**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****(-**

**∞**

**,-10**

^{100}]

**Entries: 27**

**[Indescribable]**

*Minus Sam's Number*

*"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.

*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.

*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*

*????????*

*minus Oblivion*

*-FOOT*^{10}(10^{100})

*minus BIG FOOT*

*-Rayo(10*^{100})

*minus Rayo's Number*

*-D*^{5}(99)

*minus Loader's Number*

*{{L100,10}*_{10,10}&L,10}_{10,10}

*minus meameamealokkapoowa oompa*

*meameamealokkapoowa oompa*)

*{L100,10}*_{10,10}

*minus meameamealokkapoowa*

*meameamealokkapoowa*)

*-{10*^{100}&10&10}

*minus golapulus*

*golapulus*)

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

*minus blasphemorgulus*

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

*minus humongulus*

*humongulus*)

*-{10^^^100&10}*

*minus kungulus*

*-{3&3&3}*

*minus triakulus*Here is the negative version.

*Negative triakulus*is inconceivably less than the next entry :)

*-{10^^100&10}*

*minus goppatoth*

*goppatoth*)

*-E100#^^#100*

*minus tethrathoth**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*

*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*

*godgahlah*)

*-{10,100(1)2}*

*minus goobol*

*-{10,10(1)2}*

*minus iteral*

*-G(64)*

*minus 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*

*-10*^{100}

**minus googol**

*Ordinary Negative Number Epoch***(-10**

^{100},0)

*Entries: 31*

*-19,500,000,000,000*

*Current US National Debt*

*(As of 2016)*

*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*

*-459.67*

*minus four hundred fifty-nine point six seven*

*-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*

*-6.36221590585...*

*loglog1.000001*

*double logarithm of one point zero zero zero zero zero one*

*-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:

*negative one.*

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

*-0.954242509439*

*log(1/9)*

*logarithm of one ninth*

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(1/8)

-0.903089986992...

logarithm of one eighth

log(1/7)

-0.845098040014...

logarithm of one seventh

log(1/7)

-0.845098040014...

logarithm of one seventh

log(1/6)

-0.778151250384...

logarithm of one sixth

log(1/6)

-0.778151250384...

logarithm of one sixth

loglog1.5

-0.754262201319...

double logarithm of one point five

loglog1.5

-0.754262201319...

double logarithm of one point five

**log0.2**

-0.698970004336...

logarithm of one fifth

-0.698970004336...

logarithm of one fifth

log0.25

-0.602059991328...

logarithm of one quarterlog0.25

-0.602059991328...

logarithm of one quarter

loglog2

-0.521390227654...

double logarithm of two

loglog2

-0.521390227654...

double logarithm of two

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

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

-0.47712125472...

logarithm of one third

loglog3

-0.321371236131...

loglog3

-0.321371236131...

**double logarithm of three**

log0.5

-0.301029995664...

logarithm of a half

log0.5

-0.301029995664...

logarithm of a half

**loglog4**

-0.22036023199...

double logarithm of four

-0.22036023199...

double logarithm of four

log(2/3)

-0.176091259056...

logarithm of two thirds

log(2/3)

-0.176091259056...

logarithm of two thirds

loglog5

-0.155541461208...

double logarithm of five

loglog5

-0.155541461208...

double logarithm of five

loglog6

-0.108935980359...

double logarithm of six

loglog6

-0.108935980359...

double logarithm of six

loglog7

-0.073092905527...

double logarithm of seven

loglog7

-0.073092905527...

double logarithm of seven

loglog8

-0.044268972935...

double logarithm of eight

loglog8

-0.044268972935...

double logarithm of eight

loglog9

-0.020341240467...

double logarithm of nine

loglog9

-0.020341240467...

double logarithm of nine

loglog9.9

-0.001899759965

double logarithm of nine point 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

loglog9.999999

-0.000000018861...

double logarithm of nine point nine nine nine nine nine nine

**-1/E100#^^#100**

**negative reciprocal of a tethrathoth**

*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*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 log

_{a}a^^n = a^^(n-1), for n>2, we can define this as a law for all integer values. Thus we obtain that a^^0 = log

_{a}a^^1 = log

_{a}a=1. Thus a^^0=1. Next let a^^(-1) = log

_{a}a^^0 = log

_{a}1 = 0. Thus we find that any positive integer>1, a, that a^^(-1)=0.

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

Zero

Zero

*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*

*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*

*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/G**

_{64}*reciprocal of Graham's Number*

*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/G**

_{64}*two divided by Graham's Number**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 googolminex*,

*three 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*

**10**

^{-100}**0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000**

**0000000001**

**googol-minutia***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)***10**

^{-35}

**0.00000000000000000000000000000000001**

**Planck Length**

**(in meters)**

**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)**_{[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

0.25

one quarter

log2log2

0.301029995663981195213738894724...0.301029995663981195213738894724...

*logarithm of two*

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

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**

*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...

0.333333333333333333333333333333333...

*one third*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...**

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

*log3*

0.477121254719662437295027903255115...

0.477121254719662437295027903255115...

*logarithm of three*

**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

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

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

*0.5*

*one half*

*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"*

^{434,294}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(10

^{434,294}) ~ 434,294*ln(10)+0.5772156649 ~ 999,999.46 so this is a bit of an underestimate.

**log4**

0.602059991328...

logarithm of four

0.602059991328...

logarithm of four

2/32/3

*0.6666666666666666666...*

*two thirds*

**log5**

0.698970004336...

logarithm of five

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

*log9.9999990.999999956571...logarithm of nine point nine nine nine nine nine nine*

*0.9999999999999999999999999999999..................999999999976974149....*

*w/googol-1 9s*

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

*googolduminex**googol*and

*plex*suffix by repeatedly appending it

*creating the sequence:*

*googol, googolplex, googolduplex, googoltriplex, ...*

*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}

^{}

*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.

*googoltriminex*for more insanity).

**NEITHER LARGE NOR SMALL****1**

*one*

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

*SBIIS SAIBIAN'S ULTIMATE LARGE NUMBER LIST*

*(ULNL)*

*BEGINS HERE*

*LARGE NUMBERS***(1,**

**∞)**

**Small Superuniary Epoch**

**(1,2)**

**Entries:24***large...er than 1 ... barely :/*

**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.

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

**one plus the reciprocal of a tethrathoth**

*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**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*

*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**

**0000000000**

**0000000000**

**0000000000**

**0000000000**

**00000000**

00

00

**0000000000**

**0000000000**

**0000000000**

**0000000001**

**one plus the reciprocal of a googol**

**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 ...

**1.0**

**000000**

**00000000010903970549325460813650942266345982807...**

*Time dilation factor of person who is walking versus standing still**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***1.0013784192...**

*Ratio of Neutron to Proton mass***1.0026654123...**

*Ratio of troposphere to diameter of the earth***1.01**

*One point oh one***1.05946409436...**

*twelve root of two***1.07**

*one point oh seven***1.09407190229...**

*ratio of radius of exosphere to radius of earth***1.1**

*one point one*

**1.11178201104...**

*convergence value of iterated exponentiation of one point one***1.21**

*one point two one***1.331**

*one point three three one***1.4142135623...**

square root of twosquare root of two

*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**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***1.5**

*one and a half*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*

*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*

*Palpable Epoch***[2,100]**

**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*

**2.283333333333...**

*fifth harmonic number***2.45**

*sixth harmonic number*

**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

*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*

*M*

_{2}"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

*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:

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.

**2**

^{3}-2

**6**

six

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.

**7**

*seven*

*M*

_{3}*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*).

**8**

*eight***9**

*nine**Nine*is the 3rd Square number.

**10**

ten

ten

**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.

*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*".

*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**

sixteensixteen

*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

twenty-five

**27**

*twenty-seven***2**

^{5}-2^{2}

**28**

*twenty-eight*

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

**31**

*thirty-one*

*M*_{5}

*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*

**36**

*thirty-six*

*forty-eight**forty-nine*

**64**

*sixty-four*

**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*

**100**

**One hundred**

*Terrestrial Epoch***(100,10**

^{6})**121**

*one hundred twenty-one***125**

*one hundred twenty-five***127**

*one hundred twenty-seven*

*M*_{7}

*127 is the 4th*

*mersenne prime*, expressible as 2^7-1. The next

*mersenne prime*is 8191.

**144**

*one hundred forty-four*

*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*

**196**

*one hundred ninety-six*

**216**

*two hundred sixteen***220**

*two hundred twenty*

**225**

*two hundred twenty-five*

**256**

*two hundred fifty-six***284**

*two hundred eighty-four*

**289**

*two hundred eighty-nine*

**324**

*three hundred twenty-four***343**

*three hundred forty-three***361**

*three hundred sixty-one*

**400**

*four hundred***2**

^{9}-2^{4}

**496**

*four hundred ninety-six*

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

**512**

*five hundred twelve*

**625**

*six hundred twenty-five*

**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*

**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 thousandone thousand

**1024**

*one thousand and twenty-four*

**1296**

*one thousand two hundred ninety-six***1331**

*one thousand three hundred thirty-one***1728**

*one thousand seven hundred twenty-eight*

*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*

**2197**

*two thousand one hundred ninety-seven*

**2401**

*two thousand four hundred one***2656**

*two thousand six hundred fifty-six*

**The last 4 digits of Mega are 2656.**

**2744**

*two thousand seven hundred forty-four***3125**

*three thousand one hundred twenty-five*

**3375**

*three thousand three hundred seventy-five*

**4096**

*four thousand ninety-six*

**4913**

*four thousand nine hundred thirteen***5832**

*five thousand eight hundred thirty-two*

**6561**

*six thousand five hundred sixty-one***6859**

*six thousand eight hundred fifty-nine***8000**

*eight thousand***2**

^{13}-2^{6}

**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**

**M**_{13}

*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**

*ten thousand*/ myriad**14,641**

*fourteen thousand six hundred sixty-one***20,736**

*twenty thousand seven hundred thirty-six**great great gross*, based on a continuation of the sequence,

*dozen, gross, great gross.*

**28,561**

*twenty-eight thousand five hundred sixty-one***38,416**

*thirty-eight thousand four hundred sixteen***46,656**

*forty-six thousand six hundred fifty-six*

**50,625**

*fifty thousand six hundred twenty-five*

**65,536**

*sixty-five thousand five hundred thirty-six*

**83,521**

*eighty-three thousand five hundred twenty-one***100,000**

**one hundred thousand***googolgong*.

**104,976**

*one hundred four thousand nine hundred seventy-six***130,321**

*one hundred thirty thousand three hundred twenty-one***131,071**

*one hundred thirty-one thousand seventy-one*

*M*_{17}

*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*

**248,832**

*great great great gross*

**or**

*two-hundred-forty-eight-thousand-eight-hundred-thirty-two**great great great gross*by Andre Joyce.

**524,287**

*five hundred twenty-four thousand two hundred eighty-seven*

*M*_{19}

*524,287 is the 7th*

*mersenne prime*. This is conventionally denoted as M

_{19}= 2

^{19}-1. The next

*mersenne prime*is 2,147,483,647.

**823,543**

*eight hundred twenty-three thousand five hundred forty-three*

*Astronomical Epoch***[10**

^{6},10^{1000})

*Entries: 103***1,000,000**

*million**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*

**2,985,984**

*great great great great gross*

**or**

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

*great great great great gross*by Andre Joyce.

**16,777,216**

*sixteen million seven hundred seventy-seven thousand two hundred sixteen***2**

^{25}-2^{12}

**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:*

**35,831,808**

*five-ex-great gross*

**or**

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

*five-ex-great gross*by Andre Joyce.

**100,000,000**

*one hundred million / octad**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*

**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 gros*s by Andre Joyce.

**909,090,909**

**(10^10-1)/11**

**1,000,000,000**

*/*

billionbillion

**milliard***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***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*

*M*_{31}

*Also known as M*

_{31}, this is the 8th

*mersenne prime*. It can be expressed compactly as 2

^{31}-1. The next one is 2^61-1.

**3,864,196,800**

*Age in seconds of oldest person who ever lived**billion*seconds. Hopefully a lifetime seems a lot longer now!

**2**

^{33}-2^{16}

**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*

**76,923,076,923**

**(10^12-1)/13**

**100,000,000,000**

*hundred billion / hundred milliard / ten dialogue*

**2**

^{37}-2^{18}

**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**

**1,000,000,000,000**

**trillion / billiad***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-illiad =*10^(6n)

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

**7,625,597,484,987**

*3^3^3***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*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*27*27) =**

(27*729)

(27*729)

**(27*729)**

**(27*729) =**

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

19,683*387,420,489 =

7,625,597,484,987

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

19,683*387,420,489 =

7,625,597,484,987

**8,916,100,448,256**

12^12

12^12

**588,235,294,117,647**

**(10^16-1)/17**

*integral-dekapetaseptemdecile*

**1,000,000,000,000,000**

*quadrillion / billiard*

*integral-exaundevigintile*

*quadrillion*.

*quintillion*/

*trilliad*

*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.

**2**

^{61}-2^{30}

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

*Eighth Perfect Number*

*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*

*M*_{61}

*This is the 9th*

*mersenne prime*. It is traditionally denoted M

_{61}. 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*

**1,000,000,000,000,000,000,000**

*sextillion*

**/**

*trilliard*

*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*

*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**

*septillion / quadrilliad*

*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*

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

*10th Mersenne Prime*

**M**

_{89}

**M**

_{89}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 M

_{61}'s (

*Pervushin's Number*) place for 2nd largest known prime, bumping Pervushin's Number down to 3rd. (See M

_{107}and M

_{127}). The next mersenne prime is M107 with 33 digits.

**1,000,000,000,000,000,000,000,000,000**

*octillion / quadrilliard*

*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**

*nonillion / quintilliad*

*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*

**M**

_{107}

**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***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**

*undecillion / sextilliad*

**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.

**2**

^{121}-2^{60}

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

*Ninth Perfect Number*

*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*

**M**

_{127}

**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*

*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**

*tredecillion / septilliad*

**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**

*integral-dekazettayottaseptemquadragintile*

**1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000**

*quattuordecillion / septilliard*

*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**

*quindecillion / octilliad*

*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*

*sexdecillion*or

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

**2**

^{177}-2^{88}

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

*Tenth Perfect Number*

*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**

*septendecillion / nonilliad*

**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*

*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**

*novemdecillion / decilliad*

*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*

*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.*

**2**

^{213}-2^{106}

**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*

*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*

*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*

**2**

^{253}-2^{126}

**14,474,011,154,664,524,427,946,373,126,085,988,481,573,677,491,474,**

**8**

**35,889,066,354,349,131,199,152,128**

*12th Perfect Number*

*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*

**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(2**

^{127}-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*2**

^{256}

*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*

*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.

*sexvigintillion / viginti-sextillion*

*septenvigintillion*/

*viginti-septillion*

*octovigintillion / viginti-octillion*

*novemvigintillion / viginti-nonillion*

*trigintillion*

*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**

**10**

^{100}*googol*

*or*

*ten duotrigintillion / ten triginti-deutillion*

*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.

**10**

^{101}

**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**

*quadragintillion*

**10**

^{150}

**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***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.

**2**

^{521}-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*

_{521}

**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.*

**2**

^{607}-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*

_{607}

**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*

*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!

**10**

^{200}

*gargoogol*

*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*

**10**

^{300}

*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 )

*thrargoogol*, a corruption of

*gargoogol. thrar-*is a prefix formed from combining "three" with "gar". Let

*thrar-(n) =*n^3.

**E303**

*centillion*

*googol*and

*googolplex*until much later). It was my favorite number for a time, along with the

*googolgong*(seen a little later).

**2**

^{1041}-2^{520}

2356272345726734706578954899670990498847754785839260071014302759750633728317862223973036553960260056136025556646250327017505289257804321554338249842877715242701039449691866402864453412803383143979023683862403317143592235664321970310172071316352748729874740064780193958716593640108741937564905791854949216055564697613th Perfect NumberThis is the 13thperfect number. It has 314 digits. This makes it larger that acentillionin the short scale.The next perfect number is 2^1213-2^606.

**2**

^{1213}-2^{606}

141053783706712069063207958086063189881486743514715667838838675999954867742652380114104193329037690251561950568709829327164087724366370087116731268159313652487450652439805877296207297446723295166658228846926807786652870188920867879451478364569313922060370695064736073572378695176473055266826253284886383715072974324463835300053138429460296575143368065570759537328128

*14th Perfect Number*

*perfect number.*It has 366 digits. The next perfect number is 2^2557-2^1278.

**2**

^{1279}-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*

**M**

_{1279}

**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.

**10**

^{500}

*googolding*

*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 10

^{1000}.

**32507925092532526327561017283413824652374863824571**

**2364831206**

**8325872634**

**1872540450**

**1384532106**

**4343218561**

**3284015128**

**3461328410**

**8275451257**

**6321856103**

**2841803247**

**6321806523**

**0850123857**

**6321056120**

**8653215063**

**2150623199**

**2316599123**

**6598123649**

**8236455145**

**6072357032**

**1652371652**

**1307842314**

**5213748632**

**8042384512**

**4610982346**

**8458364128**

**4321061273**

**4081642309**

**4521395629**

**8230912657**

**0932156013**

**2975412390**

**6012934632**

**1097462317**

**0523165092**

**3615076120**

**9375320984**

**3261897432**

**9802136471**

**2650924379**

**8107561029**

**8127657632**

**1095621359**

**7623150789**

**6123598172**

**3476321756**

**2310947231**

**9846547612**

**5390821650**

**7486502937**

**521093865**

*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 (10**

^{80}) or even the number of plank volumes (10

^{175}), 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.

*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.*

*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**

*centilliad*

*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.*

**1262383049660586222684174870651169998454847760535761095005091618262681**

**8413620269880155156801376138071753405453485116413864890452793160**

**516052**

**7688095259563605939964364716019515983399209962459578542172100149**

**937763**

**9385812196040727334225071800560096725409007095541095168165737795**

**933263**

**3228831487325155907785306844497786480339196258080068276001784958**

**928193**

**7637993445539366428356761821065267423102149447628375691862210717**

**202025**

**2416303031185591886783043140769438016925282469809597059016414442**

**388949**

**2862082548230343180695569022630877342682950390093052939518120873**

**959196**

**7195841536053143145775307050594328881077553168201547775**

**2**

^{2040}-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).

**2**

^{2203}-1

**1475979915214180235084898622737381736312066145333169775147771216478**

**570**

**2978780789493774073370493892893827485075314964804772812648387602**

**591918**

**1446336533026954049696120111343015690239609398909022625932693**

**502528140**

**9614983499388222831448598601834318536230923772641390209490**

**231836446899**

**6082107954829637630942366309454108327937699053999824571**

**863229447296364**

**1889062337217172374210563644036821845964963294853869**

**690587265048691443**

**4637457507280441823676813517852099348660847172579**

**408422316678097670224**

**0119902801704748944874269247421088235368084850**

**725022405194525875428753**

**4997655857267022963396257521263747789778550**

**155264652260998886991401354**

**0483809865681250419497686697771007**

*16th Mersenne Prime*

**M**

_{2203}**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.

**2**

^{2281}-1

**446087557183758429571151706402**

**101809886208632412859901111991**

**2199634046**

**85792820473369112545**

**269003989026153245931124316702**

**39575870569367936479**

**0903497461**

**147071065254193353938124978226**

**307947312410798874869040070279**

**328428810311754844108094878252**

**494866760969586998128982645877**

**5960289791**

**71536962503068429617**

**331702184750324583009171832104**

**91605015762888660637**

**2145501702**

**225925125224076829605427173573**

**964812995250569412480720738476**

**855293681666712844831190877620**

**606786663862190240118570736831**

**9018864792**

**25810414714078935386**

**562497968178729127629594924411**

**96096138671394627989**

**9275006954**

**917139758796061223803393537381**

**034666494402951052059047968693**

**255388647930440925104186817009**

**640171764133172418132836351**

*17th Mersenne Prime*

**M**

_{2281}

**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.

**2**

^{2557}-2^{1278}

54162526284365847412654465374391316140856490539031695784603920818387206994158534859198999921056719921919057390080263646159280013827605439746262788903057303445505827028395139475207769044924431494861729435113126280837904930462740681717960465867348720992572190569465545299629919823431031092624244463547789635441481391719816441605586788092147886677321398756661624714551726964302217554281784254817319611951659855553573937788923405146222324506715979193757372820860878214322052227584537552897476256179395176624426314480313446935085203657584798247536021172880403783048602873621259313789994900336673941503747224966984028240806042108690077670395259231894666273615212775603535764707952250173858305171028603021234896647851363949928904973292145107505979911456221519899345764984291328

*15th Perfect Number*

*perfect number.*It has 770 digits. The next perfect number is 2^4405-2^2202.

**2**

^{3217}-1

**259117086013202627776246767922**

**441530941818887553125427303974**

**9231618740**

**19266586362086201209**

**516800483406550695241733194177**

**44168950923880701741**

**0377709597**

**512042313066624082916353517952**

**311186154862265604547691127595**

**848775610568757931191017711408**

**826252153849035830401185072116**

**4247474618**

**23031471398340229288**

**074545677907941037288235820705**

**89235106843388298688**

**8616658650**

**280927692080339605869308790500**

**409503709875902119018371991620**

**994002568935113136548829739112**

**656797303241986517250116412703**

**5097054277**

**73477972349821676443**

**446668383119322540099648994051**

**79024162405651905448**

**3690809616**

**061625743042361721863339415852**

**426431208737266591962061753535**

**748892894599629195183082621860**

**853400937932839420261866586142**

**5032514507**

**73096274235376822938**

**649407127700846077124211823080**

**80413929808705750471**

**3825264571**

**448379371125032081826126566649**

**084251699453951887789613650248**

**405739378594599444335231188280**

**123660406262468609212150349937**

**5847822922**

**37144339628858485938**

**215738821232393687046160677362**

**909315071**

*18th Mersenne Prime*

**M**

_{3217}

**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.

**10**

^{1000}-10^{986}

*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.*

*Super-Astronomical Epoch***[10**

^{1000},10^{1,000,000})

*Entries: 57***10**

^{1000}

*googolchime*

*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).

*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.*).

**2**

^{4253}-1

**190797007524439073807468042969**

**529173669356994749940177394741**

**8826735289**

**79787005053706368049**

**835514900244303495954950709725**

**76218631122414882881**

**1920216904**

**542206960744666169364221195289**

**538436845390250168663932838805**

**192055137154390912666527533007**

**309292687539092257043362517857**

**3666246999**

**75402375462954490293**

**259233303137330643531556539739**

**92192620143860643902**

**0075174723**

**029056838272505051571967594608**

**350063404495977660656269020823**

**960825567012344189908927956646**

**011998057988548630107637380993**

**5198265823**

**89781888135705408653**

**045219655801758081251164080554**

**60905746802820330871**

**8724654081**

**055323215860189611391296030471**

**108443146745671967766308925858**

**547271507311563765171008318248**

**647110097614890313562856541784**

**1548817431**

**46033909602737947385**

**055355960331855614540900081456**

**37865906837031726769**

**6980001187**

**750995491090350108417050917991**

**562167972281070161305972518044**

**872048331306383715094854938415**

**738549894606070722584737978176**

**6864221343**

**54526989443028353644**

**037187375385397838259511833166**

**41613432369566036767**

**6897722287**

**918773420968982326089026150031**

**515424165462111337527431154890**

**666327374921446276833564519776**

**797633875503548665093914556482**

**0314822488**

**83127023777039667707**

**976559857333357013727342079099**

**06440045574183065432**

**0379350833**

**236245819348824064783585692924**

**881021978332974949906122664421**

**376034687815350484991**

*19th Mersenne Prime*

**M**

_{4253}

**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.*

**2**

^{4405}-2^{2202}

108925835505782933769822527352204898195710845430260806731890661850847015529861699629194096185890137954618268553122005576278075934240749906604670418208308712462692637816441093145096882635520557367167162420268663336080712310947045266837153759966279748493435903977995421366659882029950136638016461908026040323522955673055416399230300975265135032061993056367369528015302304949846869661814407202137283142596370146050560637811924584138655260014538407298330971714195008549808570967138705486832047797229905527391479844693621414786070688705210731238006707260231700942280931477479189470076989100981874316930302815430329007119939298429294028385221780016662922915711026408059929401645248302852815333111952344142315961493414026555024236000785821593679848950072719634751638604424172198470655832936427799590310229203462062808075234242290640128302703464967144556932428194685962217756664337548971567845131179267593598101035556288797194856901606003533460787935977037184650765997060161699831198387815042076330628949088642990048178649953764537983936521272549444151193277218276814994365984900745724698386155826514482319136775835034152778077022155694527556650483163656485683150255607805813304340005565354041331326603463935520283400612690549156956054248955102320738227613735266571701826151960481741711257652641053532399150005874999624758083445378252816th Perfect Number

*perfect number.*It has 1327 digits. The next

*perfect number*is 2^4561-2^2280.

**2**

^{4423}-1

**285542542228279613901563566102**

**164008326164238644702889199247**

**4566022844**

**00390600653875954571**

**505539843239754513915896150297**

**87839937705607143516**

**9747221107**

**988791198200988477531339214282**

**772016059009904586686254989084**

**815735422480409022344297588352**

**526004383890632616124076317387**

**4168811485**

**92486188361873904175**

**783145696016919574390765598280**

**18859903557844859107**

**7683677175**

**520434074287726578006266759615**

**970759521327828555662781678385**

**691581844436444812511562428136**

**742490459363212810180276096088**

**1114010033**

**77570363545725120924**

**073646921576797146199387619296**

**56030268026179011813**

**2925012323**

**046444438622308877924609373773**

**012481681672424493674474488537**

**770155783006880852648161513067**

**144814790288366664062257274665**

**2757871273**

**74649231096375001170**

**901890786263324619578795731425**

**69380507305611967758**

**0338084333**

**381987500902968831935913095269**

**821311141322393356490178488728**

**982288156282600813831296143663**

**845945431144043753821542871277**

**7456064478**

**58564159213328443580**

**206422714694913091762716447041**

**68967807009677359042**

**9808909616**

**750452927258000843500344831628**

**297089902728649981994387647234**

**574276263729694848304750917174**

**186181130688518792748622612293**

**3413689280**

**56634384466646326572**

**476167275660839105650528975713**

**89932021112149579531**

**1427946254**

**553305387067821067601768750977**

**866100460014602138408448021225**

**053689054793742003095722096732**

**954750721718115531871310231057**

**9026085806**

**07**

*20th Mersenne Prime*

**M**

_{4423}

**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.

^{4561}-2

^{2280}

99497054337086473442435202604522816989643863571126408511774020575773849326355529178686629498151336416502516645641699516813140394897940636561646545947753232301453603583223268085613647233768081645727669037394385696522820301535888041815559513408036145123870584325525813950487109647770743827362571822870567643040184723115825645590386313377067112638149253171843914780065137373446222406322953569124771480101363180966448099882292453452395428270875732536311539266115116490704940164192417744919250000894727407937229829300578253427884494358459949535231819781361449649779252948099909821642207485514805768288115583409148969875790523961878753124972681179944234641016960011815788847436610192704551637034472552319820336532014561412028820492176940418377074274389149924303484945446105121267538061583299291707972378807395016030765440655601759109370564522647989156121804273012266011783451102230081380401951383582987149578229940818181514046314819313206321375973336785023565443101305633127610230549588655605951332351485641757542611227108073263889434409595976835137412187025349639504404061654653755349162680629290551644153382760681862294677414989047491922795707210920437811136712794483496437355980833463329592838140157803182055197821702739206310971006260383262542900044072533196137796552746439051760940430082375641150129817960183028081010978780902441733680977714813543438752546136375675139915776

*17th Perfect Number*

*perfect number.*It has 1373 digits. The next

*perfect number is*2^6433-2^3216.

*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.*

**10**

^{1500}

*Odd Perfect Threshold*

*odd perfect numbers,*they must be largest than 10

^{1500}(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 10

^{1500}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 10

^{1500}.

^{1500 }also occurs in theoretical cosmology. It is estimated that in 10

^{1500}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)*

**2**

^{6433}-2^{3216}

33570832131986724437010877211080384841138028499879725454996241573482158450444042882048778809437690388449535774260849885573694759906173841157438424730130807047623655942236174850509108537827658590642325482494761473196579074656099918600764404702181660294469121778737965822199901663478093006075022359223201849985636144177185925402078185073015045097727084859464743635537781500284915880244886306461785982956072060013474955617851481680185988557136609224841817877083608951191123174885226416130683197710667392351007374503755403352531476227943590071651702697594241031955529898971218001214641774673134944471562560957179657881556419122102935450299751813340515170956167951095453649485576150660101689160658011770193274226308280507786835049549112576654510119670456745939890194205255175384484489909328967646988163155982471564998196261632751283127879509198074253193409580454562488664383465379885002735506153988851506645137759275553988219425439764732399824712438125054117523837438256744437055019441051006489972341609117978404563794992004873057518455748701444951238377139620494287982489529827233140637014837408856156199515457669607964052126908149265601786094447595560440059050091763547114092255371397425807867554352112542194784815494784276201170845949274674632985210421075531784918358926690395463649721452265405713484388043911634485432358638806645313826206591131266232422007835577345584225720310518698143376736219283021119287617896146885584860065048876315701088796219593640826311622273328035603309475642390804499460156797855361018246696101253922254567240908315385468240931846166962495983407607141601251889544407008815874744654769507268678051757746956891212485456261121386667407711139619071530923355823178662705374393035049022603882479742334799407130280148769298597743778193050348749740786928096033906295910199238181338557856978191860647256209708168229116156300978059197026855726877649767072684960463452763160384093838292277544911857859658328888332628525056

*18th Known Perfect Number**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 10

^{1500}, 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 10

^{1500}. 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**

**2789246520**

**53024261580301205938**

**651973985026558644015579446223**

**53592127886738069722**

**8841014691**

**598660208796189675719570183928**

**166033804761122597553362610100**

**148265112341314776825241149309**

**444717696528275628519673751439**

**5357542479**

**09321920664188301178**

**716912255242107005070906467438**

**28708514499502565861**

**9446154318**

**351137984913369177992812743384**

**043154923685552678359637410210**

**533154603135372532574863690915**

**977869032826645918298381523028**

**6936572873**

**69142264813129174376**

**213632573032164528297948686257**

**62453622180176732249**

**4056764281**

**936007872071383707235530544635**

**615394640118534849379271951459**

**450550823274922160584891291094**

**518995994868619954314766693801**

**3037176163**

**59259447974616422005**

**088507946980448713320513316073**

**91342305401988725700**

**3832980124**

**605019701346739717590902738949**

**392381731578699684589979478106**

**804282243609378394633526542281**

**570430283244238551508231649096**

**7285712171**

**70812323279048181726**

**832751011274678231741098588868**

**37085220007117334922**

**5391332230**

**075614718042900752767779335230**

**620061828601245525424306100689**

**480544658470482065098266431936**

**096038873625851074707434063628**

**6976576702**

**69925864995355797631**

**817390255089133122329474393034**

**39561613283340728316**

**6349825814**

**522686200430779908468810380418**

**736832480090387359621291963360**

**258312078167367374253332287929**

**690720549059562140688882599124**

**4581842379**

**59786347648431567376**

**092362509037151179894142426227**

**02200662864868678687**

**1018298087**

**280256069310194928083082504419**

**842479679205890881711232719230**

**145558291674679519743054802640**

**464685400273399386079859446596**

**1501752586**

**96581144756851004156**

**868773090371248253534383928539**

**75987494584970500382**

**2501248928**

**400182659005625128618762993804**

**440734014234706205578530532503**

**491818958970719930566218851296**

**318750174353596028220103821161**

**6048545121**

**03931331225633226076**

**643623668829685020883949614283**

**04847391139916696226**

**4994856368**

**523471287329479668088450940589**

**395110465094413790950227654565**

**313301867063352132302846051943**

**438139981056140065259530073179**

**0772711065**

**78349417464268472095**

**613464732774858423827489966875**

**50525043942182321913**

**5722305406**

**671537337424854364566378204570**

**165459321815405354839361425066**

**449858540330746646854189014813**

**434771465031503795417577862281**

**1776585876**

**941680908203125**

5^5^5

5^5^5

*centillion*and a

*millillion*. This number is still small enough that it can actually be computed.

**2**

^{8505}-2^{4252}

*19th Known Perfect Number*

*known perfect number.*It has 2561 digits. The next

*perfect number*is 2^8845-2^4422.

**2**

^{8845}-2^{4422}

*20th Known Perfect Number*

*known perfect number.*It has 2663 digits. The next

*known perfect number*is 2^19,377-2^9688.

**2**

^{9689}-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.

**2**

^{9941}-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.

**10**

^{3003}

millillion

millillion

*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.

**2**

^{11,213}-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.

**10**

^{5000}

*googolbell*

*googolding,*and

*googolchime.*The next largest one is

*googoltoll*.

**2**

^{19,377}-2^{9688}

*21st Known Perfect Number*

*known perfect number.*It has exactly 5834 digits.

*known perfect number*is 2^19,881-2^9940.

**2**

^{19,881}-2^{9940}

*22nd Known Perfect Number*

*known perfect number.*It has 5985 digits.

*known perfect number*is 2^22,425-2^11,212.

**2**

^{19,937}-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.

**2**

^{21,701}-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.

**2**

^{22,425}-2^{11,212}

*23rd Known Perfect Number*

*known perfect number.*It has 6751 digits.

*known perfect number*is 2^39,873-2^19,936.

**2**

^{23,209}-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.

**10**

^{10,000}

*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*.

*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*)

**2**

^{39,873}-2^{19,936}

*24th Known Perfect Number*

*known perfect number.*It has 12,003 digits.

*known perfect number*is 2^43,401-2^21,700.

**2**

^{43,401}-2^{21,700}

*25th Known Perfect Number*

*known perfect number.*It has 13,066 digits.

*known perfect number*is 2^46,417-2^23,208.

**2**

^{44,497}-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.

**2**

^{46,417}-2^{23,208}

*26th Known Perfect Number*

*known perfect number.*It has 13,973 digits.

*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.**

**2**

^{86,243}-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.

**2**

^{88,993}-2^{44,496}

*27th Known Perfect Number*

*known perfect number.*It has 26,790 digits.

*known perfect number*is 2^172,485-2^86,242.

**10**

^{30,003}

*decimillillion / myrillion*

*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 10

^{306}). 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.

**2**

^{110,503}-1

*29th Mersenne Prime*

*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.

**(See Full Decimal Expansion)**

6^6^6

6^6^6

*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.

**2**

^{132,049}-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.

**2**

^{172,485}-2^{86,242}

*28th Known Perfect Number*

*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.

**2**

^{216,091}-1

*31st Mersenne Prime*

*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.

**2**

^{221,005}-2^{110,502}

*29th Known Perfect Number*

*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.

**2**

^{264,097}-2^{132,048}

*30th Known Perfect Number*

*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.

**2**

^{276,709}

^{}*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 10

^{78}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.

^{100,000}

*googolgong*

*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*).

**2**

^{432,181}-2^{216,090}

*31st Known Perfect Number*

*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.

**2**

^{756,839}-1

**( 227,832 digits )**

*32nd Mersenne Prime**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.

**2**

^{859,433}-1

**( 258,716 digits )**

*33rd Mersenne Prime**mersenne prime.*It was discovered by David Slowinski and Paul Gage on January 4th of 1994. The next

*mersenne prime*is M1,257,787.

**M**

_{1,257,787}

**( 378,632 digits )**

*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*2**

^{1,290,000}-1

**( 388,342 digits )**

**Largest Known Twin Prime Pair**

*( As of 2016)*

**2,996,863,034,895*2**

^{1,290,000}+1

**( 388,342 digits )**

**Largest Known Twin Prime Pair**

*( As of 2016 )*

**2**

^{1,398,269}-1

**( 420,921 digits )**

*35th Mersenne Prime*

*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.

**2**

^{1,513,677}-2^{756,838}

**( 455,663 digits )**

*32nd Known Perfect Number*

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

**2**

^{1,718,865}-2^{859,432}

**( 517,430 digits )**

*33rd Known Perfect Number*

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

**~3.7598*E695,974**

7^7^7

7^7^7

*million*digits!

**2**

^{2,515,573}-2^{1,257,786}

**( 757,263 digits )**

*34th Known Perfect Number*

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

**2**

^{2,796,537}-2^{1,398,268}

**( 841,842 digits )**

*35th Known Perfect Number*

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

**2**

^{2,976,221}-1

**( 895,932 digits )**

*36th Mersenne Prime*

*mersenne prime.*Credit is given to Gordon Spence who found it on August 24th of 1997. The next

*mersenne prime*is M3,021,377.

**2**

^{3,021,377}-1

**( 909,526 digits )**

*37th Mersenne Prime*

*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***[10**

^{1,000,000},10^{10^1,000,000})

*Entries: 110***10**

^{1,000,000}

*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.}

^{}

**2**

^{5,952,441}-2^{2,976,220}

**( 1,791,864 digits )**

*36th Known Perfect Number*

*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.

**2**

^{6,042,753}-2^{3,021,376}

**( 1,819,050 digits )**

*37th Known Perfect Number*

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

**2**

^{6,972,593}-1

**( 2,098,960 digits )**

*38th Mersenne Prime*

*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.

**10**

^{3,000,003}

*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*.

**2**

^{13,945,185}-2^{6,972,592}

**( 4,197,919 digits )**

*38th Known Perfect Number*

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

**2**

^{13,466,917}-1

**( 4,053,946 digits )**

*39th Mersenne Prime*

*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.

**1,000,000**

^{1,000,000}= 10^{6,000,000}

*fzmillion*

*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.

**2**

^{20,996,011}-1

**( 6,320,430 digits )**

*40th Mersenne Prime*

*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.

**2**

^{24,036,583}-1

**( 7,235,733 digits )**

*41st Mersenne Prime*

*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.

**2**

^{25,964,951}-1

**( 7,816,230 digits )**

*42nd Mersenne Prime*

*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.

**2**

^{26,933,833}-2^{13,466,916}

**( 8,107,892 digits )**

*39th Known Perfect Number*

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

**2**

^{30,402,457}-1

**( 9,152,052 digits )**

*43rd Mersenne Prime*

*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.

**2**

^{32,582,657}-1

**( 9,808,358 digits )**

*44th Mersenne Prime*

*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.

**2**

^{37,156,667}-1

**( 11,185,272 digits )**

*45th Mersenne Prime*

*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.

**2**

^{41,992,021}-2^{20,996,010}

**( 12,640,858 digits )**

*40th Known Perfect Number*

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

**2**

^{42,643,801}-1

**( 12,837,064 digits )**

*46th Known Mersenne Prime*

*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.

**2**

^{43,112,609}-1

**( 12,978,189 digits )**

*47th Known Mersenne Prime*

*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.

**2**

^{48,073,165}-2^{24,036,582}

**( 14,471,465 digits )**

*41st Known Perfect Number*

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

**8^8^8**

**~10^15,151,335**

**2**

^{51,929,901}-2^{25,964,950}

**( 15,632,458 digits )**

*42nd Known Perfect Number*

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

**2**

^{57,885,161}-1

*48th Known Mersenne Prime*

*On January 25th of 2013 this number became the largest known prime number, as well as the largest known*

*mersenne prime*. It was discovered by Dr. Curtis Cooper as part of the GIMPS project. The next largest known

*mersenne prime*is M74,207,281.

^{60,804,913}-2

^{30,402,456}

*43rd Known Perfect Number*

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

^{65,165,313}-2

^{32,582,656}

*44th Known Perfect Number*

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

^{74,207,281}-1

*49th Known Mersenne Prime/*

*Largest Known Prime Number*

*(As of 2016)*

*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.

**2**

^{74,313,333}-2^{37,156,666}

**( 22,370,543 digits )**

*45th Known Perfect Number*

*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 10

^{31,925}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 ...

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

**2**

^{85,287,601}-2^{42,643,800}

**( 25,674,127 digits )**

*46th Known Perfect Number*

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

**2**

^{86,225,217}-2^{43,112,608}

**( 25,956,377 digits )**

*47th Known Perfect Number*

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

**2**

^{115,770,321}-2^{57,885,160}

**( 34,850,340 digits )**

*48th Known Perfect Number*

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

**2**

^{148,414,561}-2^{74,207,280}

**( 44,677,235 digits )**

*Largest Known Perfect Number*

*(As of 2016)*

*Great Internet Mersenne Primes*search. This probably won't be the last perfect number to appear on this list, so stay tuned for updates!

**10**

^{100,000,000}

*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*)

**9^9^9**

^{1,000,000,000}

*billiplexion*

*trilliplexion.*

**10^10^10**

*trialogue*

(10^10^10)^2

*gartrialogue*

*trialogue trialogues*. It is 1 followed by 20,000,000,000 zeroes. It can also be expressed as 10

^{20,000,000,000}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.

*thrartrialogue*

*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:

^{30,000,000,000}and therefore still smaller than a

*googolthrong.*

^{100,000,000,000}

*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.

**^e^π**

**Ballium's Number**_{[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...

*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

11^11^11

**10^10^12**

*Size of Hypothetical Inflationary Universe*

*also*

*trilliplexion*

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

3^3^3^3

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

**E(9.622*E12)**

12^12^12

12^12^12

**10**

^{100,000,000,000,000}

*googolgandingan*

*or*

*googolquadrigong*

*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*.

**10**

^{1,000,000,000,000,000}

*quadrilliplexion*

^{100,000,000,000,000,000}

*googolquintigong*

*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.*

^{1,000,000,000,000,000,000}

*quintilliplexion*

*guppyplex / googolsextigong*

*guppyplex*or

*googolsextigong*.

*sextilliplexion*

*googolseptigong*

*septilliplexion*

**10^10^26**

*googoloctigong*

*little foot.*

**10000**

**0000**

**0000**

**0000**

**0000**

**0000**0000

**0000^100000**

**00000**

**00000**

**00000**

**00000**

**(10**

^{32})^(10^{25})

*little foot*

*googolplex*, go fig.

^{32})^(10

^{25}), or as 10

^{3.2*10^26}and bounded by 10

^{10^27}.

**10^10^27**

*Upper Bound on "little foot"*

*also*

*octilliplexion*

*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*

*googol-duodecigong*

*duodecilliplexion*

*googol-tredecigong*

*tredecilliplexion*

*googol-quattuordecigong*

*quattuordecilliplexion*

*googol-quindecigong*

*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*).

*gogolplex / googol-sexdecigong*

*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.

*sexdecilliplexion*

*googol-septendecigong*

*septendecilliplexion*

*googol-octodecigong*

*octodecilliplexion*

*googol-novemdecigong*

*novemdecilliplexion*

*googolvigintigong*

*vigintilliplexion*

**10^10^92**

*googoltrigintigong*

*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**

*googolplex*. It's approximately equal to 10^10^98.1411176539.

**10^10^100**

*googolplex*

*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!

**(10^10^100)^2**

*gargoogolplex*

*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**

*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*

*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*.

**(10**

^{100})!*googol-bang*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!

*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*.

E(8.0723*E153)

4^4^4^4

**10^10^122**

*googolquadragintigong*

**10^10^152**

*googolquinquagintigong*

4^4^4^4

*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*

*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:

*googolplex*:

*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*

**10^10^272**

*googolnonagintigong*

**10^10^300**

*thrargoogol-plexed*

*thrargoogol*zeroes, as the name implies.

**10^10^302**

*googolcentigong*

**10^10^303**

ecetonplex

ecetonplex

*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

5^5^5^5

*ecetonplex*but still less than a

*googolplexigong*.

~10^10^36,305

6^6^6^6

**10^10^3002**

*googolmilligong*

6^6^6^6

*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*

*googolplexigong squared.*At this scale it barely matters.

**(10^10^100,000)^3**

*thrargoogolplexigong*

*googolplexigong cubed.*Yawn...

**10^10^200,000**

*gargoogolgong-plexed*

*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.

*gargoogolplexigong*and

*thrargoogolplexigong*are only

*googolplexigong^2*and

*googolplexigong^3*respectively

*, gargoogolgong-plexed*is actually

*googolplexigong*^

*googolgong*.

**10^10^300,000**

*thrargoogolgong-plexed*

*googolgong^3*zeroes.

**7^2^999,997**

**~10^10^301,029**

*Upperbound on P*_{1,000,000}

_{}*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

7^7^7^7

*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***[10**

^{10^1,000,000},^{100}10)

*Entries: 48***10^10^1,000,000**

*milliduplexion*

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

*fzmilliplexion*

*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 :/

*googolmilli-milligong*

8^8^8^8

**~10^10^369,693,099**

9^9^9^9

9^9^9^9

*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*

*tetralogue*and a

*googolduplex*.

*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*

*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**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**

ecetonduplexecetonduplex

**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*

*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**

*googolquadriplex*

*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**

*ecetonquadriplex*

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

*googolquadriplexigong*

**10^10^10^10^10^10^10**

*heptalogue***10^10^10^10^10^10^100**

**E100#6**

*googolquintiplex*

*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.

*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.*

**(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*

*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*

*triadekalogue*

*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*

*tetradekalogue*

*E1#15*

*pentadekalogue*

*E1#16*

*hexadekalogue*

*E1#17*

*heptadekalogue*

*E1#18*

*octadekalogue*

*E1#19*

*ennadekalogue*

*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****[**

^{100}10,^{10^^100}10)**10^^100**

*hectalogue / giggol**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)^(10^100)**

*giggol to the googol*

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

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

*giggol to the googolplex*

**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.*

*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

*giggol^giggol*

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

*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**

*giggol^giggol*. You can envision it as:

*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**

*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*

*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**

*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:

*giggol*^{giggol }<*grangol**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**

*grangol*. As such it acts as an upper-bound, allowing the

*grangol*to be compared to other numbers. To prove

*grangol*< 10^^102 observe

*:*

**E100#101**

*grangolplex*

*-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.

*grangol^grangol.*First observe:

*grangolplex =*10^

*grangol*it follows that it is less than

*grangol^grangol.*

**(E100#100)^(E100#100)**

*grangol to the power of a grangol*

*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:

**10^^103**

*grangol^grangol*.

**10^^257**

*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.

**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:

**2[5]**

*Mega / two in a circle*

*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^^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*.

**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.*

**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*

**3^^^3**

*tritri*

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

**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^^(10**

^{100}

**)**

*googol-stack*

*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.

*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:

*googol*=

*googol*Es

*googol*10s

*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.*

*googol-stack*. In fact it's greater than 10^^(

*googol+1)*but less than 10^^(

*googol+2*).

**(10**

^{100})^^(10^{100})

*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^((10**

^{100})^^(10^{100}))

*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)*:

*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.

*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)**

*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*

*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**

*googolquadriplexidex***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^^100}10,100^^^100]

*Entries: 15*

*giggolplex*

*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**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*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*

*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*

*giggolduplex*

*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:

**((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

*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)!**

*Ended's Salad Factorial*

*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.

*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**

*gaggol*or 10^^^100, relatively speaking. Next up ... the Ackermann Numbers ...

*Primitive-Recursive Epoch***(100^^^100,E100##100]**

*Entries: 34***E3#100#99**

**10^^^101**

*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*

*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 is an upperbound on a**

10^^^102

*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^^^10216^^^102

*2^^^408*

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^^^^33^^^^3

**((...((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)

*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.*

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

**10^^^10^^^100**

*gaggolplex*

*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*

*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*

*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*

*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

*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*

*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)!*

*Twasbrillig's Up-arrow Salad*

*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...*