PART 3 (1010^1,000,000,000,000 ~ 10^(100)10)
The Tetrational Range (1010^1,000,000,000,000 ~ 10^^100)
2.718... x 10(10^1,000,000,000,012) + 1,000,000,000,000
It is usually impossible to compute the leading digits of powers greater than about 10^10^10^9, but an exception is numbers of the form (10n + 1)^^2, whose first digits converge to the decimal expansion of e. This number begins with the first 1 trillion digits of e, which is the most that have been computed as far as I am sure.
~ 101.39*10^3,638,334,640,023
The reptend in the reciprocal of 3^^4. Its decimal expansion isÂ
794903529673899.............61678811111111111......111111111190601464078......72789922222222222...22222223017125751...8390103333333333333......505677.
~ 106.002*10^3,638,334,640,023
3^^5. The last 16 digits are ...6939489660355387, and the first digits are just beyond reach using the currently known method. However, I have tried to find a series for log_10(3) that I may be able to use to calculate the first digits of this number.
~ 105.3635*10^9,622,088,391,634
12^^4, the last 3 digits are ...416.
~ 101.592*10^22,212,093,154,093,428,529
16^^4. Can also be expressed as (2^^3)^^4. The last 5 digits are ...15616.
~ 103.1101*10^355,393,490,465,494,856,486
f3(4) in the fast-growing hierarchy. The last 7 digits are ...9756544.
~ 1010^(1.364*10^26)
20^^4. The last digits are ...22607743740081787109376 followed by 20^^3 zeroes.
~ 1010^(8.852147*10^33)
The first of Skewes' numbers.
~ 1010^(1.3111*10^40)
2vvvv4 in down-arrow notation, the last 7 digits are ...5123456.
~ 1010^(1.724*10^40)
f1(7, 2) in my reptend hierarchy. The decimal expansion of this number is 399020666104666......0856598393. Can also be expressed as r(7, 2, 1) using the generalized reptend function I invented.
~ 1010^10^44
30^^4. The last digits are ...47865522 followed by 3030 zeroes and then a 1, followed by 30^^3 zeroes.
~ 1010^(3.485689*10^76)
2^2^2^2^2^3. The last 5 digits are ...68736.
1010^10^100
A googolduplex.
1010^(10^100 + 100)
A googolplex raised to the power of itself. This seems barely larger than a googolduplex, but it is actually a googolduplex raised to the googolth power!
~ 1010^(1.992373902852*10^619)
4 in a square in Steinhaus-Moser notation, and also the third step in calculating the mega. The last 5 digits of this number are ...43456.
~ 1010^(1.3357404*10^2184)
5^^5, the megafugafive. Can also be expressed as 5^^^2. The last 12 digits are ...618408203125.
~ 1010^(6.031226*10^19727)
2^^7. The last 14 digits are ...66659621748736. Also three more than Ackermann(4, 4).
~ 1010^2*10^36305
6^^5. The last 6 digits are ...238656.
~ 1010^(3.1*10^695974)
7^^5. The last 10 digits are ...5965172343.
~ 1010^(6.002*10^3638334640023)
3^^6. The last 20 digits are ...13333445425126595387.
~ 1010^10^10^25043
The SpongeTechX mixed factorial of 5, denoted 5*. The mixed factorial is defined as follows: 1* = 1, (n+1)* = n* [n] (n+1), where [n] denotes the n-th hyperoperator (n-2 arrows). 2* = 1+2 = 3, 3* = 3*3 = 9, 4* = 9^4 = 6561, and 5* is this enormous value. The last 100 digits are ...9120536122025539013648088650686937941157296717201747816753170112920483215947823395227334944489504161.
~ 1010^10^2.06*10^36305
6^^6, megafugasix. The third Pickover superfactorial. The last 10 digits are ...9127238656.
10^^6
3^^7
4^^7
~ (10 ^)4 1.33574*102,184
5^^7.
~ (10 ^)4 6.031226*1019,727
2^^9. The last 10 digits are ...7112948736.
~ (10 ^)4 2.06*1036,305
6^^7.
~ (10 ^)4 3.177*10695,974
The megafugaseven. The last 12 digits are ...511565172343.
10^^7
3^^8
~ (10 ^)6 19,727.78
2^^10. The last 10 digits are ...4232948736.
10^^8
10^^9
10^^10
A power tower of 10 10s. Can be expressed as 10^^^2. Referred to as a dekalogue, or a decker.
~ (10 ^)10 4.554695*10117
4 vvvv 4 in down-arrow notation, a number that I dubbed the "tiny tritet". The last 10 digits are ...7044345856.
~ (10 ^)23 33.265
24^^24, the fourth Pickover superfactorial. The last three digits are ...976.
~ (10 ^)24 8.072304726*10153
4^^27, the tetrational factorial of 4.
The High Tetrational Range (10^^100 ~ 10^^^3)
~ (10 ^)98 914,494,741,655,235,482,395,976,221,307,727,480,566,986,739,826,149
The reptoogol. The largest googolism I've seen (defined by anyone) where the leading digits can still be known. This number begins with 63614547728..., and its first 1.409*1048 digits form the decimal expansion of 41152263374485596707818933^98. This is then followed by more than 4.558*1050 consecutive zeroes, and then the digits 1214278...
~ (10 ^)98 9.968*1038,160,085,469,014,705,624,435,882,736,067
f3(100) in the fast-growing hierarchy.
(10 ^)100 100
A grangol. Can be thought of as a googol-99-plex.
~ (10 ^)119 249.8196
120^^120, the fifth Pickover superfactorial.
~ (10 ^)195 4.829*10183230
An expofaxul. The last digits are ...22474614575178186752 followed by 2*(199!1) zeroes.
~ (10 ^)255 1.992373902852*10619
The mega in Steinhaus-Moser notation, equal to 2 in a pentagon. The last 29 digits are ...10112922449731993539660742656.
~ (10 ^)718 5.437*102057
720^^720, the sixth Pickover superfactorial.
~ (10 ^)65,531 6.03122606*1019,727
2^^^4. Can also be expressed as 2^^^^3. The last 20 digits are ...98615075353432948736.
~ (10 ^)386,201,103 1.4515*10301
The output of pete-8.c in Bignum Bakeoff, equivalent to f16*17^6(999), where f(n) = 9*2n. The last 6 digits are ...647744.
~ (10 ^)7,625,597,484,983 6.00225*103,638,334,640,023
3^^^3, a number known as the tritri. Can also be expressed as 3^^^^2. The last 7625597484986 digits are the same as those of Graham's number. This number appears frequently when evaluating expressions with the base of three in notations extending on up-arrows; it is equal to 3 -> 3 -> 3 in Conway chained arrow notation and {3, 3, 3} in BEAF.
~ 10^^(2.6448990072*1022,212,093,154,093,428,529)
The hypermega, a mega-like googolism I first invented back in July of 2015. The last 43 digits are ...4062488962851476125888204176575436964233216.
~ (10 ^)(10^10^100) - 5 4.829*10183,230
The exponential factorial of a googolplex. This number was given the name zootzootplex by Andrew Schilling when he was four years old.
The Pentational Range (10^^^3 ~ 10^^^(10^^^2))
~ 2^^(2^^5137)
The most recent upper bound for the solution to Graham's problem. This number is exactly (2^^5138)*((2^^5140)^^(2*(2^^5137))), and its last digits are ...7296.
3^^(3^^7,625,597,484,987)
3^^^4.
4^^^4
5^^^4
Another variation of the mega, where the triangle is n^(n^(n^n)) instead of just n^n. The last 10 digits are ...4926452736.
6^^^4
7^^^4
8^^^4
9^^^4
10^^^4
3^^^5
4^^^5
~ 10^^(10^^(10^^1010^(3.55*10^20)))
f4(4) in the fast-growing hierarchy. The last 7 digits are ...5652864.
While ideas such as Robert Munafo's concept of classes eventually become obsolete in terms of categorizing numbers, there is one idea for classifying numbers that remains useful for categorizing numbers of any size: the order type in the fast-growing hierarchy.
~ 10^^(10^^(10^^(1010^10^619)))
4 in a pentagon in Steinhaus-Moser notation.
5^^^5
~ 10^^(10^^(10^^1010^1.33574*10^2184))
The boogafive. This number is equal to a power tower of 5s where the number of 5s is itself a power tower of 5s with as many 5s as a power tower of 5s with 55^5^3125 5s!
~ 5^^(5^^(5^^((5 ^)19 2504)))
The tiny tripent, equal to 5 vvvvv 5 in down-arrow notation.
6^^^6
10^^^6
2^^^8
Three more than the gagfive.
7^^^7
8^^^8
9^^^9
10^^^10
~ 10^^(10^^(10^^(10^^(10^^(10^^(10^^(10^^(10^^((10 ^)9 100,000,000,011)))))))))
The megiston, equal to 10 in a pentagon in Steinhaus-Moser notation.
10^^^10^^^2
The Hexational Range (10^^^(10^^^2) ~ 10^^^^10^^^^2)
~ 10^^^(mega + 1)
2 in a hexagon in Steinhaus-Moser notation, sometimes referred to as A-ooga or megision. The last 8 digits are ...79341056.
~ 10^^^((10 ^)65531 6.031226*1019727)
2^^^^4.
~ 10^^^(10^^7,625,597,484,986)
3^^^^3, the first term in the sequence defining Graham's number.
~ 10^^^(10^^^(10^^^(10^^^(10^^^5))))
6^^^^6, the boogasix.
The Higher Hyperoperator Range (10^^^^100 ~ 10^(100)10)
2^^^^^4
The largest number covered in Sbiis Saibian's up-arrow article.
5^^^^^5
The tripent, equal to {5, 5, 5} in Bowers' array notation.
7^^^^^7
The boogaseven.
6^^^^^^6
The trihex.
8^^^^^^8
The boogaeight.
7^^^^^^^7
The trisept.
9^^^^^^^9
The booganine.
~ 10^^^^^^^^^11
f10(10) in the fast-growing hierarchy, a number that Denis Maksudov calls dekalum. The last 7 digits of this number are ...7034240. This is an example of a number that can be expressed using fw(n), which has a growth rate on par with increasing the number of arrows in an expression using Knuth arrows.
2^^^^^^^^^^^^3
The first term in the sequence defining Little Graham.
10^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^100
The boogol.
PART 4 >>