Pentation

In mathematics, pentation (or hyper-5) is the fifth hyperoperation. Pentation is defined to be repeated tetration, similarly to how tetration is repeated exponentiation, exponentiation is repeated multiplication, and multiplication is repeated addition. The concept of "pentation" was named by English mathematician Reuben Goodstein in 1947, when he came up with the naming scheme for hyperoperations.

The number a pentated to the number b is defined as a tetrated to itself b - 1 times. This may variously be denoted as ${\displaystyle a[5]b}$, ${\displaystyle a\uparrow \uparrow \uparrow b}$, ${\displaystyle a\uparrow ^{3}b}$, ${\displaystyle a\to b\to 3}$, or ${\displaystyle {_{b}a}}$, depending on one's choice of notation.

For example, 2 pentated to 2 is 2 tetrated to 2, or 2 raised to the power of 2, which is ${\displaystyle 2^{2}=4}$. As another example, 2 pentated to 3 is 2 tetrated to the result of 2 tetrated to 2. Since 2 tetrated to 2 is 4, 2 pentated to 3 is 2 tetrated to 4, which is ${\displaystyle 2^{2^{2^{2}}}=65536}$.

Based on this definition, pentation is only defined when a and b are both positive integers, though progress has been made to allow for any value of a.