- Not to be confused with Ackermann ordinal.
The Ackermann numbers are a sequence defined with the original definition of Ackermann function (not to be confused with the well-known Robinson's definition) as A(n) = A(n+2,n,n) where \(n\) is a positive integer. It can be expressed with arrow notation as:[1][2]
\[A(n) = n\underbrace{\uparrow\uparrow...\uparrow\uparrow}_nn\]
The first few Ackermann numbers are \(1\uparrow 1 = 1\), \(2\uparrow\uparrow 2 = 4\), and \(3\uparrow\uparrow\uparrow 3 =\) tritri. More generally, the Ackermann numbers diagonalize over arrow notation, and signify its growth rate is approximately \(f_\omega(n)\) in FGH and \(g_{\varphi(n-1,0)}(n)\) in SGH.
The \(n\)th Ackermann number could also be written \(3\)\(\&\)\(n\) or \(\lbrace n,n,n \rbrace\) in BEAF.
Last 20 digits[]
Below are the last few digits of the first ten Ackermann numbers.[3]
- 1st = 1
- 2nd = 4
- 3rd = ...04575627262464195387 (tritri)
- 4th = ...22302555290411728896 (tritet)
- 5th = ...17493152618408203125 (tripent)
- 6th = ...67965593227447238656 (trihex)
- 7th = ...43331265511565172343 (trisept)
- 8th = ...21577035416895225856 (trioct)
- 9th = ...10748087597392745289 (triennet)
- 10th = ...00000000000000000000 (tridecal)
Approximations in other notations[]
| Notation | Approximation |
|---|---|
| Hyper-E notation | \(\textrm En\#\#n\) |
| BEAF | \(\lbrace n,2,1,2 \rbrace\) (exact value) |
| Fast-growing hierarchy | \(f_\omega(n)\) |
| Hardy hierarchy | \(H_{\omega^\omega}(n)\) |
| Slow-growing hierarchy | \(g_{\varphi(\omega,0)}(n)\) |
Sources[]
- ↑ Conway and Guy. The Book of Numbers. Copernicus. 1995. ISBN 978-0387979939 p.60
- ↑ Ackermann Number
- ↑ Fish. Last 20 digits of Ackermann numbers 2024-08-29.