Tetration

Tetration is an operation that can be understood as repeated exponentiation. Tetration sometimes appears in CA-related discussions, as a way of expressing very large lifespans or bounds on lifespans.

Definition

If a, n are positive integer numbers, then the nth tetration of a (usually denoted either a ⇈ n or ⁿa) is the value of the expression

(i.e. the "power tower" with n copies of a, where the exponentiations are evaluated right-to-left).

The number a is called the base, and the number n is called the height.

The nth tetration of a can also be defined recursively:

a ⇈ 1 = a,

a ⇈ (n + 1) = a^{a ⇈ n}.

For example,

10 ⇈ 2 = 10¹⁰ = 10,000,000,000

10 ⇈ 3 = 10¹⁰¹⁰ = 10^{10,000,000,000}

3 ⇈ 3 = 3³³ = 3²⁷ = 7,625,597,484,987

See also

• Engineered diehard#Sawtooth-based designs

Further reading

• "Telling the tale of two tetrations". a blog by biggiemac42 (April 17, 2022).
• "Tetrational machines". Complex Projective 4-Space (June 23, 2022).
• "Miscellaneous discoveries". Complex Projective 4-Space (July 23, 2023).
• D. E. Knuth (1976) Mathematics and Computer Science: Coping with Finiteness
• "Tetration". Wikipedia.