Down-arrow notation is the same as up-arrow notation, only left-associative instead of right-associative.
We will begin with weak tetration. a vv b is equal to (...(((aa)a)a)...)a with b copies of a, or aa^(b-1).
2 vv 2 = 22 = 4
2 vv 3 = (22)2 = 42 = 16
2 vv 4 = 256
2 vv 5 = 65,536
2 vv 6 = 4,294,967,296
2 vv 7 = 18,446,744,073,709,551,616
2 vv 8 = 340,282,366,920,938,463,463,374,607,431,768,211,456
3 vv 2 = 33 = 27
3 vv 3 = 19,683
3 vv 4 = 7,625,597,484,987 (this is the same as 3^^3)
3 vv 5 = 443,426,488,243,037,769,948,249,630,619,149,892,803
This is the same as the double exponentials we saw back in the iterated exponential numbers article.
Weak pentation is when things really get interesting:
2 vvv 2 = 2 vv 2 = 4
2 vvv 3 = (2 vv 2) vv 2 = 4 vv 2 = 256
2 vvv 4 = ((2 vv 2) vv 2) vv 2 = ((4 vv 2) vv 2) = 256 vv 2
2 vvv 5 = (((2 vv 2) vv 2) vv 2) vv 2 = 22048 vv 2 = 22^2059 = 3984434...365056 (a number with more than 10619 digits)
2 vvv 259 is exactly equal to the mega, a number we will cover later. Here are some base 3 examples:
3 vvv 2 = 3 vv 3 = 19683
3 vvv 3 = (3 vv 3) vv 3 = 1968319683^2 = 196833^18 = 33^20 ~ 2.1981970178*101,663,618,948
3 vvv 4 = ((3 vv 3) vv 3) vv 3 = (33^20)(9^3^20) ~ 1010^3327237896
Even just 3 vvv 3 is a number with over a billion digits in its decimal representation, and 3 vvv 4 is greater than the largest number I have directly computed the leading digits of.
10 vvv 2 = 10 vv 10 = 101,000,000,000
10 vvv 3 = (10 vv 10) vv 10 = (101,000,000,000)(10^9,000,000,000) = 1010^9,000,000,009
Next we will cover weak hexation.
2 vvvv 2 = 2 vvv 2 = 2 vv 2 = 4
2 vvvv 3 = 4 vvv 2 = 4 vv 4 = 340,282,366,920,938,463,463,374,607,431,768,211,456
2 vvvv 4 = (4 vvv 2) vvv 2 = 22^(2^135 - 121) ~ 1010^1.31*10^40 (the last 50 digits are ...39868488354763628686817342417799816556931455123456)
3 vvvv 2 = 3 vvv 3 = (3 vv 3) vv 3 = 33^20 ~ 2.198197*101,663,618,948
3 vvvv 3 = (3 vvv 3) vvv 3 = 33^20 vvv 3 = (33^20 vv 33^20) vv 33^20 = ((33^20)((3^3^20)^((3^3^20) - 1) ) vv 33^20 ~ 1010^10^10^1663618948
4 vvvv 2 = 4 vvv 4
4 vvvv 3 = (4 vvv 4) vvv 4 ~ (10 ^)7 117.66 (the last 30 digits are ...410421154027352957496491769856)
4 vvvv 4 = (((4 vvv 4) vvv 4) vvv 4 ~ (10 ^)11 117.66
4 vvvv 4 is a number that I call the tiny tritet. Its last 30 digits are ...508014316085166613967044345856.
In general, a vvv b is approximately equal to (and "slightly" less than) a^^(b+1), while a vvvv b is approximately equal to (and, once again, "slightly" less than) a^^(a*(b-1) + 1), and a vvvvv b is approximately (a ^^)b - 1(a(b - 1) + 1)) Thus 5 vvvvv 5 ~ 5^^(5^^(5^^(5^^21))) (the former being a number I dubbed the tiny tripent).
And we can even define the "weak Ackermann numbers":
1v1 = 1
2vv2 = 4
3vvv3 (tiny tritri) = 21,981,970,178,167,422,...,704,003 (1,663,618,949 digits)
4vvvv4 (tiny tritet) ~ E117#11 (the last 30 digits are ...508014316085166613967044345856).
5vvvvv5 (tiny tripent) ~ 5^^5^^5^^5^^21 (the last 4 digits are ...3125)
It is also possible to define mixed up-arrows to indicate the left-associativity of some operators and the right-associativity of others. The first arrow doesn’t matter, while subsequent arrows have substantial impact on the result.
The first arrow gives the associativity of the multiplication in the computation (which has no effect), the second gives the associativity of exponentiation, and all further arrows give the associativity of the operation formed by the arrows up to (and not including) it.
a ↑↑↓ b = (…((a ↑↑ a) ↑↑ a)…) ↑↑ a w/ b copies of a
3 ↑↓↑ 3 = 3 ↑↓ (3 ↑↓ 3) = 3 ↓↓ (3 ↓↓ 3) = 3 ↓↓ 19683 = 33^19682 ~ 102.39*10^9390
2 ↑↓↑ 4 = 2 ↑↓ (2 ↑↓ (2 ↑↓ 2)) = 2 ↑↓ (2 ↑↓ 4) = 2 ↑↓ 256 = 2 ↓↓ 256 = 22^255 ~ 101.7428*10^76
3 ↑↓↑↓ 3 = (3 ↑↓↑ 3) ↑↓↑ 3 = 33^19682 ↑↓↑ 3 = 33^19682 ↑↓ (33^19682 ↑↓ 33^19682) = 33^19682 ↓↓ (33^19682 ↓↓ 33^19682) = NN^((N^N^(N-1))-1) where N = 33^19682 ~ 33^3^3^3^3^19682 >~ 10^^7
3 ↓↑↓↑ 3 = 3 ↓↑↓ (3 ↓↑↓ 3) = 3 ↓↑↓ 33^205,891,132,094,652 = 3 tetrated to 3 33^205,891,132,094,652- 1 times ~ , the last 7 digits are …3838403. This can also be expressed as 3(*^^3)(3(*^^3)2).
The number I originally called the tiny tritet in August 2015 is actually 4 ↓↓↑↑ 4. The last digits of the number are …102204192334540131270656.
Three arrow operators
2 ↑↑↓ 2 = 4
2 ↑↑↓ 3 = 256
2 ↑↑↓ 4 = 22048 ~ 3.2317006*10616
2 ↑↑↓ 5 = 22²⁰⁵⁹ ~ 101.9923739028520154*10^619
3 ↑↑↓ 2 = 7625597484987
3 ↑↑↓ 3 = 33^205,891,132,094,652
3 ↑↑↓ 4 = 33^(3^(3^205891132094652 + 205891132094652) + 205891132094652) ~> 3 ↑↑ 7
2 ↓↓↑ 2 = 4
2 ↓↓↑ 3 = 256
2 ↓↓↑ 4 = 22²⁵⁵ ~ 101.742844*10^76
2 ↓↓↑ 5 = 22^(2^2^255 – 1)
3 ↓↓↑ 2 = 19683
3 ↓↓↑ 3 = 33^19682 ~ 102.39422*10^9390
3 ↓↓↑ 4 = 33^(3^3^19682 – 1)
Since we already covered regular and weak pentation, we will move on to four arrows, where there are 6 operators we haven’t yet covered.
Four arrow operators
2 ↑↑↓↓ 2 = 4
2 ↑↑↓↓ 3 = 4 ↑↑↓ 2 = 4 ↑↑ 4
2 ↑↑↓↓ 4 = (4 ↑↑↓ 2) ↑↑↓ 2 = (4 ↑↑ 4) ↑↑ (4 ↑↑ 4)
3 ↑↑↓↓ 2 = 3 ↑↑↓ 3 = (3 ↑↑ 3) ↑↑ 3 = 33^205891132094652
3 ↑↑↓↓ 3 = (3 ↑↑↓ 3) ↑↑↓ 3 = ((33^205891132094652) ↑↑ (33^205891132094652)) ↑↑ (33^205891132094652)
3 ↑↑↓↓ 4 = ((3 ↑↑↓ 3) ↑↑↓ 3) ↑↑↓ 3 = (N ↑↑ N) ↑↑ N where N is the previous number
2 ↓↓↓↑ 3 = 2 ↓↓↓ 4 = 22048 ~ 3.2317006*10616
2 ↓↓↓↑ 4 = 2 ↓↓↓ 22048 = 2 raised to the power of itself 22048-1 times = …………………8072723456 (approximately 10^^(3.2317*10^616)
2 ↓↓↓↑ 5 = 2 ↓↓↓ the previous number = …………………5949843456
3 ↓↓↓↑ 2 = 3 ↓↓↓ 3 = 33,486,784,401
3 ↓↓↓↑ 3 = 3 ↓↓↓ (3 ↓↓↓ 3) = (n -> nn²)3³⁴⁸⁶⁷⁸⁴⁴⁰¹-1(3) = …767704003
2 ↓↓↑↑ 3 = 2 ↓↓↑ 4 = 2 ↓↓ (2 ↓↓ 4) = 22²⁵⁵
2 ↓↓↑↑ 4 = 2 ↓↓↑ 22²⁵⁵ ~> 2 ↑↑ (22²⁵⁵+1 – 3)
2 ↓↓↑↑ 5 = 2 ↓↓↑ the previous number ~> 2 ↑↑ (2*the previous number – 3)
3 ↓↓↑↑ 2 = 3 ↓↓↑ 3 = 3 ↓↓ (3 ↓↓ 3) = 33¹⁹⁶⁸² ~ 102.39422*10⁹³⁹⁰
3 ↓↓↑↑ 3 = 3 ↓↓↑ (3 ↓↓↑ 3) = 3 ↓↓↑ 33¹⁹⁶⁸² is between 3 ↑↑ (2*33¹⁹⁶⁸²- 2) and 3 ↑↑ (2*33¹⁹⁶⁸²- 1), the last digits are …1865683683.
I found the last digits of (3 ↑↓↑ 3) ↓↓↓ 3 are …2992299683 , and the number is approximately 1010^10^10^10^2.39422*10^9390. I originally believed this to be 3 ↑↓↑↓ 3, but was mistaken. I kept this number solely because of the interesting pattern in the terminating digits.
9 ↑↓↓↑ 9 = 9 ↑↓↓ (9 ↑↓↓ (9 ↑↓↓ (9 ↑↓↓ (9 ↑↓↓ (9 ↑↓↓ (9 ↑↓↓ (9 ↑↓↓ 9)))))))
9 ↓↓↓ 2 ends in …956241807657083622930379891209, and 9 ↓↓↓ 3 ends in …664966456988767556072540390409. 9 ↓↓↓ 9 ends in …160206880796418169893364633609.