this number is actually "slightly smaller". The real value is closer to 10^10^10^963.5185

e^e^e^e^7.705

2nd Skewes' Number

This is the exact value of 2nd Skewes' Number. It can be approximated in base-10 power tower form as 10^10^10^963.5185, which we can write in Hyper-E as E963.5185#3. The exact value of 2nd Skewes' Number is rarely stated in secondary sources and usually the approximations 10^10^10^963 or more crudely 10^10^10^1000 are used.

10^10^10^1000

2nd Skewes' Supremum

This is a very rough estimate typically cited as 2nd Skewes' Number. This is actually an upper-bound on the actual value (see 2nd Skewes' Number)

10^10^10^100,000

googolduplexigong

10^10^10^1,000,000

millitriplexion

10^10^10^10^10

pentalogue

10^10^10^10^100

googoltriplex

This number is mentioned on Jonathan Bowers' infinity scraper page along with googol, googolplex, and googolduplex.

10^10^10^10^303

ecetontriplex

10^10^10^10^100,000

googoltriplexigong

10^10^10^10^10^10

hexalogue

10^10^10^10^10^100

E100#5

This number is listed on Jonathan Bowers' Infinity Scrapers page as googolquadraplex. The spelling googolquadriplex is also sometimes used and has been popularized by myself as it's easier to continue by adapting the same latin prefixes used in the -illions. googolquadruplex is also sometimes used. This number is the first in the sequence googol,googolplex,googolduplex,googoltriplex,...etc. with no standard spelling. Bowers' may be solely responsible for the googolquadraplex spelling, and this spelling can therefore be considered a Bowerism.

10^10^10^10^10^303

10^10^10^10^10^100,000

10^10^10^10^10^10^10

heptalogue

10^10^10^10^10^10^100

E100#6

googolquintiplex

This number is called "googolquinplex" by Bowers, making googolquinplex a small bowerism. Oddly it doesn't show up in the "Googol Group" on his Infinity Scrapers page, but it can be found on his -illions page for "size comparison" with Bowers large -illions.

I coined the term googolquintiplex for this number to allow for easier extensibility: googolsextiplex, googolseptiplex, googoloctiplex, etc. It can be written concisely in Hyper-E Notation as E100#6.

(10^(10^(10^10!)!)!)!

Crazyjimbo's Factorial-Power Tower

This was the 13th valid entry in the "My Number is Bigger" competition. However it was already beaten by the previous two entries, thus it is not considered an "official" competitive entry. At this point it's beaten by a long shot by Twasbrillig's 10^^512.

Each factorial adds roughly another 10 to the stack so that you get about 10^^8 instead of 10^^4, although it is smaller since 10! < 10^10. Note that Factorials have higher priority than exponents, in which case there is no ambiguity here. This number is approximately E6#7, more precisely it's less than E3,628,809#6.

(10^(10^(10^11!)!)!)!

Rodan's Factorial-Power Tower

This is the 14th valid number entered in the "My Number is Bigger" competition. It still ranks well below the 11th and 12th entry, making it non-competitive. It's approximately E7#7.

10^10^10^10^10^10^10^10

octalogue

10^10^10^10^10^10^10^10^10

ennalogue

10^10^10^10^10^10^10^10^10^10

dekalogue / decker

This is one of Jonathan Bowers' original googolism's to appear on his list of Infinity Scrapers. He calls this number decker. I use the name dekalogue instead, and it also leads to a series of other extended names by changing the suffix.

E1#11

endekalogue

E1#12

dodekalogue

E1#13

13^^13

This number was the 11th valid entry in the "My Number is Bigger" competition shortly after Rodan's 10^10^10^10. This number was entered by Blatm in the form D^^D where "D" was hexadecimal for 13.

E1#14

E1#15

E1#16

E1#17

E1#18

E1#19

E1#20

icosalogue

E1#30

triantalogue

E1#40

terantalogue

E1#50

penantalogue

E1#60

exatalogue

E1#70

eptatalogue

E1#80

ogdatalogue

E1#90

entatalogue

Hyper Tetranomical Epoch
[10010,10^^10010)
Entries: 56

10^^100

hectalogue / giggol

The giggol is the first of Jonathan Bower's original extensions to the googol series. It is also the 2nd largest number to appear on Robert Munafo's Number list, so we could say the very large numbers begin here. The giggol can be written in Hyper-E Notation as E1#100.

(10^^100)^2

gargiggol

This might seem like a significant improvement over the giggol, like having a googolplex googolplexes is a lot more than a googolplex. It might seem that (10^10^10^...^10)^2 is the same as 10^10^10^...^10^2, but in fact a giggol squared is much smaller! (See next entry)

(10^^100)^3

thrargiggol

This number is still vastly smaller than even E2#100. Simply observe:

(10^^100)^3 = (E1#100)^3 = 10^(3E1#99) < 10^10^(1+E1#98) < ... < E(1+E1#1)#99

= E11#99 < E100#99 = E2#100

(10^^100)^(10^100)

giggol to the googol

Surely this is greater than E2#100? Nope, not even close:

(10^^100)^(10^100) = 10^(E1#99 * E100) = 10^10^(100 + E1#98) < 10^10^10^(1+E1#97)

= E(1+E1#97)#3 < E(1+E1#96)#4 < ... < E(1+E1#1)#99 = E11#99 < E100#99 = E2#100

(10^^100)^(10^10^100)

giggol to the googolplex

Believe it or not, this is barely an improvement over the last entry. We simply get:

(10^^100)^(10^10^100) = 10^10^(E100 + E1#98)

< 10^10^10^(1+E1#97) < ... < E11#99 < E100#99 < E2#100

E11#99

This number is 10^10^10^ ... ^10^10^11 w/99 10s. It's what you would get if you changed the top most exponent in a giggol with 11. Amazingly, despite the fact that this seems like a very minor improvement, it's better than even raising the giggol to the googolplex.

E2#100

This number is 10^10^10^ ... ^10^10^2 w/100 10s. It looks like giggol^2 but it's actually a lot larger. This serves as a lower-bound of megafuga-hundred.

100^^100

megafuga-hundred

In Alistair Cockburn's number system, megafuga-n = n^^n. So naturally megafuga-hundred = 100^^100. It's obvious that this must be larger than 10^^100 (giggol) but probably not by as much as you might think. In fact this number is less than 10^^101. In fact it's even less than E3#100.

E3#100

This number is 10^10^10^ ... ^10^10^3 w/100 10s. It looks like giggol^3 but it's actually much much larger. This serves as an upper-bound for megafuga-hundred.

10^^101

giggol-plexed

It is a common rookie response to take whatever the largest named number they know is, and simply have 1 followed by that many zeroes. By the time we get up to numbers of this size however things are moving much much faster, so it turns out to be not all that competitive a response. The reason for the prevalence of this kind of response is probably because f(n) = 10^n is the fastest growing function most people know of.

In Hyper-E this number can be represented as E1#101, or E10#100. E10#100 acts as a lower-bound for giggol^giggol (see next entry).

(10^^100)(10^^100)

fzgiggol

This was a number I used to illustrate how much larger a grangol is than a giggol. Even if you raised a giggol to it's own power, you still would come up vastly short from a grangol. giggol^giggol is also "ever so slightly" greater than 10^^101. This implies that giggol^giggol ~ 10^giggol. However it must be remembered that this is only an approximation. In truth "ever so slightly" is really a huge unimaginable gulf. You would need to raise 10^^101 to the 10^^99th power to get giggol^giggol, so it is really much much much larger in the ordinary sense. The following calculations confirm that giggol^giggol is indeed bounded by 10^^101 and grangol:

giggol^giggol = (E1#100)^(E1#100) = (10^E1#99)^(E1#100) = 10^(E1#99 * E1#100) [Law of Exponents]

= 10^(E1#100 * E1#99) = (10^E1#100)^(E1#99) = (E1#101)^(E1#99) = (10^^101)^(10^^99) > 10^^101

: 10^^101 < giggol^giggol

giggol^giggol = (E1#100)^(E1#100) = 10^(E1#99 * E1#100) = 10^10^(E1#98 + E1#99)

< 10^10^(2E1#99) < 10^10^(10E1#99) = 10^10^(10 * 10^(E1#98))

= 10^10^10^(1+E1#98) = E(1+E1#98)#3

< E(1+E1#97)#4 < E(1+E1#96)#5 < ... < E(1+E1#1)#100 = E(1+10)#100 = E11#100 < E100#100 = grangol

((10^^100)^(10^^100))^(10^^100)

giggol to the giggol raised to the giggol

As amazing as this number sounds, it isn't even as large as E(1+E1#98)#3, let alone E11#100. In my original proof for giggol^giggol < grangol there was an implicit proof that in fact (giggol^giggol)^giggol < E(1+E1#98)#3. This suggests that grangol is much much larger than a giggol than even my initial proof would suggest!

E(1+E1#98)#3

This number is a relatively exacting upper-bound on giggol^giggol. You can envision it as:

10^10^10^(1+10^10^10^10^ ... ^10^10^10)

where there are 98 10s inside the parenthesis. In other words, this number is a power tower of tens 101 terms high, with a +1 occurring at the 4th position heading upwards. In comparison with E11#100 (see next entry) this seems like a good bound, but in truth even if you raised giggol^giggol to the power of a giggol you still would come up vastly short of this number!

E11#100

This number acts as an upper-bound on giggol^giggol. It proves conclusively that it is less than a grangol. Even this upper-bound is actually a huge overestimate.

(100)[3]

This is f_3(100). This number is approximately E32#100, so it lies between a giggol (E1#100), and grangol (E100#100).

E100#100

grangol

A "grangol" is a number I myself coined in 2011. The name is derived from combining the words "grand" and "googol", thus a grangol is short for "grand googol". It's an example of a number expressible using my Hyper-E notation. Hyper-E notation is a modern equivalent of a notation I devised as a kid. For simple expressions like above let:

Ea#b = 10^10^10^ ... ^10^a w/b 10s

A grangol is therefore 10^10^10^...^10^10^100 w/100 10s. One way to think of a grangol is as a continuation of the googol tradition. We begin by creating a "googol series". The first member of this series is the eponymous googol, or 10^100. The 2nd member of the series is a googolplex, or 10^10^100. The 3rd member is a googolduplex or 10^10^10^100, and so on. A grangol is defined as the 100th member of the googol series. A grangol turns out to be "just a little" larger than Jonathan Bower's "giggol". In fact, it can be shown that:

giggolgiggol < grangol

That is, a giggol raised to a giggol is still smaller than a grangol. A giggol in Hyper-E is equal to E1#100. We can work it out as follows:

giggol^giggol = (E1#100)^(E1#100) = (10^E1#99)^(E1#100) =

10^(E1#99*E1#100) < 10^(E1#100*E1#100) = 10^(E1#100)^2 =

10^(10^E1#99)^2 < 10^(10^E1#99)^10 = 10^10^(10*E1#99) =

10^10^(10*10^E1#98) = 10^10^10^(1+E1#98)

= E(1+E1#98)#3 < E(1+E1#97)#4 < E(1+E1#96)#5 < ... etc. ... < E(1+E1#1)#100 =

E(1+10)#100 = E11#100 < E100#100 = grangol

Thus we conclude that giggol^giggol < grangol.

10^^102

This is the smallest power tower of 10s larger than a grangol. As such it acts as an upper-bound, allowing the grangol to be compared to other numbers. To prove grangol < 10^^102 observe:

grangol = E100#100 < E10,000,000,000#100 = E(10^10)#100 = E1#102 = 10^^102

E100#101

grangolplex

This was a number I coined to illustrate the inadequacy of the -plex prefix to capture higher level recursions. It is true that Bower's has used the -plex prefix to refer to any type of recursion, however based on the googolplex many have concluded that n-plex always means 10^n. If that is so than a grangolplex is an inadequate name for E100#100#2 (see grangoldex). Following the above definition it follows that grangol-plex = 10^grangol = E(E100#100) = E100#101.

This number is greater than 10^^102, but less than grangol^grangol. First observe:

E100#101 > E10#101 = E1#102 = 10^^102

Since grangolplex = 10^grangol it follows that it is less than grangol^grangol.

(E100#100)^(E100#100)

grangol to the power of a grangol

As usual, this number isn't "much larger" than a grangol, at least in terms of power tower height. It must be greater than 10^^102 since grangol^grangol > 10^grangol = E100#101 > E10#101 = E1#102 = 10^^102. However it must be less than 10^^103. This is a little more involved, but can easily be established as follows:

(E100#100)^(E100#100) = 10^(E100#99 * E100#100) = 10^10^(E100#98 + E100#99)

< 10^10^10^(1+E100#98)

= E(1+E100#98)#3 < E(1+E100#97)#4 < E(1+E100#96)#5 < ... < E(1+E100#1)#100

= E(1+E100)#100 < E(E101)#100 = E101#101 < E(10^10)#101 = E1#103 = 10^^103

10^^103

An upper-bound for grangol^grangol.

10^^257

This is a lower-bound that is commonly used for a Mega since it is the largest integral power tower of 10s which is less than a Mega. This lower-bound is still much much bigger than a grangol, proving that a Mega is also larger.

It can be written in Hyper-E as E1#257 or E10#256.

E619#256

This is a more accurate lower bound on the Mega which attempts to narrow down the top most exponent. Written in full it looks like:

10^10^10^10^ ... ... ^10^10^10^10^619 w/256 10s

2[5]

Mega / two in a circle

This number goes by various names, "Mega", "Zelda", "two in a circle" or "two in a pentagon". It is among the "classic" large numbers along with a googolplex, and Graham's Number. It was first defined by Hugo Steinhaus using his own custom operator notation...(READ MORE)

E620#256

This is a more accurate upper-bound on the Mega. It looks like:

10^10^10^10^ ... ... ^10^10^10^10^620 w/256 10s

10^^258

This is a common upper-bound on the Mega. It is the smallest expression in the form of 10^^N , where N is a positive integer, that is greater than the Mega.

It can be written in Hyper-E as E1#258 or E10#257 or E10,000,000,000#256.

500^^500

Alternative Interpretation of Blatm's D^^D

In the infamous "My Number is Bigger" thread, Gmalivuk pointed out that D^^D was vague and could be interpreted as 13^^13 using hexadecimal or 500^^500 using roman numerals. This latter interpretation is much much larger, though still smaller than Twasbrillig's Power Tower, the 12th competitive entry.

10^^512

Twasbrillig's Power Tower

This is the 12th valid entry in the "My Number is Bigger" competition. The original posting of this number was deleted unfortunately but traces of it remain in the responses. This was the first entry to officially break the Tetrational Epoch barrier. This is also the last in an uninterrupted string of larger and larger entries. After this some smaller entries are entered in the competition before this "Large Number in play" is overcome. (See Crazyjimbo's Factorial power tower for entry 13 ).

256^^512

A weak upper-bound on the Mega based on the Left Associative Tetrates Lemma.

10^^100,000

giggolgong

This number is the result of combining one of Bower's numbers with my -gong suffix. If a googolism, call it g, can be defined as f(100), then g-gong is defined as f(100,000). A giggol = 10^^100, and therefore the giggolgong is 10^^100,000. A giggolgong is a power tower of 10s 100,000 terms high! This number is way way bigger than a Mega, yet it's vanishingly small compare to 256^^(2^256), a naive upper-bound on the Mega.

In Hyper-E this number can be written as E1#100,000.

10^^100,001

A lower bound on a grangolgong.

E100,000#100,000

grangolgong

The grangolgong is equal to 10^10^10^ ... ^10^10^100,000 w/100,000 10s. It lies between E10#100,000 = E1#100,001, and E10,000,000,000#100,000 = E1#100,002.

10^^100,002

A upper bound on a grangolgong.

1,000,000^^1,000,000

megafugamillion

This is a number that came up in the xkcd forum "My number is, in fact, bigger!", an unofficial sequel to the "My Number is Bigger!" thread. The new competition was initiated by Vytron. This number was defined by "Earthling on Mars" as a part of a larger naive attempt to beat the number <10,10,googol> using iterated power towers.

This number is larger than 10^^1,000,000 and so is also larger than 10^^100,002. On the other hand, 1,000,000^^1,000,000 < (10^^2)^^1,000,000 < 10^^1,000,002. At the same time we have 10^^1,000,002 is less than (3^^3)^^1,000,002 < 3^^1,000,005 and therefore much less than 3^^7,625,597,484,987 = 3^^^3.

3^^^3

tritri

Jonathan Bowers' tritri is a relatively small pentational number. Expanding it reveals it's formidable size. We have...

3^^^3 = 3^^3^^3 = 3^^3^3^3 = 3^^3^27 = 3^^7,625,597,484,987

It's a power tower of 3's 7,625,597,484,987 terms high. This makes it unfathomably larger than 10^^100,002, but vanishingly small compare to 2^^(2^256).

2^^(2^256)

This is a naive upper-bound on the Mega. It is roughly equal to a power tower of 10s E77 terms high. This makes it less than a googol-stack, but more than a grangolgong.

10^^(10100)

googol-stack

This is another number I found on Cantor's attic. n-stack is defined as 10^^n. In other words, n-stack is a power tower of 10s "n" terms high. Having a power tower of a googol tens certainly seems pretty impressive. Yet this is still small for a pentational number. It is by necessity much larger than a "Mega" since a mega must be less than a power tower of tens only 258 terms high. Yet this number must also be vanishingly small compared to a grangoldex, because a grangoldex is greater than a "stack" of tens a grangol terms high, where a grangol is the 100th member of the googol series!

Therefore between the googol-stack and the grangoldex must be a vast sea of numbers!

E100#1#2

googoldex

The googoldex is a number I coined to illustrate just how many kinds of numbers can be named between my numbers using Hyper-E Notation. Hyper-E Notation has the advantage of more easily defining numbers between numbers in other systems.

The -dex prefix simply takes some number of the form Ea#b and returns Ea#b#2. So we let googol = E100#1, and so googoldex becomes E100#1#2. What does this mean? Working it out we obtain:

E100#1#2 = E100#(E100#1#1) = E100#(E100) = E100#googol =

EEEEEEEEEEEEE ... ... EEEEEEEEEEEEEEEE100 w/googol Es

= 10^10^10^10^ ... ... ^10^10^10^10^100 w/googol 10s

In other words, a googoldex is a power tower of 10s a googol terms high topped off with a 100. It's the googolth member of the googol series.

It's just "a little" larger than a googol-stack. In fact it's greater than 10^^(googol+1) but less than 10^^(googol+2).

(10100)^^(10100)

megafugagoogol

A megafugagoogol is only slightly larger than a googoldex. To see why consider the folowing:

googol^^googol > E(10^100)#(10^100-1) = E100#(10^100) = googoldex.

10^((10100)^^(10100))

megafugagoogol-plexed

This is a non-standard way to interpret "megafugagoogolplex". One can image the "megafuga" being applied before the "plex". To distinguish the cases I coin this number as megafugagoogol-plexed. This number is only slightly larger than a googoldex and vastly smaller than a googolplexidex. This can be demonstrated easily using the Left Associate Polyates Lemma (LAPL):

10^((10^100)^^(10^100)) < 10^((10^10^10)^^(10^100))

10^((10^^3)^^(10^100)) < 10^10^^(3+10^100) = 10^^(4+10^100)

< 10^^(10^10^100) = E1#(googolplex) < E100#(googolplex) = E100#(E100#2) = E100#2#2 = googolplexidex

:: megafugagoogol-plexed << googolplexidex

E100#2#2

googolplexidex

This number is a power tower of 10s a googolplex terms high topped off with a 100.

E100#(1+10^10^100)

googolplexidexiplex

This number serves a a lower-bound on the megafugagoogolplex. It is also an example of a combinatorial googolism. There is a wealth of numbers that can be derived from various googological systems, an only a tiny fraction of them are ever explicitly stated. These tend to aggregate in certain vicinities, because of the vast differences of power of the different word components.

(E100#2#2)^(E100#2#2)

googolplexidex to the power of a googolplexidex

This number is incredibly close to a megafugagoogolplex, yet it is still slightly smaller.

E(1+10^100)#(10^10^100)

This number is an upperbound on googolplexidex^^2.

E(100+10^100)#(10^10^100)

A very accurate lower-bound on a megafugagoogolplex.

(10^10^100)^^(10^10^100)

megafugagoogolplex

This number is derived from the work of Alistair Cockburn. It combines his megafuga- prefix with the number googolplex. n-plex = 10^n while megafuga-n = n^^n. Note that the definition here is actually ambiguous. Is a megafugagoogolplex equal to megafuga(googolplex) or plex(megafuga(googol). That is, we can read it grammatically as meaning either megafuga-"googolplex" or as "megafugagoogol"-plex. These result in different numbers (See megafugagoogol-plexed). The original intent of Cockburns work however is that the prefixes are being applied after the suffixes. Thus a megafugagoogolplex is intended to mean googolplex^^googolplex. This number has come up independently from a few sources , usually as an example of the largest kind of number the average person would think of to try to trump Graham's Number (psst ... it doesn't even come close. See Graham's Number far below). That being the case it's nice to have a name, any name, for this number.

This number is pretty insanely huge, although it has more to do with the height of the stack than the terms being a googolplex. It confers just enough benefit so that it goes slightly past a googolplexidex. A googolplexidex is a power tower with a googolplex+1 terms, where as a megafugagoogolplex has only a googolplex terms, but the megafugagoogolplex ends up being ever so slightly larger (from a googologist's perspective) mainly because of the leading exponent. It can be shown that megafugagoogolplex lies between E100#(googolplex+1) and E100#(googolplex+2), and is greater than googolplexidex^googolplexidex. For a worked out proof click here.

E(101+10^100)#(10^10^100)

This is an accurate upper-bound on the megafugagoogolplex.

E100#(2+10^10^100)

googolplexidexiduplex

This number serves as a "weak" upper-bound on a megafugagoogolplex. It's also an example of a combinatorial googolism.

E100#3#2

googolduplexidex

This number is a power tower of 10s a googolduplex terms high topped off with a 100. This number is still vastly smaller than a giggolplex.

E100#4#2

googoltriplexidex

E100#5#2

E100#6#2

googolquintiplexidex

E100#7#2

googolsextiplexidex

E100#8#2

googolseptiplexidex

10^^10^^10

tria-teraksys

This number is 10^^^3. It is also the 13th official competitor in the "My Number is Bigger" competition, and the 15th valid number. This number is the 3rd entry by Gmalivuk, the starter of the competition. It is after this number that Rodan "shuffles out".

E100#9#2

googoloctiplexidex

E100#10#2

googolnoniplexidex

Pentational Epoch
[10^^10010,100^^^100]
Entries: 15

10^^10^^100

giggolplex

The giggolplex is a number coined by Jonathan Bower's as an extension of the googol naming conventions. A giggolplex is a power tower of 10s a giggol terms high, where a giggol is itself a power tower of 10s 100 terms high. It can be notated as:

giggolplex = 10^10^10^ ... ^10^10^10 w/giggol 10s

E100#100#2

grangoldex

Here's a HUGE number, very similar in spirit to the once great googolplex. A grangoldex is a power tower of 10s a grangol terms high, with a 100 on top of all this! Another way to think about it is that a grangoldex is the grangolth member of the googol series. We can also notate it as:

grangoldex = 10^10^10^ ... ^10^10^10^100 w/grangol 10s

This number is just slighly larger than a giggolplex. This can be seen since giggol < grangol, it follows that a grangoldex has more 10s than a giggolplex.

(1,000,000^^1,000,000)^^(1,000,000^^1,000,000)

mungo

This is a googolism coined by "Earthling on Mars" as part of a naive attempt to beat <10,10,googol> using power towers. He describes 1,000,000^^1,000,000 as a "mega" and mungo as mega^^mega. It turns out however not be any faster an iteration than pentation. From the Knuth-Arrow theorem (see my paper "A Theorem for Knuth-Arrows") it follows that...

(1,000,000^^1,000,000)^^(1,000,000^^1,000,000) < 1,000,000^^(1,000,000+1,000,000^^1,000,000)

< (10^^2)^^(1,000,000+1,000,000^^1,000,000) < 10^^(1,000,002+1,000,000^^1,000,000)

< 10^^(1,000,002+10^^1,000,002) << 10^^10^^1,000,003

This means this number is still massively smaller than 10^^10^^10^^100 or giggolduplex. On the other hand, grangoldex = E100#100#2 = E100#(E100#100) < E1#(2+E1#102) < E1#(E1#103) = 10^^10^^103, and this is much smaller than (1,000,000^^1,000,000)^^(1,000,000^^1,000,000), so we know a mungo is larger than grangoldex. Thus we can say...

grangoldex << mungo << giggolduplex

10^^10^^10^^100

giggolduplex

A Bowerism found in the "Giggol Group".

E100#100#3

grangoldudex

grangoldudex = E100#100#3 = E100#(E100#100#2) = E100#grangoldex = EEE...EEE100 w/grangoldex Es.

The grangoldudex is smaller than 10^^^5. This can be seen as follows:

E100#100#3 = E100#(E100#(E100#100)) < E100#(E100#(E10,000,000,000#100)) = E100#(E100#(E1#102))

< E100#(E1#(2+E1#102)) < E1#(2+E1#(2+E1#102)) < E1#(E1#(3+E1#102)) < E1#(E1#(E1#103)) = E1#103#3

< E1#(10^^10)#3 = E1#1#5 = 10^^^5

((1,000,000^^^2)^^^2)^^^2

humungo

This googolism was coined by "Earthling on Mars"  as part of the "My number is, in fact, bigger" thread. Written in this form we can see that...

humungo < 1,000,000^^^6 < (3^^^2)^^^6 < 3^^^8

Since humungo is less than 3^^^8 we know it's less than 3^^^10.

3^^^10

Blatm's Pentational Number

The 14th competitor in the "My Number is Bigger" competition, and the 16th valid number. This number was entered by Blatm by reversing the order of the arguments of Gmalivuk's number. We are now well into pentational numbers. At this point elementary arithmetic expressions with exponents and factorials can no longer compete.

4^^^10

Xooll Shrug's

The 15th competitive entry in the "My Number is Bigger" competition, and the 17th valid entry. Xooll entered in response to Blatm's Pentational Number. Xooll typed *shrug* after it, as if to say, what's the big deal? But this response isn't terribly competitive and the real competition has only begun.

10^^^10

deka-teraksys

This number is equal to 10^^10^^10^^10^^10^^10^^10^^10^^10^^10. To envision this number imagine Stage 1 as "10", Stage 2 as "10^10^10^10^10^10^10^10^10^10", Stage 3 as "10^10^ ... ^10^10" w/Stage 2 10s, ... and go all the way to Stage 10. This massive number is bigger than even the grangoldudex, but still smaller than the Megiston.

In Hyper-E this can be written as E1#1#10.

(10^^^10)!

Technically this is the 16th competitive entry in the "My Number is Bigger" competition, and the 18th valid entry. It was entered by User Ended. Most of the strength of the number comes from pentation. At this point the numbers are so big that adding a factorial is so negligible that we can ignore it as a "salad factorial". To understand this, realize that at this scale N! ~ 10^N. Furthermore we have 10^^^10 = 10^^(10^^^9). So 10^^^10 is a power tower of 10s 10^^^9 terms high. 10^^^9 is an inconceivably vast number. From this we can gather that (10^^^10)! ~ 10^(10^^^10) = 10^^(1+10^^^9) ~ 10^^(10^^^9) = 10^^^10. So it has virtually no effect.

10^^^11

This number serves as a weak lower-bound for the Megiston.

10[5]

Megiston / ten in a circle

The Megiston is the lesser known of the two numbers Hugo Steinhaus defined with his circle notation. It is much much larger than a Mega, taking advantage of the full power of the circle operator, which is roughly on par with pentation. This number is much more difficult to bound than the Mega, due to various technical difficulties. It isn't too difficult however to show that it must lie somewhere between 10^^^11 and 10^^^12.

*Assuming side note: Bowers' incorrectly calls this number megaston on his infinity scrapers page.

10^^^12

ten pentated to twelve

10^^^100

gaggol

This is Jonathan Bowers' gaggol, defined as 10^^^100. This makes it a very large pentational number and a very small hexational number. This number is larger than a megiston, but is "slightly" smaller than a greagol. In Hyper-E it can be expressed as E1#1#100.

100^^^100

This number serves as a benchmark for the largest pentational number. Although this designation is arbitrary, a pentational number is usually understood as anything of the form a^^^b where a and b are relatively small arguments and the result is a number not already included in a smaller class of numbers. Hence if we let 100 be the limit of a "relatively small" argument, then 100^^^100 is the largest pentational number. Interestingly, this number is not that much larger than a gaggol or 10^^^100, relatively speaking. Next up ... the Ackermann Numbers ...

Primitive-Recursive Epoch
(100^^^100,E100##100]
Entries: 34

E3#100#99

This number serves as an upperbound on 100^^^100. Consider that 100^^n < E3#n. Therefore 100^^100 < E3#100, 100^^^3 = 100^^100^^100 < 100^^E3#100 < E3#(E3#100) = E3#100#2, 100^^^4 = 100^^100^^^3 < 100^^E3#100#2 < E3#(E3#100#2) = E3#100#3 ... and in general 100^^^n < E3#100#(n-1). Since E3#100#99 < E100#100#100 it follows that 100^^^100 < greagol. It can also be shown that 100^^^100 < E3#100#99 < 10^^^101. For a more detailed proof click here.

10^^^101

This serves as both an upper-bound on 100^^^100 and a lower bound on a greagol. The lower-bound is easier to demonstrate. Simply observe that 10^^^101 = E1#1#101 = E1#(E1#1#100) = E1#(E1#(E1#1#99)) = E1#(E1#1#99)#2 = E1#(E1#(E1#(E1#1#98))) = E1#(E1#1#98)#3 = ... = E1#(E1#1#1)#100 = E1#10#100 < E100#100#100 = greagol.

E100#100#100

greagol

A greagol, short for "great googol", is the 100th member of the grangol series. It is larger than and comparable to Jonathan Bowers' gaggol.

E1#102#100

This number is larger than a greagol, but less than 10^^^102. Hyper-E allows one to express more intermediate values than Knuth Up-arrow notation.

10^^^102

This is an upperbound on a greagol. In Hyper-E it can be written E1#1#102. This is one of the intervening steps in the proof that greagol << Folkman's Number.

16^^^102

This is another step in the proof greagol << Folkman's Number.

2^^^408

This is the final step in the proof greagol << Folkman's Number. By converting the base of the pentation to 2, it's made immediately apparent that this number must be less than Folkman's Number of 2^^^(2^901). Simply consider 2^^^408 < 2^^^512 = 2^^^(2^9) << 2^^^(2^901)

2^^^(2^901)

Folkman's Number

This moderately sized Ackermann class number was mentioned in the same article by Martin Gardner where he introduced the world to "Graham's Number" (See article here). Folkman was looking for a graph containing no K4s that forces a monochromatic K3 when it's two-colored. He devised an example of such a graph ... but it would contain 2^^^(2^901) points! This number is insanely large. Yet it's still smaller than G(1) of Graham's Number. Folkman's Number is somwhere between a greagol and G(1). Roughly speaking, the reason is because a greagol ~ 2^^^100 (actually larger) where as G(1) ~ 2^^^(3^^7,625,597,484,987) (actually larger). For a full proof click here.

G(1)

3^^^^3

This is 3 hexated to the 3rd. Evaluating  it we have:

3^^^^3 = 3^^^3^^^3 = 3^^^3^^3^^3 = 3^^^3^^3^3^3 =

3^^^3^^3^27 = 3^^^3^^7,625,597,484,987 =

3^^^3^3^3^3^ ... ^3^3^3^3 w/7,625,597,484,987 3s after 3^^^ =

3^^3^^3^^3^^3^^ ... ^^3^^3^^3^^3^^3

w/3^3^3^ ... ^3^3^3 3s
w/7,625,597,484,987 3s
To imagine this, let stage 1 = 3. Let stage 2 = 3^3^3 or 7,625,597,484,987, let stage 3 = 3^3^ ... ^3^3 w/7,625,597,484,987 3s, and in general each new stage is a power tower of 3s with the previous stage number of terms. 3 hexated to the 3rd is Stage 3^3^3^ ...^3^3^3 w/7,625,597,484,987 3s. This number is also G(1), the first member of graham's sequence (See G(64)).

((...((1,000,000^^^2)^^^2)...)^^^2)^^^2

w/((1,000,000^^^2)^^^2)^^^2-1 "^^^2"s

Earthling on Mars Number

This is the final form of Earthling on Mars's Number. This was his attempt to come up with a number larger than <10,10,googol> using power towers. This number however can be demonstrated to be in the hexational range, much much smaller than <10,10,googol>. Firstly we can observe that...

Earthling on Mars Number < 1,000,000^^^(2*1,000,000^^^6) < 10^^^(2+2*1,000,000^^^6)

< 10^^^(3*1,000,000^^^6) < 10^^^(3*10^^^8) < 10^^^(10*10^^^8) < 10^^^(10^^10^^^8)

= 10^^^10^^^9 < 10^^^10^^^10 = 10^^^^3 < 10^^^^10 = <10,10,4>

So Earthling on Mars Number is less than even <10,10,4>. In fact it's less than 10^^^10^^^10 making it smaller than a gaggolplex. On the other hand we have...

Earthling on Mars Number > 10^^^10^^10^^10^^10 = 10^^^10^^^4 > 3^^^3^^^3 = 3^^^^3.

So the Earthling on Mars Number is bigger than 3^^^^3 or G(1).

10^^^10^^^100

gaggolplex

This number is massively larger than G(1), yet at this stage it starts becoming more obscure why. The reason is because 3^^^3 (tritri) is vastly smaller than 10^^^100 (gaggol). Therefore 3^^^(3^^^3) << 10^^^10^^^100.

E100#100#100#2

greagolthrex

10^^^10^^^10^^^100

gaggolduplex

Another Bowerism on the Infinity scrapers' page. In E# this can be written as E1#1#100#3 making it definitely smaller than E100#100#100#3 (though googologically in the same neighborhood basically). It is still vastly larger than a greagolthrex. Just observe:

E100#100#100#2 = E100#100#greagol < E100#(E100#1#1)#greagol = E100#1#(1+greagol)

E100#(E100#1#greagol) < E(10^10)#(E100#1#greagol) = E1#(2+E100#1#greagol)

= E1#(2+E100#(E100#(E100#(E100#(E100#(... E100#(E100#1)...))))))))

E1#(E1#(E1#(E1#(E1#(E1#(E1#(E1#(E1#( ... E1#(E101#1) ... ))))))))))

E101#1 < E(10^10)#1 = E1#3

so we get an upperbound of:

E1#3#(1+greagol)

which is less than E1#(E1#1#1)#(1+greagol) = E1#1#(2+greagol) = E1#1#(2+E100#100#100)

we can again ascend the 2 up through the various power towers eventually reaching the innermost level...

E101#100 which is still less than E(10^10)#100 = E1#102 so we get...

E1#1#(E1#102#100) and since 102 < 10^^10 = 10^^^2 = E1#1#2 we have:

E1#1#(E1#(E1#1#2)#100) = E1#1#(E1#1#102)

which is E1#1#102#2 which is less than E1#1#100#3 since this is equal to E1#1#gaggol#2.

E100#100#100#3

greagolduthrex

E100#100#100#100

gigangol

A gigangol, short for "gigantic googol", is the 100th member of the greagol series.

E100#100#100#100#2

gigangoltetrex

E100#100#100#100#3

gigangoldutetrex

E100#100#100#100#100

gorgegol

The gorgegol, short for "the gorged googol", is the 100th member of the gigangol series.

E100#100#100#100#100#2

gorgegolpentex

E100#100#100#100#100#3

gorgegoldupentex

E100#100#100#100#100#100

gulgol

The gulgol, short for the *gulp* googol, is the 100th member of the gorgegol series.

E100#100#100#100#100#100#2

gulgolhex

E100#100#100#100#100#100#3

gulgolduhex

E100#100#100#100#100#100#100

gaspgol

The gaspgol, short for "gasp googol", is the 100th member of the gulgol series.

E100#100#100#100#100#100#100#2

gaspgolheptex

E100#100#100#100#100#100#100#3

gaspgolduheptex

E100#100#100#100#100#100#100#100

ginorgol

The ginorgol, short for "ginormous googol", is the 100th member of the gaspgol series.

E100#100#100#100#100#100#100#100#2

ginorgoloctex

E100#100#100#100#100#100#100#100#3

ginorgolduoctex

10^^^^^^^^^^10

Tridecal

This number can also be written as <10,10,10> using linear array notation. This googolism was coined by Jonathan Bowers'.

(10^^^^^^^^^^10)! * (10^^^^^^^^^10)!^^^^^^(10^^^^^^10)!

This is the 17th record setting entry in the "My Number is Bigger" competition. This number is a big jump from the 16th record setter. However it's a salad number. The factorials and multiplication add little to the number and it ends up being not much larger than a Tridecal. This is also the 19th number listed in the forum.

(11^^^^^^^^^^11)! * (11^^^^^^^^^11)!^^^^^^(11^^^^^^11)!

Blatm's Finesse

This is the 18th record setting entry in the "My Number is Bigger" competition, and the 20th official entry. This was Blatm's response to Twasbrillig's Up-arrow Salad. We can see that people recognize intuitively that salad numbers are sloppy because Blatm responded by saying "Not one for Finesse, are you?". Blatm simply took Twasbrillig's number and converted to base 11. This makes for a (slightly) larger value, though at this point this likely to be clobbered by whoever is willing to type out more up-arrows, such as a screen fill. As it turns out, this is the last number in the Primitive Recursive Epoch, and the next entry is much much larger!

g(1)

2^^^^^^^^^^^^3

This is the first value in the sequence used to construct Little Graham, the original Graham's Number used in RL Graham's 1971 paper "Ramsey Theory for n-parameter sets".

E100##100

gugold

This number is roughly equivalent to 10^^^...^^^100 w/100 ^s = 10<100>100 = <10,100,100>. In actuality it's larger than this. Since Jonathan Bower's boogol = 10<100>10 < 10<100>100 < E100##100, it follows that boogol < gugold.

A gugold is my smallest googolism to use Extended Hyper-E Notation. It can be written in ordinary Hyper-E as:

E100#100#100#100# ... ... ... ... #100#100#100#100#100 w/100 100s

Continue on to Part II for some even more tremendous numbers that are inexpressible even using 100 generation primitive recursive functions...