Definition:Ackermann-Péter Function
Definition
The Ackermann-Péter function $A: \Z_{\ge 0} \times \Z_{\ge 0} \to \Z_{> 0}$ is an integer-valued function defined on the set of ordered pairs of positive integers as:
- $\map A {x, y} = \begin{cases} y + 1 & : x = 0 \\
\map A {x - 1, 1} & : x > 0, y = 0 \\ \map A {x - 1, \map A {x, y - 1} } & : \text{otherwise} \end{cases}$
Also defined as
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Some sources define the Ackermann function as $A: \Z_{> 0} \times \Z_{> 0} \to \Z_{> 0}$ where:
- $\map A {x, y} = \begin{cases} 2 y & : x = 1 \\
x & : x > 1, y = 1 \\ \map A {x - 1, \map A {x, y - 1} } & : \text{otherwise} \end{cases}$
Examples
- $\begin {array} {c|c|c|c}
\map A {m, n} & m = 0 & m = 1 & m = 2 & m = 3 \\ \hline n = 0 & 1 & \map A {0, 1} & \map A {1, 1} & \map A {2, 1} \\ n = 1 & 2 & \map A {0, \map A {1, 0} } & \map A {1, \map A {2, 0} } & \map A {2, \map A {3, 0} } \\ n = 2 & 3 & \map A {0, \map A {1, 1} } & \map A {1, \map A {2, 1} } & \map A {2, \map A {3, 1} } \\ n = 3 & 4 & \map A {0, \map A {1, 2} } & \map A {1, \map A {2, 2} } & \map A {2, \map A {3, 2} } \\ n = 4 & 5 & \map A {0, \map A {1, 3} } & \map A {1, \map A {2, 3} } & \map A {2, \map A {3, 3} } \\ \end{array}$
which leads to:
- $\begin{array}{c|c|c|c}
\map A {m, n} & m = 0 & m = 1 & m = 2 & m = 3 \\ \hline n = 0 & 1 & 2 & 3 & 5 \\ n = 1 & 2 & 3 & 5 & 13 \\ n = 2 & 3 & 4 & 7 & \map A {2, 13} \\ n = 3 & 4 & 5 & 9 & \map A {2, \map A {3, 2} } \\ n = 4 & 5 & 6 & 11 & \map A {2, \map A {3, 3} } \\ \end{array}$
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Examples
No, you're right, I really don't know what I'm doing, at all.
- $\begin{array}{c|c|c|c}
\map A {m, n} & m = 1 & m = 2 & m = 3 & m = 4 & \cdots & m = k \\ \hline n = 1 & 2 & 2 & 3 & 4 & & k \\ n = 2 & 4 & \map A {1, \map A {2, 1} } & \map A {2, \map A {3, 1} } & \map A {3, \map A {4, 1} } & & \map A {k - 1, \map A {k, 1} } \\ n = 3 & 6 & \map A {1, \map A {2, 2} } & \map A {2, \map A {3, 2} } & \map A {3, \map A {4, 2} } & & \map A {k - 1, \map A {k, 2} } \\ n = 4 & 8 & \map A {1, \map A {2, 3} } & \map A {2, \map A {3, 3} } & \map A {3, \map A {4, 3} } & & \map A {k - 1, \map A {k, 3} } \\ n = 5 & 10 & \map A {1, \map A {2, 4} } & \map A {2, \map A {3, 4} }\ & \map A {3, \map A {4, 4} } & & \map A {k - 1, \map A {k, 4} } \\ \vdots & & & & & & \\ n = j & 2 j & \map A {1, \map A {2, j - 1} } & \map A {2, \map A {3, j - 1} } & \map A {3, \map A {4, j - 1} } & & \map A {k - 1, \map A {k, j - 1} } \\ \end{array}$
which leads to:
- $\begin{array}{c|c|c|c}
\map A {m, n} & m = 1 & m = 2 & m = 3 & m = 4 & \cdots & m = k \\ \hline n = 1 & 2 & 2 & 3 & 4 & & k \\ n = 2 & 4 & 4 & 8 & \map A {3, 4} & & \map A {k - 1, k} \\ n = 3 & 6 & 8 & 2^8 & \map A {3, \map A {4, 2} } & & \map A {k - 1, \map A {k, 2} } \\ n = 4 & 8 & 16 & 2^{2^8} & \map A {3, \map A {4, 3} } & & \map A {k - 1, \map A {k, 3} } \\ n = 5 & 10 & 32 & \map A {2, \map A {3, 4} } & \map A {3, \map A {4, 4} } & & \map A {k - 1, \map A {k, 4} } \\ \vdots & & & & \\ n = j & 2 j & 2^j & \map A {2, \map A {3, j - 1} } & \map A {3, \map A {4, j - 1} } & & \map A {k - 1, \map A {k, j - 1} } \\ \end{array}$
![]() | There is believed to be a mistake here, possibly a typo. In particular: This needs to be checked, because "in the book" it has that $\map A {3, 4}$ works out at $2^{65,536} = 2^{2^{16} }$ not $2^{2^8}$. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by reviewing it, and either correcting it or adding some explanatory material as to why you believe it is actually correct after all. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Mistake}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
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Also known as
The Ackermann-Péter function is also known as the Ackermann function.
However, there are a number of different similar functions which go by this name, so the full appellation can be argued as being more useful.
Also see
- Results about the Ackermann-Péter function can be found here.
Source of Name
This entry was named for Wilhelm Friedrich Ackermann and Rózsa Péter.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $2^{65536}$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $2^{65,536}$
- Weisstein, Eric W. "Ackermann Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AckermannFunction.html
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