pentation

Pentation is the fifth hyperoperation, extending the sequence that includes addition, multiplication, exponentiation, and tetration. It is commonly denoted as a ↑↑↑ b in Knuth's up-arrow notation and represents an iteration of tetration.

Formally, for integers a ≥ 1 and b ≥ 1, pentation is defined by a ↑↑↑ 1 = a and a ↑↑↑ (n+1) = a ↑↑ (a ↑↑↑ n), where a ↑↑ b denotes tetration: a ↑↑ b = a^(a^(...)) with b copies of a. Thus pentation applies tetration to the result of a previous pentation step, in a recursive manner.

Examples illustrate the rapid growth. 2 ↑↑↑ 2 = 2 ↑↑ 2 = 4. Another example: 3 ↑↑↑ 2 = 3 ↑↑ 3 = 3^(3^3) = 3^27 = 7,625,597,484,987.

Pentation grows extraordinarily fast; for any base a > 1 and increasing height b, its values outpace both exponentiation and tetration by enormous margins. As a theoretical construct, it is mainly of interest in the study of hyperoperations, large-number phenomena, and related areas in number theory and combinatorics. It does not have widespread practical applications in standard mathematics.

Extensions to non-integer heights are not standardized. Some research explores continuous or analytic extensions of tetration, but pentation for non-integer heights is less developed and not part of a universally accepted framework.