hexation

Hexation is a hyperoperation, specifically the seventh hyperoperation in the sequence defined by iteration. It follows exponentiation and precedes tetration. Hyperoperations are a way to generalize arithmetic operations. The sequence begins with addition, then multiplication, then exponentiation, then tetration, and so on.

Hexation can be represented using Knuth's up-arrow notation. If addition is represented by a single up-arrow (a + b), multiplication by two up-arrows (a x b = a^b), tetration by three up-arrows (a ^^ b), then hexation is represented by four up-arrows (a ^^^^ b).

The definition of hexation, denoted by a ^^^^ b, involves repeated application of tetration. Specifically, a ^^^^ b is a tetrated to itself b times. For example, 2 ^^^^ 3 means 2 tetrated to itself 3 times, which is 2 ^^ (2 ^^ 2). To calculate this, we first find 2 ^^ 2, which is 2^2 = 4. Then we calculate 2 ^^ 4, which is 2^(2^(2^2)) = 2^(2^4) = 2^16 = 65536. Therefore, 2 ^^^^ 3 = 65536.

As with other higher hyperoperations, hexation grows extremely rapidly. Even small integer values for the operands can result in astronomically large numbers that are difficult or impossible to compute directly. Its primary use is in theoretical mathematics and in discussions about the limits of computation and number growth.