## ⓘ Knuths up-arrow notation

Knuths up-arrow notation is a way of expressing very big numbers. It was made by Donald Knuth in 1976. It is related to the hyperoperation sequence. The notation is used in Grahams number.

One arrow represents exponentiation, 2 arrows represent tetration, 3 for pentation, etc.:

- Exponentiation a ↑ 1 b = a b = a × a × ⋯ × a ⏟ b t i m e s {\displaystyle a\uparrow ^{1}b=a^{b}=\underbrace {a\times a\times \cdots \times a} _{b\ times}} a multiplied by itself, b times.
- Tetration a ↑ 2 b = a ↑ ↑ b = b a = a ⋅ ⋅ a) ⏟ b t i m e s = a ↑ 1 a ↑ 1. ↑ 1 a) ⏟ b t i m e s {\displaystyle a\uparrow ^{2}b=a\\uparrow b={^{b}a}=\underbrace {a^{a^{\cdot ^{\cdot ^{a)}}}}} _{b\ times}=\underbrace {a\uparrow ^{1}a\uparrow ^{1}.\uparrow ^{1}a)} _{b\ times}} a exponentiated by itself, b times.
- etc
- Third level a ↑ 3 b = a ↑ ↑ ↑ b = a ↑ ↑ a ↑ ↑ a ↑ ↑ … a …) ⏟ b t i m e s {\displaystyle a\uparrow ^{3}b=a\\\uparrow b=\underbrace {a\\uparrow a\\uparrow a\\uparrow \ldots a\ldots)} _{b\ times}}

This notation is used to describe the incredibly large Grahams Number.

- exponentiation, tetration, etc. They are often written using Knuth s up - arrow notation Natural number level hyperoperations can be defined recursively
- big even to write in scientific notation In order to be able to write it down, we have to use Knuth s up - arrow notation We will write down a sequence

3 (number) |

40 (number) |

42 (number) |

7 (number) |

Acre |

Attack model |

Backpropagation |

Basis (linear algebra) |

Bernoulli distribution |

Bertrands postulate |

Binomial distribution |

Block cipher |

Block size (cryptography) |

Brute force attack |

Cardinal number |

Carmichael number |

Chain rule |

Chi-square distribution |

Chinese remainder theorem |

Chosen-plaintext attack |

Ciphertext-only attack |

Communication Theory of Secrecy Systems |

Composite number |

Cone |

Confusion and diffusion |

Consecutive integer |

Converse (logic) |

Cryptosystem |

De Moivres formula |

Decagon |

Differential cryptanalysis |

Digon |

Diophantus |

Divergence |

Dodecahedron |

Double factorial |

EFF DES cracker |

Equilateral triangle |

ESTREAM |

Euclidean algorithm |

Eulers totient function |

Euler–Mascheroni constant |

Exponential distribution |

Exponentiation by squaring |

Googolplex |

Great circle |

Hexacontagon |

Hexadecagon |

Hexagon |

Hilberts tenth problem |