Conway Chained Arrow Notation - Graham's Number

Graham's Number

Graham's number itself can not be expressed accurately in Conway chained arrow notation, but by defining the intermediate function, we have: (see functional powers), and

Proof: Applying in order the definition, rule 3, and rule 4, we have:

(with 64 's)

(with 64 's)

(with 64 's)

(with 65 's)

(computing as above).

Since f is strictly increasing,

which is the given inequality.

With chain arrows it is very easy to specify a much larger number. For example, note that

which is much greater than Graham's number.